1993 PAPER ONE
2 x 1
1.a) Solve 3
3
x 1
Ans:
3 1 0
x
b) Solve the simultaneous equations
x + 2y − 3z = 0
3x + 3y − z = 5
x − 2y + 2z = 1
Ans: (a) x = -1, x = 0; (b) x = 1, y = 1, z = 1
2. When the quadratic expression ap2 + bp + c is divided
by p − 1, p − 2 and p + 1, the remainders are 1, 1 and
25 respectively. Determine the values of a, b and c.
Hence the factors of the expression. (Ans: a = 4, b = 12, c = 9; factors: (2p – 3) and (2p – 3)
b) Express
+
− 4x − 3 in the form
(x2 + x − 2) Q(x) + Ax + B; where Q(x) is a
polynomial in x and A and B are constants. Determine
the values of A and B and the expression Q(x).
(Ans: A = -3, B = 3), Q(x) = 2x + 3
2x3
5x2
3. i) Show that ln2r, r = 1, 2, 3 is an arithmetic
progression.
ii) Find the sum of the first 10 terms of the progression.
(Ans: 38.1231)
iii) Determine the least value of m for which the sum of
the first 2m terms exceeds 883.7. (Ans: m = 25)
b) Given that the equations y2 + py + q = 0 and y2 + my +
k = 0 have a common root. Show that (q − k)2 = (m −
p)(pk − mq).
4. Solve the simultaneous equations
z1 + z2 = 8
4z1 − 3iz2 = 26 + 8i
Using the values of z1 and z2, find the modulus and
argument of z1 + z2 − z1z2 (Ans: z1 = 8 + 2i; z2 = -2i)
5. Use the Maclaurin’s theorem to show that the
x 2
x 3x 3 .
expansion e-x sin x up to the term in x3 is
3
Hence evaluate e 3 sin to 4 decimal places
3
(Ans: 0.334)
6. Differentiate with respect to x:
i) tan
1
1 2 x2
ii) cos x
(Ans:
2 cos x
6 12 x 2
6x
2x
2
4
(Ans: 1 32 x 4 x
2x
ln cos x x tan x
b) Write down the expression of the volume v, and
surface area s of a cylinder of radius r and height h
If the surface area is kept constant, show that the volume
of the cylinder will be maximum when h = 2r
7. Find: i)
ln x
2
4 dx
x 2
x ln x 2 4 2 x 2 ln
c
x 2
ii)
dx
3 2cos x
2
Ans: 5
tan 1 5 tan 2x c
1
iii) Use the substitution of x to evaluate
u
Ans:
2
x
1
dx
x2 1
3
8. A curve is given by the parametric equations x = 4
cos 2t, y = 2 sin t.
i) Find the equation of the normal to the curve at
t5
6 (Ans: y = 4x – 7)
ii) Sketch the curve for t
2
2
iii) Find the area enclosed by the curve and the y –
axis (Ans: 7.6425 sq. units)
9. Show that cos 3θ = 4cos3 θ − 3 cos θ.
3
Hence solve the equation 4 x 3x
3
0
3
(Ans: 0.9492, -0.2037, -0.746)
b) Find all the solutions of the equation
5cos x 4sin x 6 in the range -1800 ≤ x ≤1800
(Ans: -18.2, -59.1)
10. Given that tan-1(α) = x and tan-1(β) = y by expressing
α and β as tangent ratios, of x and y, and manipulating
1
the ratios, show that x y tan 1 . Hence or
otherwise:
(i) Solve for x in
1
1
1
tan 1
tan x 1 tan 2 ,
x 1
(Ans: x = -2, 2)
(ii) Without using table or calculators, determine the
1
1
1
1
1
1
value of tan 2 tan 5 tan 8
(Ans:
4
)
T(t2,
11.a) A tangent from the point
2t) touches the curve
y2 = 4x. Find:
(i) The equation of the tangent (Ans: ty – x – t2 = 0)
(ii) The equation of line L parallel to normal at (t2, 2t)
and passing through (1, 0), (Ans: y + tx – t = 0)
(iii) The point of intersection X of line L and the
tangent. (Ans: (0, t)
b) A point P(x, y) is equidistant from X and T. Show that
the locus of P is t4 – 3t – 2(x + y) = 0
The proof is not valid. Loci is t4 – 3t – 2(xt + y) = 0
12. The equation of a circle, centre O is given by x2 + y2
+ Ax + By + C = 0, where A, B and C are constants.
Given that 4A = 3B, 3A = 2C and C = 9,
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(a) Determine i) the coordinates of the centre of the
circle (Ans: (-3, -4)
ii) The radius of the circle (Ans: 4 units)
b) A tangent is drawn from the point Q(3, 2) to the
circle. Find
(i) the coordinates of P, the point where the tangent
meets the circle
(Ans: (-4.16, -0.17) OR (0.83, -5.16))
(iii) the area of the triangle QPO
(Ans: 14.96 sq. units)
13. A and B are points whose position vectors are a = 2i
+ k and b = i − j + 3k respectively. Determine the
position vector of the point P that divides AB in the
1
ratio 4: 1 (Ans: 6i 4 j 16k )
5
b) Given that a = i − 3j + 3k and b = -i − 3j + 2k,
determine
(i) the equation of the plane containing a and b
(Ans: -3x + 5y + 6z = 0)
x 4 y z 1
makes with the
(ii) the angle the line
4
3
2
plane in (i) above (Ans. 19.4°)
14. The table below shows the marks scored in General
paper by some students in mock examination from a
certain school:
Marks
Number of students
31 – 40
12
41 – 50
18
51 – 60
14
61 – 70
8
71 – 80
6
81 – 90
2
a) i) Draw a histogram to represent the scores
ii) From your histogram, estimate the mode. (46.5)
b) Calculate the mean, median and standard deviation
(Ans: 52.833, 50.5, 13.646)
1993 PAPER TWO
1. Show that the equation 3x3 + x – 5 = 0 has a real root
between x = 1 and x = 2
(a) Use linear interpolation to find the first
approximation to this root (Ans: 1.045)
(b) Using the Newton Raphson formula twice, find
the value of the root correct to 2 dps . (Ans: 1.09)
2 (a). When x = 0.8, ex = 2.2255 and
e-x = 0.4493 correct to 4decimal places.
(i) Round off the values of ex and e-xto 2decima
places.
(Ans: 2.23, 0.44)
(ii) Truncate the values of ex and e-x to 2 decimal
places
(Ans: 2.22, 0.44)
(iii)If the maximum possible error in the values ex
and e-x is 0.00005, what are the corresponding
maximum and minimum values of the quotient ex/
e-x? Give your answers correct to 3 decimal
places). (Ans: 4.953)
(b) Show that the iterative formula for solving the
equation x3 = x + 1 is xn 1 1
1
xn
. Starting with
xo = 1, find the solution of the equation to four
significant figures. Draw a flowchart that computes
and prints the root of the equation. (Ans:
1.324678713)
3 a) Forces 2i – 3j, 7i + 9j, -6i – 4j, -3i –2j act on a
lamina at points (1, -1), (1, 1), (-1, -1),
(-1, 1) respectively. Determine
(i) The resultant of the forces (Ans: (0i + 0j)
(ii) The sum of their moments about (0, 0) what effect
do the forces have on the body? (Ans: 4 units)
b) Two beads A and B start together from a point O and
slide down in a vertical plane along smooth straight
wires inclined at angles 30º and 60º respectively. The
wires are on the same side of the vertical. Taking i and j
as unit vectors in the horizontal and vertical directions
respectively, Show that the acceleration of the bead B
relative to bead A is
g
j ; where g is the acceleration
2
due to gravity.
4 ABCD is a square lamina of side a from which a
triangle ADE is removed. E being a point in CD of a
distance t from C.
(i) Show that the center of mass of the remaining lamina
a 2 at t 2
is at a distance
from BC
3(a t )
(ii) Hence, show that if this lamina is placed in a vertical
plane with CE resting on a horizontal table,
equilibrium will not be possible if t is less than
a ( 3 1)
2
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5a A particle moving with simple harmonic motion has
speeds of 5ms-1 and 8ms-1 at distances 16m and 12m
respectively from its equilibrium position. Find the
amplitude and the period of the motion.
(Ans: 20m, t = 4π seconds
b) A particle of mass 3kg is moving on the curve
described by r = 4sin 3ti + 8cos 3tj where r is the
position vector of the particle at time t.
(i) Determine the position and velocity of the particle
at the time t = 0. (Ans: -8j, 12i)
(ii) Show that the force acting on the particle is -27r.
6(a) A body of mass 10 kg rests on a smooth horizontal
plane. Horizontal forces of magnitudes 2 3 , 16, 5 and
F Newtons act on the body in the directions 0300, 1200,
00 and 2700 respectively. Given that the acceleration of
the body is 3 ms2. Find the value of F and give the
direction of the acceleration.
(Ans: F = 46.829, 𝜃 = 87.577°)
b) A car of mass 1500kg is pulling a trailer of mass
600kg up a road inclined at an angle = sin (0.1) . The
resistance to motion for both the trailer and car is 0.15 N
per kg. If they are retarding at 0.5ms-2, find:
(i) The tractive force exerted by the engine;
(Ans: 1323N)
(ii) The tension in the coupling between the car and the
trailer. (Ans: 378 N)
7. A particle of mass m is placed on a smooth face of a
wedge, which stands on a smooth horizontal plane.
The face of the wedge is inclined at an angle α to the
horizontal and the mass of the wedge is 4m. If the
system is released from rest, show that the speed of
the particle relative to the wedge after one second is
5 g sin
4 sin 2
8. A particle is projected with speed v at an angle
above the horizontal. If the particle passes through the
point P(x, y)
(i) Write down expressions of x and y in terms of v, θ
and time t. (Ans: y v sin
x
1 x2
g 2
)
v cos 2 v cos 2
(ii) Hence or otherwise show that
y x tan
gx 2
2v 2 cos 2
; where g is the acceleration
due to gravity.
(iii) Show that the range on the horizontal through the
v 2 sin 2
point of projection is R
g
b) A mortar fires shells at different angles of projection
from point O. If the speed of projection is 50g
where g is the acceleration due to gravity and the
shell is projected so as to pass through the point B(10,
20)
i) Find the possible angles of projection.
(Ans: 81.87°, 71.57)
ii) Deduce that the difference between the
corresponding times taken to travel from O to B is
10 2 5
5
9. a) The rate of change of atmospheric pressure, P with
respect to altitude, h in kilometers is proportional to
the pressure. If the pressure at 6000 meters is half of
the pressure at P0 at sea level. Find the formula for
the pressure at any height. (Ans: P P0 e0.0001155h )
b) Solve the differential equation
dy
( x 2 1) y 2 1 0, x 0, y 1 .
dx
(Ans: tan 1 ( y) tan 1 ( x) )
4
10. A random variable X has the probability density
function
k 1 x 2 ; 0 x 1,
f x
; Other wise.
0
Where k is a constant. Find:
i) The value of the constant k
(Ans: k = 1.5)
3
)
8
19
iii) The variance of X (Ans:
)
320
ii) The mean of X
(Ans:
b) The number of times a machine breaks down every
month is a discrete random variable X with a
probability density function
1
x ; x 0,1, 2,3,...
k
P X x 4
0 ; Other wise.
Where k is a constant
Determine the probability that the machine will not
63
)
break more than two times a month. (Ans:
64
12. The life time of a bulb is normally distributed with a
mean of 800 hours and standard deviation of 80 hours.
The manufacturer guarantees to replace bulbs which
blow after less than 660 hours.
i) What percentage of bulbs will he have to replace
under the guarantee? (Ans: 4.01%)
ii) The manufacturer is only willing to replace a
maximum of 1% of the bulbs. What should be the
guaranteed lifetime of the bulbs? (Ans: 613.92 hrs)
iv) Instead of reducing the guaranteed lifetime as in
(ii), the mean lifetime was increased by superior
technology. What should be the new mean so
that only 1% are replaced if the guaranteed
lifetime remains 660 hours but the standard
deviation is reduced to 70 hours?(Ans: 822.82)
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13. Nkeza makes 5 practice runs in the 100 m sprint. A
run is successful if he runs it in less than 11 seconds.
There are 8 chances out of 10 that he is successful.
Find the probability that
a) i) He records no success at all, (Ans. 0.003)
ii) He records at least 2 successes (Ans: 0.9933)
b) If he is successful in the 5 practice runs, he makes
two additional runs. The probability of success in
either additional runs is 0.7. Determine the
probability that Nkeza will make 7 successful runs in
total. (Ans: 0.16073)
14. Ten shops in Kampala which attract a similar
number and type of customers are ranked in times of
quality of service, size of verandah and price of
items. Rank 1 indicates best service, largest verandah
and lowest price of commodities. The results,
including monthly average sales and are given below:
Shop Quality of Size of
Price of
Sales
service verandah commodities (kg)
A
3
3
6
20
B
7
5
10
10
C
4
10
7
31
D
6
7
2
47
E
8
2
4
37
F
2
1
5
38
G
5
8
3
38
H
9
6
8
15
I
10
4
10
21
J
1
9
1
42
a) By calculation, determine whether the price of
commodities or the size of the verandah is the more
important factor affecting sales.
(Ans ρ = 0.8758; Since there is a highly positive
correlation between price of commodities and sales,
therefore, it is the price of commodities other than
the size of the verandah that affects sale.)
b) Is there any evidence that the size of the verandah
influences the quality of service?
(Ans ρ = 0.-0.1879; Null hypothesis; H0; = 0 (there
is no correlation between size of the verandah and
quality of service)
Alternative; H1; | | > 0 (there is correlation between
the two)
But from the tables, | | = 0.65
Since | |= 0.188 < 0.65
Therefore there is no sufficient evidence at 5% level
of significance to show that the size of the verandah
influences the quality of service.
c) Is there evidence that a shop with lower priced
commodities offer poor quality service (e.g. by
employing fewer sales people)?
Ans: Since | |= 0.558 < 0.65, there is no sufficient
evidence at 5% level of significance to conclude
that a shop with lower priced commodities offer
poor quality services
1994 PAPER ONE
1. a) Solve 2
x 1 x 4 1
13
(Ans: 5, 9 )
b) solve the simultaneous equations:
x+y+z=2
x 2 y y 2z 2x z
3
4
5 (Ans: 1, -2, 3)
2) a) If log 2 x log 4 x log16 x
21
, find x
16
3
(Ans: x = x 2 4 )
b) In an arithmetic progression u1 + u2 + u3 + ...., u4 =
15 and u16 = -3. Find the greatest integer N such
that uN ≥ 0. Determine the sum of the first N terms
of the progression. (Ans: 136.5)
3. a) Given that α and β are the roots of the equation
x2 + px + q = 0, express (α – β2) (β – α2) in terms of
p and q. Deduce that for one root to be the square of
another, p3 − 3pq + q2 + q = 0 must hold
b) Determine the expansion of
( x 4)
in
( x 2 1)
ascending powers of x up to the term containing xr
for |x| <1.
(Ans: Hence
5 3 1r
x4
4 x 4 x 2 x 3 .....
2
x 1 x 1
xr
Provided |x| < 1 i.e. -1 < x < 1)
4. a) Given that z = 3 + 4i, find the value of the
expression z
b) Given that
25
. (Ans: 6)
z
z 1
2 , show that the locus of the
z 1
complex number is
x2 y 2
10 x
1 0 .
3
Sketch the
locus.
5. a) Express
sin 2 cos 2 1
in terms of tan θ.
2 2 sin 2
(Ans:
b)
c)
1
tan 1
)
Find the general solution of the equation √3 sin θ –
2
)
cos θ + 1 = 0 (Ans: 2n
3
Factorise cos θ – cos 3θ – cos 7θ + cos 9θ in the
form Acosk θ sinl θ sinm θ, where A, k, l and m are
constants. (Ans: 4cos5 sin 3 sin 2 )
6. Given that sin x + sin y = λ1 and cos x + cos y = λ2,
show that
i.
tan
( x y ) 1
2
2
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2 2 1 2
ii. cos (x + y) =
2 2 1 2
b) Solve the simultaneous equations:
cos x + 4siny = 1
4secx − 3 cosec y = 5 for values of x and y
between 00 and 3600
(Ans: x 78.5, 281.5 , y 11.5, 168.5 )
7. Prove that the tangents to the parabola y2 = 4ax at the
points P(ap2, 2ap) and Q(aq2, 2aq) meet at the point
T(apq, a(p + q)).
i) If M is the midpoint of PQ, prove that TM is
bisected by the parabola
ii) If P and Q vary on the parabola in such a manner
that PQ is always parallel to the fixed line y = mx,
show that T always lies on the fixed line parallel to
the x-axis
8. The coordinates of a point P(x, y) on the curve are
given parametrically by the equations x = a cos θ, y =
b sin θ where a and b are constants and θ is the
parameter.
Find:
i. The Cartesian equation for the curve and identify the
x2 y2
curve (Ans: 2 2 1 )
a
b
ii. The equation of the tangent to the curve at the point
where the parameter θ = φ.
(Ans: aysinφ + bxcosφ − ab = 0)
iii. The relation between φ1 and φ2 if the tangents at the
points (a cos φ1, b sin φ1), (a cos φ2, b sin φ2) are at
right angles to one another.
(Ans: tan 1
b
2
a tan 2
2
)
9. a) Differentiate:
i. eax sin bx. (Ans: eax sin bx a b cot bx )
ii.
( x 1) 2 ( x 2)
, giving your answer in the
( x 3)3
simplest form (Ans:
b) Given that y e
tan 1 x
dy x 1 5 x 9
)
dx
x 34
, show that
2
d y
dy
(2 x 1)
0.
dx2
dx
Hence or otherwise, determine the first four nonzero terms of the Maclaurin’s expansion of y.
(1 x 2 )
(Ans: 1 x
x 2 x3
.... )
2
6
2
10. a) Evaluate
( x tan x)dx.. (Ans: 1.0003 (4 dps))
1
The accuracy can be improved by increasing the
number of sub-intervals.
b) Use the trapezium rule with six sub-intervals to
estimate
sin
x
dx. Correct to 3 decimal places.
3
0
Find the error in your estimation and suggest how the
accuracy of your result can be improved. (Ans:
(0.0033)
11. a) Determine the equation of the normal to the curve
y
1
at the point x = 2. Find the coordinates of the
x
other point where the normal meets the curve
again.
(Ans: (18 , 8) )
b) Find the area of the region bounded by the curve
y
1
, the x-axis and the lines
x 2 x 1
x = 1, x = 2
(Ans: 0.1823 sq units)
12. Show that the curve y
x 1
x 2x
2
has no turning
points. Sketch the curve. Give the equations of the
asymptotes.
13. A vector XY of magnitude a units makes an angle of α
with the horizontal. Another vector YZ beginning from
the end point Y, inclined at an angle β to the same
horizontal axis is of magnitude b units. If θ is the angle
between the positive directions of the two vectors,
where θ = β − α is acute, show that he resultant vector
XZ has a magnitude xz equal to
(a 2 b 2 2ab cos )
units and is inclined at an angle sin 1 b sin
to the
xz
horizontal. Hence or otherwise calculate the magnitude
and direction of the resultant vector of the vectors XY
and YZ, inclined at 300 and 750 to the horizontal and of
magnitude 9 and 6 units respectively
(Ans: 47.7642)
14. The table below shows the weights of Freshers in
1991/1992 academic year who underwent medical
examination at the university hospital
Weight (in kg) Number of students
44 – 44
3
45 – 49
10
50 – 54
15
55 – 59
10
60 – 64
4
65 – 69
5
70 – 74
4
75 – 79
6
80 – 84
1
a) Calculate,
i) The mode (Ans: 52)
ii) The median and mean weight of the students.
(Ans: 55, 57.948)
b) Draw a cumulative frequency graph. Hence deduce
the interquartile range of the weights (Ans: 16)
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15. a) Four and five digit numbers greater than 6000 are
obtained by arranging the digits 3, 4, 5, 6, 7. How many
of these are
i) Odd numbers (Ans: 30)
ii) Even numbers (Ans 48)
b) Nine out of twelve members of the school’s
Geography club are to be taken out on a study tour.
Given that there are seven boys and five girls and at
least three girls have to go for the tour.
(i) Find the number of ways in which the selection of
students can be done (Ans: 16 ways)
(ii) If there are two sisters in the club, who are
definitely selected to go, in how many ways can the
remaining students be selected? (Ans: 161 ways)
1994 PAPER TWO
1.(a) Given the following table of values:
x 0 5 10 15 20
t 0 12 25 39 54
Use linear interpolation to find
i. t when x = 12 (Ans: 30.6)
ii. x when t = 45 (Ans: 17)
b) Show graphically that there is only one positive real
root of the equation xe-x – 2x + 5 = 0.
Using the Newton-Raphson method, find this root
correct to one decimal place (Ans: 2.5967)
2 a) Given x = 3, y = 12, z = 6, all to the nearest integer,
find the maximum value of
i)
x y
,
z
ii)
x y
xy
, iii) .
z
y z
(Ans: 0.8696, 7.9545, -1.4649)
b) A trader in tea and coffee makes an annual profit in
tea of Sh. 1080 million with a margin of error of ±10%
and an annual loss in coffee of Sh. 560 million with a
margin of error of ±5%
i. Find the range of values representing his real
income. (Ans: 384 ≤ P ≤ 656)
ii. Given that his annual income tax is Sh. 75 million,
express this as a percentage of his gross income
giving your answer as a range of value.
(Ans: 11.4% to 19.5%)
3. To a motorist travelling due north at 40 kmh1 the
wind appears to come from the direction N 600 E at 50
kmh-1.
i. Find the true velocity of the wind.
(Ans: 45.82 kmh-1, N 70.9W)
ii. If the wind velocity remains constant, but the
speed of the motorist is increasing, find his speed
when the wind appears to be blowing from the
direction N 450 E. (Ans: 58.3 kmh-1)
4. Strings AC and BC are both of natural length 5l. AC is
inelastic and BC has a modulus of elasticity of λ. A
and B are attached to points in a horizontal line,
distance 5l apart. A mass M is attached to C and the
system is in equilibrium in a vertical plane with BC of
length 6l. Find λ and the tensions in CA and CB.
(Ans: λ = 17.165, 3.433M N; 7.347M N)
5. a) A bullet travelling at 150 ms-1 will penetrate 8 cm
into a fixed block of wood before coming to rest.
Find the velocity of the bullet when it has
penetrated 4 cm of the block. (Ans: 106ms-1)
b) A particle of mass 2 kg, initially at rest at (0, 0, 0) is
2t
acted upon by the force t N
3t
t
Find i) its acceleration at time t. (Ans: a 12 t
3t
2
- 377 -
ii) its velocity after 3 seconds.
(Ans: v i j
9
2
9
4
27
4
k)
iii) the distance the particle has traveled after 3
seconds. (Ans: 8.4 m)
6. a) A, B ,C and D are the points (0, 0), (10,0), (7, 4) and
(3, 4) respectively. If AB, BC, CD and DA are made
of a thin wire of uniform mass, find the coordinates
of the centre of gravity. (Ans: 5, 1.5)
b) i) If instead AB is a uniform lamina, find its centre of
gravity, G. (Ans: 5, 1.7)
ii) If the lamina is hung from B, find the angle AB
makes with the vertical. (Ans: 18.8°)
7. a) A particle P of mass M lies on a smooth horizontal
table and is attached to two light elastic strings fixed to
the table at points A and B. The natural lengths of the
strings are AP = 12l, PB = 5l and their moduli of
elasticity are mg and
5mg
respectively. AB = 12l.
2
Show that when P is in equilibrium, AP = 6l.
P is now held at C in the line AB with AC = 5l and
then released. Show that the resulting motion is simple
Harmonic with period 4
l . Find the maximum
3g
1
speed. (Ans:
3gl )
2
8. Anon – uniform beam AB, 4.5m long is balanced
horizontally on two supports P and Q such that AP =
0.4m and QB = 0.6m. When a mass of 20kg is placed at
either end, the beam is on the point of toppling.
Find (i) the distance from A at which the weight of the
beam acts. (Ans: 1.8m)
(ii) Weight of the beam. (Ans: 56 N)
(iii) The distance from A at which the 20kg mass
must be placed for the reactions of the supports
to be equal. (Ans: 2.25m from A)
dt
t cot 2 cos
d
g
9. a) Solve the differential equation
10. The packets of omo sold in a shop are of four
categories namely, small, medium, large and giant. On
a particular day, the stock is such that the ratio of small:
medium: large: giant is equal to 4:2:1:1. The costs of
the packets are in the ratio small: medium: large: giant
equal to 350:500:800:1400 respectively.
a) 30 packets are sold randomly on that particular day,
the total cost of the sales being s shillings.
Calculate i) the expected value of s. (Ans: 17250)
ii) the standard deviation of s.
(Ans: 10310.7953)
b) Ten packets are picked at random. Determine the
probability that six are medium size packets.
(Ans: 0.0162)
11. A continuous random variable X has a probability
density function:
f(x) = kx(3 – x)
for 0 ≤ x ≤ 2,
f(x) = k(4 – x)
for 2 ≤ x ≤ 4,
f(x) = 0
Else where.
Find i) The value of k, (Ans: k = 3/16)
ii) The mean, (Ans: 1.75)
iii) F(x), the cumulative distribution function,
(Ans:
0;
3 3 2 1 3
x 3 x ;
F ( x) 16 2
2
34 x 323 x 12 ;
1
x0
0 2
2 4
x4
iv) P(1 ≤ x ≤ 3) (Ans. 0.6875)
12. a) Two biased tetrahedrons have each their faces
numbered 1 to 4. The chances of getting any
one face showing uppermost is inversely
proportional to the number on it. If the two
tetrahedrons are thrown and the number on the
uppermost face noted, determine the probability
that the faces show the same number.
(Ans: 0.328)
(b) If it is a fine day, the probability that Alex goes to
play football is 9 10 and the probability that Bob
goes is 3 4 . If it is not
fine, Alex’s probability is 1 2 and Bob 1 4 . Their
2.
decisions are independent. In general it is known
iven that t = 3 when
1
that it is twice as likely to be fine as not fine.
(Ans: t 2 cosec 5 cos 2 )
i) determine the probability that both go to play
b) The mass of a man together with his parachute is
(Ans: 59/120)
70 kg. When the parachute is fully open, the system
experiences an upward force proportional to the
ii) if they both go to play, what is the probability
velocity of the system. If the constant of
that it is a fine day? (Ans: 54/59)
proportionality is 1/10 and the system is descending
13. a) Among the spectators watching a football match,
at the speed of 10 ms-1 when the parachute opens
80% were the home teams’ supporters while 20%
out, determine the speed of the parachute three
were the visiting teams’ supporters. If 2500 of the
minutes later. (Ans: 7.738 ms-1)
spectators are selected randomly, what is the
- 378 -
probability that there are more than 540 visitors in
this sample? (Ans: 0.0215)
b) A factory manager states that the average time
taken to make one unit of a product is 48 minutes.
A sample of 49 trials were taken and the average
time was 49 minutes with a standard deviation of 2
minutes.
i. Test the manager’s claim at 99% level of
confidence.
ii. Determine the 80% confidence limits of the
mean production time per unit.
(Ans: 47.634 and 48.366.)
14. a) In many government institutions, officers
complain about typing errors. A test was
designed to investigate the relationship between
typing speed and errors made.
Twelve typists A, B, C, L, .... L were picked at
random to type the same text. The table below
shows the ranking of the typists according to
speed and errors made. [N.B. lowest ranking in
errors indicate least errors made]
TYPIST: A B C D E F G H I J K L
SPEED: 3 4 2 1 8 11 10 6 7 12 5 9
ERROR: 2 6 5 1 10 9 8 3 4 12 7 11
Calculate the rank correlation coefficient: Test the
assertion made by officers and comment on your
result. [σ = 0.71 and τ = 0.58 are Spearman’s and
Kendall’s’ correlation coefficients respectively, at
1% level of significance based on 12 observations]
(Ans: 0.8182)
b) The cost of travelling a certain distance away
from the city centre is found to depend on the
route and the distance a given place is away
from the centre. The table below gives the
average rates of travel charged for distances to
be traveled away from the city centre:
Distance, s(km)
9 12 14 21 24
Rates charged, r (shs) 750 1000 1150 1200 1350
30
33
45
46
50
1250 1400 1750 1600 2000
i. Plot the above data on a scatter diagram and draw
line of best fit through the points of the scatter
diagram.
ii. Determine the equation of the line in (i) above in the
form r = βs + ά, where ά and β are constants. Use
your result to estimate to the nearest shilling the
cost of travelling a distance of 40 km.
1250
s 475.6 ; r = 1695/-)
(Ans: r
41
1995 PAPER ONE
SECTION A (40 marks)
1. Solve the simultaneous equations:
2x + 4y = 12
3(2)x − 2(2)2y = 16. (Ans: x = 2, y = 1)
Hence show that (4)x + 4(3)2y = 100.
b) Given that α and β are roots of the quadratic
equation ax2 + bx + c, determine an equation whose
roots are α + β and α3 + β3. Hence or otherwise
solve the equations
α+β=2
α3 + β3 = 26
(Ans: α = 3 when β = -1 or α = -1 when β = 3)
2. The first term of an arithmetic progression (A.P) is 73
and the 9th is 25. Determine
i) The common difference of the A.P. (Ans: -6)
ii) The number of terms that must be added to give
a sum of 96. (Ans: 96)
b) A geometric progression (G.P.) and an arithmetic
progression (A.P) have the same first term. The
sums of their first, second and third terms are 6,
10.5 and 18 respectively. Calculate the sum of their
fifth terms. (Ans: 57)
3. a) Determine the possible values of x in the equation
log2x + logx64 = 5. (Ans: 8)
b) Jack operates an account with a certain bank which
pays a compound interest rate of 13.5 % per
annum. He opened the account at the beginning of
the year with sh. 500,000 and deposits the same
amount of money at the beginning of every year.
Calculate how much he will receive at the end of 9
years. After how long will the money have
accumulated to sh. 3.32 million? (Ans: 4.6 years)
4 a) Express each of the following complex numbers
2 6i
4i
z1 = (1 − i) (1 + 2i), z2
and z3
in the
1 i
3i
form a + bi. (Ans: 3 + i, 2i, 2 – 2i resp.)
ii) Find the modulus and argument of Z1Z2Z3 given in
(a) (i) above. (Ans: 8 + 16i, 63.4349°)
b) Find the square root of 12i – 5.
(Ans: 2 + 3i or -2 – 3i)
5. a) Express sin θ + sin 3θ in the form m cos θ sin n θ
where m and n are constants. (Ans: 2 sin 2θ cos θ)
b) Find the general solution of
cos 7θ + cos 5θ = 2cos θ.
(Ans: 2n , 2n , 2n )
2
3
sin A sin 4 A sin 7 A
tan 4 A .
c) Prove that
cos A cos 4 A cos 7 A
6. Show that f x
x x 5
x 3 x 2
has no turning points.
- 379 -
Sketch the curve y = f(x). If g x
1
, sketch the
f x
curve of y = g(x) on the same axes. Show the
asymptotes and where f(x) and g(x) intersect.
7. The tangent at any point P ct , c t on the hyperbola
xy = c2 meets x and y axes at A and B respectively. O
is the origin.
a) Prove that
i) AP = PB
ii) The area of triangle AOB is constant
b) If the hyperbola is rotated through an angle of -450
about O, find the new equation of the curve.
(Ans: x2 – y2 = 2c2)
a
b
c
2 R where ABC
8. a) Prove that
sin A sin B sin C
has all the angles acute and R is the radius of the
circumcircle.
b) From the top of vertical cliff 10 m high, the angle of
depression of ship A is 100 and of ship B 150. The
bearings of A and B from the cliff are 1620 and
202½0 respectively. Find the bearing of B from A.
(Ans: 301.5°)
d
a x a x ln a
9) a) i) Show that
dx
(ii) Find
3
2 x 1
dx. (Ans:
3
2 x 1
ln 3
1
2 x 1 ln 3 c )
b) A shell is formed by rotating the portion of the
parabola y2 = 4x for which 0 ≤ x ≤ 1 through two
right angles about its axis.
Find:
i)
the volume of the solid formed.
(Ans: 6.2832 cubic units)
ii) The area of the base of the solid formed.
(Ans: 12.5664 sq units)
10. Express
x3 3
( x 2)( x 2 1)
as partial fractions.
Hence or otherwise find
x3 3
( x 2)( x 2 1)
dx .
(Ans: x ln( x 2) 12 ln( x 2 1) tan 1 x c )
b) Use the trapezium rule to evaluate the integral
3
x
x2 1 dx using five sub-intervals. Give your
2
answer correct to 4 decimal places. Find the error in
your estimation. (Ans: 0.0001)
dy
11. a) i) If x2secx − xy + 2y2 = 15, find
.
dx
y x 2 sec x tan x 2 x sec x
)
(Ans:
4y x
ii) Given that y = θ − cos θ; x = sin θ; show that
d 2 y 1 sin
.
dx 2
cos3
b) Determine the maximum and minimum values of x2e-x
(Ans: 0, 0.5413)
12.a) Obtain the first two non- zero terms of the
2
Maclaurin’s series for sec x. (Ans: 1 x2 )
b) Show by Taylor’s expansion that the first four terms
2
3
of cos x h cos x h sin x h2 cos x h6 sin x ,
where h and x are in radians and h rather small. Use
the expansion to evaluate cos 3.90 correct to two
decimal places, using x = 0
13. The position vector of the points A and B with
3
4
respect to the origin O are 1 and 1
2
2
respectively. Determine the equation of the line AB.
3
1
2
4
(Ans: r 1 2 )
b) Find the equation of the plane OPQ where O is the
origin and P and Q are the points whose position
0
1
vectors are 3 and 0 respectively.
0
2
(Ans: -2x + z = 0)
c) i) Given that R is the point at which line AB meets
the plane OPQ, find the coordinates of R.
(Ans: R(7,-7,14))
ii) Show that the point S (1, -1, 2) lies on OR
14 The data below shows the amount of cotton (in
1000’s of bales) produced by Grower’s unions over a
certain period of time.
70
41 34 55 45 66 73 77 80 30
50
45 72 50 27 70 55 70 85 70
30
50 60 53 40 45 35 55 20 81
25
51 35 62 60 30 45 35 50 89
53
23 28 65 68 50 65 34 35 76
i) Beginning with the 20 − 29 class and using class
intervals of equal widths, construct a frequency
table for the data
Using the frequency table,
ii) draw a cumulative frequency curve for data and
hence estimate the median production. (Ans: 54)
iii) calculate the mean and standard deviation of the
production. (Ans: 53.7, 17.9822)
15. b) i) Evaluate 80 P5 80 C6 (Ans: 9.6)
- 380 -
ii) Solve for n in n C4 nC2 (Ans: 6)
iii) A committee of five students to comprise the school
council is to be selected from eight male students and
five female students. Find how many possible
committees can be obtained. (Ans: 1280 ways)
1995 PAPER TWO
SECTION A
1. a)
START
N=N
N=0
T=0
N=N+1
Pmax
T = T+ 6A + 1
8 mg
1 9 2
ii. Given that the force acts horizontally in a vertical
plane through a line of greatest slope and that the
particle is on the point of sliding down the plane,
show that the minimum force required to maintain
4 mg
the particle in equilibrium is Pmin
1 3 2
PRINT N,T
IS
N=10?
4. A particle P starts from a point with a position vector
2j + 2k with a velocity j + k. A second particle Q starts
at the same time from the point whose position vector
is –11i – 2j–7k with a velocity of 2i + j + 2k . Find
i) the time when the particles are closest together
(Ans: 6.2)
ii) the shortest distance between the particles.
(Ans: 5.079 units)
iii) How far each particle has travelled by this time.
(Ans: 8.2j + 8.2k; 1.4i + 4.2j +5.4k)
5(a). A particle of mass 2m rests on a rough plane
inclined to the horizontal at an angle of tan-1 (3),
where is the coefficient of friction between the
particle and the plane. The particle is acted upon by a
force of P Newtons.
i. Given that the force acts along the line of greatest
slope and that the particle is on the point of slipping
up, show that the maximum force possible to
maintain the particle in equilibrium is
NO
YES
STOP
Perform a dry run of the flow chart shown above. What
is the outcome in words? After the dry run, state the
relationship between N and T. (Ans: N3 = T)
(b) By sketching the graphs of 2x and tan x, show that the
equation 2x = tan x has one real root between x = 1.1
and x = 1.2. Use linear interpolation to find the value
of the root correct to 2 decimal places. (Ans: 1.17)
2.(a) Show that the Newton Raphson formula for finding
the root of the equation 2x3 + 5x − 8 = 0 is
4 xn 8
3
6 xn 2 5
.
b) Taking the first approximation to the root of the
above equation as 1.2, draw a flow diagram which
reads and prints the number of iterations and root.
Carry out a dry run of the flow chart and obtain the
root with an error of 0.001. (Ans: 1.087)
6. Two uniform rods AB, AC, each of weight W and
length 10 cm are smoothly hinged at A. the ends B
and C rest on a smooth horizontal plane. An
inextensible string joins B and C and the system is
kept in equilibrium in a vertical plane with the string
taut. An object of weight 2w climbs the rod AC to a
point E such that AE = 8cm. given that angle BAC =
2. Determine in terms of w and
14 w
i) the reaction at the ends B and C. (Ans:
)
5
ii) the tension in the string. (Ans:
Hence show that the reaction at the hinge A is given
by
w
49 tan 2 4
10
7. A particle of mass
4
3
m is attached to one end of a
string of length, l, the other end being attached to a
fixed point A. the particle falls from rest at B at the
same horizontal level as A. if AB =
i)
3. In the gulf waters, a battleship steaming northwards at
16 km-1is 5 km southwest of a submarine.
Find two possible courses which the submarine could
take in order to intercept the battleship if its speed is
12km -1. (Ans: N 64.50W and N25.50W)
14 w
)
5
3
l,
2
Show that the impulse on the string when it
2
tightens is m gl .
3
- 381 -
ii) Find the inclination of the string to the vertical
when the kinetic energy of the particle is 3 of
2
that at the point when the string first tightens.
(Ans: 48: 59°)
8 a) A bullet travelling at 150 ms-1 will penetrate 8 cm
into a fixed block of wood before coming to rest.
Find the velocity of the bullet when it has
penetrated 4 cm of the block. (Ans: 106ms-1)
(b) A particle of mass 2 kg, initially at rest at (0, 0, 0)
2t
is acted upon by the force
t N
3t
Find i) its acceleration at time t.
(Ans: v
t2
t2
3t 2
i j
kc )
2
4
4
ii) its velocity after 3 seconds.
9
2
9
4
(Ans: v i j
27
k)
4
iii) the distance the particle has traveled after
3 seconds. (Ans: 8.4m)
10. A note contains one 200sh note, three 100sh notes
and n 50sh notes. A note is selected at random from
the bag, its value noted and then replaced. The
process is repeated many times. If the average of the
values of the notes after many trials is 110sh,
determine
i) the value of n, (Ans: n = 1)
ii) the expected value of the sum of two notes
selected at random without replacement.
(Ans: 220)
11. a) In an examination, only two papers, namely
mathematics and physics were done. The failure rates
were 45% and 40% respectively.
The number of candidates who sat for the
examination was 2000. Find the probability that a
candidate selected at random
i) failed both mathematics and physics.
(Ans: 0.18)
ii) passed both mathematics and physics
(Ans: 0.33)
iii) passed mathematics and failed physics.
(Ans: 0.22)
b) Determine the number of candidates who passed both
papers in other grades given that 21.8% and 22.9%
passed with distinction in mathematics and physics
respectively. (Ans: 246)
c) When visiting a friend, john may go by road, air or
rail. The probabilities of using road, air or rail are 0.3,
0.8 and 0.6 respectively. The corresponding
probabilities of arriving on an agreed time are 0.2, 0.8
and 0.1 respectively. Find the probability of having
used the road given that he arrived on time.
(Ans: 0.0789)
12 A random variable X has probability density function
2
x a ; a x 0
3a
f x
1 2a x ; 0 x 2a
3a
Where a is constant. Determine:
i)
ii)
iii)
iv)
the value of a. (Ans: a = 1)
the median of x. (Ans: 0.2679)
P[(x ≤ 1.5)/(x > 0)]. (Ans: 0.9375)
the cumulative distribution function F(X). Sketch
the graph of F(X)
0
x 1
1 2
( x 2 x 1) 1 x 0
(Ans: F ( x) 3
2
1
6 (2 4 x x ) 0 x 2
1
x2
13. A total population of 700 students sat an
examination for which the pass mark was 50. The
marks were normally distributed. 28 students scored
below 40 marks while 30 scored above 60.
i) Find the mean mark and standard deviation of the
students. (Ans: 50.3 and 5.891)
ii) What is the probability that a student chosen at
random passed the examination? (Ans: 0.5199)
iii) Suppose the pass mark is lowered by 2 marks, how
many more students will pass? (Ans: 92)
14. In a certain commercial institution, a speed and error
typing examination was administered to 12 randomly
selected candidates A, B, C ....L of the institution. The
table below shows their speeds (y) in seconds and the
number of errors in their typing scripts (x)
No. of
A B C D E F G H
errors(x) 12 24 20 10 32 30 28 15
Speed
(y)in S 130 136 124 120 153 160 155 142
I
J K L
18 40 27 35
145 172 140 157
i) Plot the data on a scatter diagram.
ii) Draw the line of best fit on your diagram and
comment on the likely association between speed
and the errors made.
iii) Determine the equation of your line in the form
y = xk + b where k and b are constants.
26
x 102.7 )
(Ans: y 15
iv) By giving rank 1 to the fastest student, and the
student with the fewest errors, rank the above data
and use them to calculate the rank correlation
coefficient. Comment on your result. (Ans: 0.839)
- 382 -
1996 PAPER ONE
8
2i )
3
(Ans:
SECTION A (40 marks)
1. Solve 3(32x) + 2(3x) – 1 = 0. (Ans: -1)
2. Express as equivalent fraction with a rational
denominator
2
(Ans:
3 6 15
)
6
2 3 5
x 1 x 2
3. Solve the inequality
x2 x3
(Ans: 3 x 7 or x > 2)
6
4. Find how many terms of the series
1
1
1
1 2 3 ..... must be taken so that the sum
5 5
5
will differ from the sum to infinity by less than 10-6
(Ans: 9)
5. Solve the simultaneous equations
2x – 5y + 2z = 14
9x + 3y – 4z = 13
7x + 3y – 2z = 3 (Ans: x = 1, y = -4 and z = -4)
6. Find the orthocenter (the point of intersection of the
altitudes) of the triangle with vertices at A (-2, 1), B
(3, -4) and C (-6, -1). (Ans: O (-2, -4))
7. Differentiate with respect to x, expressing your results
3 5 cos x
as simply as possible : sin 1
5 3 cos x
4
(Ans:
)
5 3cos x
1
8. Evaluate
2
sin 2 x cos xdx.
0
(Ans:
2
)
3
10. Given that z = √3 + i, find the modulus and argument
of
i. z2 (Ans: 4, π/3)
calculator, that tan( A B ) 9 3 8 2
5
13. a) Prove that (sin2θ – sinθ)(1 + 2cosθ) = sin 3θ.
b) A vertical pole BAO stands with its base O on a
horizontal plane, where BA = c and AO = b. a point
P is situated on a horizontal plane at distance x
from O and the angle APB = θ.
Prove that tan
i)
cx
x b 2 bc
2
iii. show in an Argand diagram the points
1
z
b) In an Argand diagram, P represents a complex
number z such that 2|z – 2| = |z – 6i|
Show that P lies on a circle; find
i. the radius of this circle: (Ans: 4.2164 units)
ii. the complex number represented by its centre.
x 1 x 9
x 1 x 9
Determine the turning point of the curve.
(Ans: minima - 3, - 1 , maxima -(-3, -4))
4
ii)
Determine the equation of the asymptotes of
the curve. (Ans: x = 1, x = -9, y = 1)
iii) Sketch the curve
15. a) Find the general solution of the equation
(Ans: ½, -π/6)
representing complex numbers z, z2 and
12. a) Given that 7tanθ + cot θ = 5secθ, derive a
quadratic equation for sin θ.
Hence or otherwise, find all values of θ in the
interval 00 ≤ θ ≤ 1800 which satisfy the given
equation, giving your answers to the nearest 0.1°,
where necessary.
(Ans: θ = (19.50, 300, 1500, 160.50))
b) The acute angles A and B are such that cosA = ½, sin
B = 1/3. Show without the use of tables or
14. A curve is given by y
b) The sum of p terms of an arithmetic progression is
q and the sum of q terms is p; find the sum of p + q
terms.
(Ans: x = -(p +q))
1
z
(Ans: Centre- 113 , 8 , radius = 4.71 units)
26
26
As P takes different positions on the horizontal
plane, find the value of x for which θ is greatest.
(Ans: 1826' )
SECTION B
9. a)Find x if log x 8 log x2 16 1 (Ans: x = 2)
ii.
11. a) Find the equation of the circle circumscribing the
triangle whose vertices are A (1, 3), B (4, -5) and
C (9, -1). Find also its centre and radius.
113
8
41
0)
x y
(Ans: x 2 y 2
13
13
13
b) If the tangent to the circle, at A (1, 3) meets the xaxis at P (h, 0) and the y-axis at Q (0, k), find the
values of h and k.
x
dy
2 y ( x 2)e x . (Ans: y e x cx2 )
dx
b) The rate of cooling of a body is given by the
dT
k (T 10) Where T is the
equation
dt
temperature in degree Centigrade, k is a constant,
and t is the time in minutes.
When t = 0, T = 90 and when t = 5, T = 60. Find T
when t = 10. (Ans: 41.25°)
- 383 -
16. a) In the triangle ABC, P is the point on BC such that
BP : PC = λ: μ
Show that (λ + μ)AP = λ AC + μAB.
b) three non collinear points A, B and C have
position vectors a, b and c respectively with
respect to an origin O. The point M on AC is such
that AM : MC = 2:1 and the point N on AB is such
that AN : NB = 2: 1.
i) Show that BM = 1/3 a – b + 2/3c, and find a
similar expression for CN.
(Ans: 13 a 23 b c )
ii) The lines BM and CN intersect at L. Given
that BL = rBM and CL = sCN, where r and s
are scalars, express BL and CL in terms of r, s,
a, b and c. (Ans: 13 sa 23 sb sc )
iii) Hence by using triangle BLC, or otherwise,
find r and s. (Ans: r 3 5 and s 3 5 .)
1996 PAPER TWO
SECTION A
1. One end of a light inextensible string of length of
75cm is fixed to a point on a vertical pole. A particle
of weight 12N is attached to the other end of the
string. The particle is held 21cm away from the pole
by a horizontal force. Find the magnitude of this
force and the tension in the string.
(Ans: 12.5N, 3.5N)
2. A particle with position vector 4i + 10j + 20k moves
with a constant speed of 5ms-1 in the direction of the
vector 4i + 7j + 4k. Find its distance from the origin
after 9seconds. (Ans: 85 m)
3. A cyclist travels 1.25km as he accelerates uniformly
at a rate of Qms-2 from a speed of 15kmh-1. Find the
value of Q. (Ans: 16.7ms-2)
4. In an experiment the following observations were
recorded
T:
0
12
20
30
θ:
6.6
2.9
-0.1
-2.9
Use linear interpolation to find
i)
θ when T = 16 (Ans: 1.4)
ii)
T when θ = –1 (Ans: 23.21)
5. A balanced coin is tossed three times and the number
of times X a “Head” appears is recorded. Complete
the following table
N
0
1
2
3
Event {TTT}
{HHT,HTH,THH}
P(X=n)
Determine the average or the expected number of
heads to appear. (Ans: 1.5)
6. In a certain year in the mid -1980s, the production of
tea in the common wealth as per the following
countries was as shown below.
Country
Production of tea in
millions of kg
Bangladesh 41
India
635
Indonesia
108
Kenya
140
Malawi
40
Sri Lanka
212
Tanzania
17
Uganda
7
Give a pie chart representation of the data
7. A bicycle dealer imports 40% and 60% of spare parts
from countries A and B respectively. The
percentages of parts produced defective in the
countries are 0.3% and 0.5% respectively. A spare
part is drawn at random from sample of part
imported from A and B. Find the probabilities that
i) it is defective and is from country B. (Ans: 0.003)
ii) it is defective. (Ans: 0.0042)
8. A population consists of 15 numbers 2, 4, 7, 3, 5, 6, 3,
6, 10, 7, 8, 9, 3, 4, 3. Find:
- 384 -
i) the mode (Ans: 3)
ii) the median. (Ans: 5)
iii) the mean and standard deviation of the
population. (Ans: 5.333, 2.3851)
SECTION B
9. (a) Find the position of the center of gravity of three
particles of masses 1kg, 5kg and 2kg which lie on
the y-axis at points (0,2), (0,4) and (0,5)
respectively. (Ans: (0, 4))
(b) The area enclosed by the curve y = x2 and the lines
y = 0, x = 2 and x = 4 , lying in the first
quadrant is rotated about the x–axis through
one revolution . Find the co-ordinates of the center
of gravity of the uniform solid so formed.
(Ans: 3.39, 0)
10. Initially two ships A and B are 65 km apart with B due
east of A . A is moving due east at 10 km h-1 and B
due south at 24kmh-1 .The two ships continue moving
with these velocities. Find the least distance between
the ships in the subsequent motion and the time taken
to the nearest minute for such a situation to occur
(Ans: 60 km, 57.6 minutes)
11.(a) A conical pendulum consists of a light inextensible
string AB of length 50cm fixed at A & carrying a
bob of mass 2kg at B. The bob describes a
horizontal circle about the vertical through A with a
constant angular speed of A rad m–1.
Find the tension in the string. (Ans: T = ω2)
(b) A smooth surface is inclined at 300 to the horizontal.
A body A of mass 2kg is held at rest on the surface
by a light elastic string which has one end attached
to A and the other to a point on the surface 1.5m
away from A up a line of greatest slope. If the
modulus of the string is 2g N, Find its natural
length . (Ans: 1m)
12. A block of mass 6.5kg is projected with a velocity of
4ms-1 up a line of greatest slope of a rough plane.
Calculate the initial kinetic energy of the block.
(Ans: 52J)
The coefficient of friction between the block and the
plane is 2/3 and the plane makes an angle with the
horizontal where sin = 5/13. The block travels a
distance d m up the plane before coming
instantaneously to rest. Express in terms of d
i)
14. In a survey of newspaper reading of members of staff
of a university, it is found that 80% read NEW VISION
(N), 50 percent read MONITOR (M) and 30% read the
EAST AFRICAN (E). Further, 20% read both M and N,
15% read both N and E and 10% read both M and E.
a) If a member of staff is chosen at random from the
university, find the probabilities
i) that the member reads none of the three papers.
ii) the member is one of those who read at least one
of the three papers.
b) Estimate the number of members of staff who read at
least two papers if the total number is 500
c) What is the probability that given that a member of
staff reads two papers, he reads all the three?
(Ans: impossible to obtain answers for this
question)
15. A certain factory produces ball bearings. A sample
of the bearings from the factory produced the
following results
Diameter of bearings Frequency
in mm
91 – 93
4
94 – 96
6
97 – 99
34
100 – 93
40
103 – 102
13
106 – 108
3
i) Determine the mean and diameter of the sample
bearings. (Ans: 99.83 mm, 9.3411)
ii) Estimate the mean surface area of the bearings
produced by the factory. (Ans: 31293.33)
16. The following table gives the marks obtained in
Calculus, Physics and Statistics by seven (7)
students
Calculus 72 50 60 55 35 48 82
Physics 61 55 70 50 30 50 78
Statistics 50 40 62 70 40 40 60
Draw scatter diagrams and determine rank correlation
coefficients between the performances of the students in
i. Calculus and Physics. (Ans: 0.902)
ii. Calculus and Statistics. (Ans: 0.643)
Give interpretations to your results.
the potential energy gained by the block in coming
to rest. (Ans: 25d J)
ii) the work done against friction by the block in
coming to rest. Hence calculate the value of d.
(take g = 10ms-1). (Ans: d = 0.8)
13. i)Show that the iterative formula for solving the
2
equation 2x2 – 6x – 3 = 0 is xn 1 2 xn 3
4 xn 6
2
ii) Show that the positive root for 2x – 6x – 3 = 0 lies
between 3 and 4. Find the root correct to 2 decimal
places. (Ans: 3.44)
- 385 -
1997 PAPER ONE
SECTION A
1. Solve the equation 4cosx – 2cos2x = 3 for 00 ≤ x ≤ π.
(Ans: x = 60°)
2
2. Find the values of k for which the equation x x 1
x 1
has repeated roots. What are the repeated roots?
(Ans: 0 and 0 or 2 and 2)
3. By reducing to echelon form, solve the simultaneous
equations
x+y+z=0
x + 2y +2z = 2
2x + y 3z = 4. (Ans: x = -2, y = -1 and z = 3)
4. Given that
x = θ – sin θ,
y = 1 – cosθ,
dy
cot .
2
dx
Show that
5. ABCD is a square inscribed in a circle x2 + y2 – 4x –
3y = 36. Find the length of diagonals and the area of the
square. (Ans: 84.5 Sq. Units)
6. Find. i)
sin
2
xdx (Ans:
ii)
tan
3
xdx (Ans:
1
1
x sin 2 x C )
2
2
1
tan 2 x ln cos x C )
2
7. Find the distance of the point (-2, 0, 6) from the
plane 2x − y + 3z = 21. (And: 1.8708 units)
8. Determine the volume of the solid generated when
the region bounded by the curve y = cos 2x and the xaxis for values of x between 0 and ¾ is rotated about
the x-axis. (Ans: 1.232 cubic units)
i. Find the vector equation of line AC
(Ans: r 2i 3j 10i 8j )
ii. Determine the coordinates of D if ABCD is a
parallelogram. (Ans: (5, 0))
iii. Write down the vector equation of the line
through which point B perpendicular to AC and
find where it meets AC.
(Ans: r i 2 j t (4i 5j) )
2
13. Express f ( x) 2 x x 14 in partial fractions.
(4 x 2 1)( x 3)
3
Hence evaluate
f ( x)dx . (Ans: 0.7440)
1
14. Find the equation of the chord joining the points
c
ct1 , and ct 2 , c on a hyperbola. Hence
t1
t 2
c
deduce the equation of the tangent at ct , .
t
(Ans: x t1t2 y c t1 t2 0 )
Find the equation of the tangents to the hyperbola x =
4t, y = 4/t which passes through point (4, 3)
(Ans: x + 4y − 16 = 0 and 9x + 4y − 48 = 0)
15. Show that the tangents at (-1, 3) and (1, 5) on the
curve y = 2x2 + x + 2 passes through the origin. Find
the area enclosed between the curve and these two
tangents. (Ans: 4 3 sq .units )
16. a) Use Maclaurin’s theorem to expand ln(1 + sin x)
as far as the term in x3. (Ans: x
SECTION B
9. If logba = x, show that b a
log a b
1
x
x 2 x3
.... )
2
6
1
and deduce that
1
log b a
b) Expand 1 x 3 as far as the x3. Use your expansion
to deduce 3 24 correct to three significant figures.
(Ans: 2.88)
b) Solve i) logn4 + log4n2 = 3 (Ans: n = 2 or 4)
ii) 2 2 x 1
3
2 x 1 (Ans: x = (0, 1.585))
2
10(a). Given the complex numbers z1 =1– i; z2 = 7 + i
represent z1z2 and z1 – z2 on the Argand diagram.
Determine the modulus and argument of
z1 z 2
z1 z 2
(Ans: 0.6325, -124.69520)
b) If z is a complex number in the form (a + bi),
z 1 2
solve
i. (Ans: Z = 1 ± i 2 )
z 1
11.a) Prove that sin3θ = 3sinθ – 4sin3θ.
b) Find all the solutions to 2sin3θ = 1 for θ between 00
and 3600. Hence find the solutions of
8x3 – 6x + 1 = 0. (Ans: x = (1.734, 0.766, -0.9367))
12. The points A, B and C have position vectors (-2i +
3j), (i – 2j) and (8i – 5j) respectively.
- 386 -
1997 PAPER TWO
SECTION A
1. A bag contains 5 white, 3 red and 2 green counters. 3
counters are drawn without replacement. What is the
probability that there
i) is no green counter, (Ans: 0.4667)
ii) are 2 white counters and a green counter?
(Ans: 1/6)
2. Given below is a table of corresponding values of x
and y
x
0
8
12
20
y
9.2
6.0
4.4
1.5
Use linear interpolation to find
(i)
y when x = 15 (Ans: 3.3125)
(ii)
x when y = 5.0 (Ans: 10.5)
3. A particle with vector 10i +3j+5k moves with
constant speed of 6ms-1 in the direction I +2j +2k.
Find its distance from the origin after 5 seconds.
(Ans: 39.42 m)
4. Particles of mass 4, 5 and 6kg are placed at (0, 0), (4,
3) and (5, -2) respectively in the x-y plane , Find the
co-ordinates of their centre of mass . (Ans: ( 10 3 , 1 5 ))
5. The yields of 13 plots in 1000’s of kg were 16, 7, 10,
3, 11, 5, 8, 14, 18, 4, 11, 14 and 90
Find the: (i) Mean (Ans: 10,000)
(ii) Standard deviation (Ans: 44549 )
6. A box of mass 2kg is at rest on a plane inclined at
25 to the horizontal. The coefficient of friction
between the box and the plane is 0.4. What minimum
force applied parallel to the plane would move the box
up the plane?
(Ans: 15.388N)
4
7. The probability of winning a game is . Ten games
5
are played. What is the:
(i) mean number of successes. (Ans: 8)
(ii) variance (Ans: 1.6)
(iii) probability of at least 8 successes in the ten
games. (Ans: 0.6778)
8. A mass of 3kg is at rest on a smooth horizontal table
.It is attached by a light inextensible string passing
over a smooth fixed pulley at the edge of the table to
another mass of 2kg, which is hanging freely. The
system is released from rest. Determine the resulting
acceleration and the tension in the string.
SECTION B
9. (a) The table below shows the likelihood of where A
and B spend Saturday evening:
A
i)
B
Goes to dance
1
Visits neighbour
1
Stays at home
1
2
3
6
2
1
1
3
ii) If we know that they both go out, what is the
probability that both go to dance? (Ans: 12/25)
b) Four competitors throw a die in turn. What is the
probability that
(i) they all score more than a 4) (Ans: 1/81)
(ii) two get less than a 3? (Ans: 2/3)
(iii) the total score is 23? (Ans: 1/324)
10. The probability density function of a random
variable X is given by
k ( x 2) ; 1 x 0
f x 2k (1 x); 0 x 1
0
; else where
i. Sketch the function
ii. Find k and the mean of x. (Ans: k = 3/5, -2/15)
iii. Find the probability P(0 < x < ½x/ x > 0)
(Ans: ¾)
11. The heights of students in S 1 were according to
the following frequency table
Class
151-153
154-156
157-159
160-162
163-165
166-168
f
2
14
13
13
2
1
(i) Estimate the mean and standard deviation of the
height of students. (Ans: 158.133, 3.222)
(ii) Determine and plot the cumulative frequency
distribution for the students’ heights. Hence
estimate the median, lower and upper quartiles
for the heights of the students.
(Ans: = 158 cm, 155.3 cm, 160.5 cm)
12. Using the iterative formula show that the 4th root of the
number N is xn 1
3
N
. Hence show that
xn
4
4 xn3
(45.7)¼ 2.600 (correct to 3dps)
13.a) Two particles are moving towards each other along
a straight line. The first particle has a mass of 0.2kg
and moving with a velocity of 4ms-1, and the second
has a mass of 0.4kg moving with a velocity of 3ms-1.
On collision, the first particle reverses its direction and
moves with a velocity of 2.5ms-1 . Find:
I) velocity of the second particle after collision
(Ans: 0.25 ms-1)
II) percentage loss in kinetic energy. (Ans: 81.25%)
b The diagram shows particle A of mass 0.5kg
attached to one end of a light inextensible string
passing over a fixed light pulley and under a
movable light pulley B. The other end of the string
is fixed as shown below.
6
6
Find the probability that both go out. (Ans: 25/36)
- 387 -
1998 MARCH PAPER ONE
SECTION A
1. Solve simultaneously
1 1 1
x y 6
B
x(5 − x) = 2y (Ans: (x, y) = (9, -18) and (2, 3))
A
0.5kg
(i) What mass should be attached at B for the system
to be in equilibrium. (Ans: 1 kg)
(ii)If B is 0.8kg what are the accelerations of particle
A and pulley B? (Ans: g/14 ms-2)
14. A particle of masses½ kg is released from rest and
slides down a rough plane inclined at 30 to the
horizontal. It takes 6 seconds to go 3 meters.
(i) Find the coefficient of friction between the
particle and the plane (correct to the 2d.p)
(Ans: 0.557)
(ii) What minimum horizontal force is needed to
prevent the particle from moving? (Ans: 0.0855)
15. Two uniform rods AB, BC of masses 4kg and 6kg
respectively are hinged at B and rest in a vertical
position on a smooth floor as shown. A and C are
connected by a rope.
B
A
4cm
log3 x
. Hence given that
1 log3 2
2. Prove that log6 x
log32 = 0.631, find without using tables or calculator
log6 4 correct to 3 significant figures.
3. Show that cos 4
tan 4 6 tan 2 1
tan 4 2 tan 2 1
4. The distance S m of a particle from a fixed point is
given by S t 2 t 2 6 4t t 1 t 1 , where t is the
time. Find the velocity and acceleration of the particle
when t = 1s. (Ans: 0 ms-2, 8 ms-1)
1
5. Using the substitution 2x + 1 = p, find
xdx
2 x 13
0
6. ABCD is a quadrilateral with A(2, -2), B(5, -1),
C(6, 2) and D(3, 1). Show that the quadrilateral is a
rhombus. (Ans: 1/8)
7. The points P(4, -6, 1), Q(2, 8, 4), and R(3, 7, 14) lie
in the same plane. Find the angle formed between PQ
and QR.
(Ans: 84.5°)
8. Use Maclaurin’s expansion to express
ln(1 + x)2 in ascending powers of x up to the term in
B
x4 (Ans: 2 x x 2
2 x3 x 4
)
3
2
8cm
a)Find the reactions between the rods and the floor at A
and C when the rope is taut. (Ans: 14g/3, 16g/3)
b)If now a body is attached a quarter of the way up CB,
and the reactions are equal, find the mass of the body.
(Ans: 1kg)
16 At noon, a boat A is 30 km from boat B and its
direction from B is 2860. Boat A is moving from
North East direction at 16kmh-1 and boat B is moving
in the Northern direction at 10kmh-1. By scale drawing
or otherwise determine when they are closest to each
other. What is the distance between them then?
(Ans: 2:26 pm, 11.54 km)
SECTION B
9. When f(x) = x3 − ax + b is divided by x + 1, the
remainder is 2 and x + 2 is a factor of f(x). Find a
and b (Ans: a = 5 and b = -2)
b) If the roots of the equation x2 + 2x + 3 = 0 are α
and β, form the equation whose roots are
α2 − β and β2 − α. (Ans: x2 + 2 = 0)
10. a) Show the region represented by |z − 2 + i| < 1 on
an Argand diagram
b) Express the complex number z = 1 − 3i in
modulus argument form and hence find z2 and
in the form a + bi. (Ans:
1
z
1 1 i 3
)
z 4
4
11 (a) Differentiate with respect to x
i) 2 x x (Ans: 2 x x ln( x 1) )
ii) sin 3 2x (Ans: 6sin 2 2 x cos 2 x )
b) Find the equation of the tangent and normal to the
curve y 4x3 6x2 3x at the point (1, 1)
(Ans: y = 3x – 2)
12.(a) Find the solution of 3 cot θ + cosec θ = 2 for
- 388 -
0 ≤ θ ≤ 3600. (Ans: θ = 72.40°, 220.2°)
b) Solve 2 sin x = sin(x − 600) for -1800 ≤ x ≤ 1800.
(Ans: x = -300, 1500)
2
2
13. P(ap , 2ap) and Q(aq , 2aq) are two points on the
parabola y2 = 4ax, PQ is a focal chord. Prove that
pq = -1 and hence that if the tangents at P and Q
intersect at T, the locus of T is given by x + a = 0.
PM and QN are perpendiculars onto x + a = 0, s =
^
^
(a, 0). Prove that M S N = 900 = P T Q .
4x2
14. a) Integrate
1 x
6
with respect to x
4
(Ans: sin 1 x3 c )
3
x2 1
1 x3 4 x 2 3xdx (Ans: 0.3489)
3
b) Evaluate
15. Solve a) y
y = uv.
b)
dy
2 x y by using the substitution
dx
(Ans:
y
x
2
1 = k)
2 y
x
dy
y tan x cos 2 x
dx
1
(Ans: y sin x sec x sin 3 x sec x k sec x )
3
4
1
16. Given that OP 3 and OQ 0 , find the
5
2
coordinates of the point R such that PR : PQ 1: 2
and the points P, Q and R are collinear.
(Ans: R(2.5, -1.5, 3.5)
b) Show that the vector 5i − 2j + k is perpendicular to
the line r i 4 j t (2i 3j 4k ) .
c) Find the equation of the plane through the point
with position vector 5i − 2j + 3k perpendicular to
the vector 3i +4j − k. (Ans: 3x + 4y − z = 4)
1998 MARCH PAPER TWO
SECTION A
1. A discrete random variable, X, has the following
probability distribution:
X
1
2
3
4
3
1
1
P(X)
½
16
16
4
4 A particle of weight 8N is attached to point B of a light
inextensible string AB. It hangs in equilibrium with
point A fixed and AB at angle of 300 to the downward
vertical. A force F at B acting at right angle to AB,
keeps the particle in equilibrium. Find the magnitude
of F and the tension in the string. (Ans: 4N, 4 3N )
5. There are 3 black and 2 white balls in each of the two
bags. A ball is taken from the first bag and put in the
second, then a ball is taken from the second into the
first, what is the probability that there are now the same
number of black and white balls in each bag as there
were to begin with?
(Ans: 3/5)
6. Using the same graph show that the curves
x3
and 2x + 5 have a common real root. Using the
Newton Raphson’s formula twice, find the positive
root of the equation x3 –2x – 5 = 0 giving your
answer correct to 2dps. (Ans: 2.09)
7. The flow chart below shows the social security
numbers (SSN) and the monthly wage (W shillings) of
an employee. P represents the net pay
START
SSN, W
IS
W <400?
YES
T=0
NO
IS
400≤W≤
3000?
YES Excess I=W-400
Excess II = 0
NO
EXCESS I = 2600
EXCESS II = W−3000
Tax=Excess x 0.125
TAX = EXCESS I X 0.125
+EXCESS II X 0.25
P = W − TAX
WRITE
SSN, W, T, P
Find the mean and variance of X (Ans: 2.1875), 0.6523)
2. A carton of mass 0.4 kg is thrown across a table with
a velocity of 25 ms-1. The resistance of the table to its
motion is 50N. How far will it travel before coming to
rest? What must be the resistance if it travels only 2
meters. (Ans: 2.5m, 62.5N)
3. A motorcycle decelerated uniformly from 20 km–1 to
8kmh-1 in travelling 896m. Find the rate of deceleration
in ms-2. (Ans: 0.0145ms-2)
STOP
Copy and complete the following
SSN
W
T
280 – 04
380
.....
180 – 34
840
.....
179 – 93
4500 .....
80 – 66
5,550 .....
P
.....
.....
.....
.....
- 389 -
385 – 03
8000
.....
.....
8. 64% of the students at A’ level take science subjects
and 36% do Arts subjects. The probability of them
being successful is ¾ for Science students and 5/6 for
Arts students. Find the probability that a student
chosen at random will fail. (Ans: 0.22)
SECTION B
9(a). Given that the values x = 4, y = 6 and z = 8 each
has been approximated to the nearest integer. Find
the maximum and minimum values of
y
(i) , (Ans: 1.857142857, 1.2222)
x
(ii)
zx
y
(Ans: 0.90909090, 0.46153846)
(iii) (x + y)z. (Ans: 93.5, 67.5)
(b). A Company had a capital of sh.500 million. The
profit in a certain year was
sh.25.8 million in
section A of the company and sh.14.56 million in
section B. There was a possible error of 5% in
section A and an 8% error in B.
Find the
maximum and minimum values of the total profits
of the sections as a percentage of the capital.
(Ans: 8.56%, 7.58%)
10. (a) Show that the Newton Raphson’s formula for
finding the smallest positive root of the equation
3 tan x x 0 is
6 xn 3 sin 2 xn
6 2 cos 2 xn
(b) By sketching the graphs of y = tan x, y
x
or
3
otherwise, find the first approximation to the
required root and use it to find the actual root
correct to 3dps (Hint work in radians).
(Ans: 2.456)
11. The diameter of a sample of oranges to the nearest
cm were:
Diameter(cm) 8
9
10 11 12 13 14
Frequency
9
15 21 32 19 13 11
i) Calculate the mean and standard deviation.
(Ans: 11)
ii) Assuming the distribution is normal, find the
minimum diameter if the smallest 10% of the
oranges are rejected for being too small.
(Ans: 1.6633)
12. A pupil has ten multiple choice questions to answer.
There are four alternative answers to choose from. If a
pupil answers the questions randomly, find the
i) probability that at least four answers are correct.
(Ans: 0.2241)
ii) most likely number of correct answers (Ans: 2)
b) Otim’ chances of passing Physics are 0.60, of
Chemistry 0.75 and of Mathematics 0.80.
i) Determine the chance that he passes one subject
only. (Ans: 0.17)
ii) If it is known that he passed at least two
subjects, what is the probability that he failed
Chemistry? (Ans: 0.4815)
13. A random variable X has a distribution probability
function given by
0 x 1
kx,
2
f ( x) k (4 x ), 1 x 2
0,
Elsewhere
i) Find the constant k (Ans: 6/13)
ii) Determine E(X) and var (X). (Ans: 1.1923, 0.1399)
iii) Find the cumulative distribution function, F(x) and
sketch it.
0
3 2
x
13
Ans: F ( x)
3
24 x - 2 x -19
13
1
x 0
0 x 1
1 x 2
x 2
14. At the points A (0, - 4), B (2, 1), C (1, 3) and D ( 1
,
5
4,-2), there are forces
2
3
1
2
and N
4
4
respectively
(i) Prove that the resultant is a couple and find its
moment
(ii) If the force at D is halved, find the magnitude of
the resultant force. Find also the equation of the
line of action of the resultant. (Ans: y = 2x – 13)
15.(a) A particle is projected vertically upwards from a
point O with speed
4
v , After it has travelled a distance
3
2
x above O, on its upward motion, a second particle is
5
projected vertically upwards from the same point and
with the same initial speed.
Given that the particles collide at a height
2
x above O,
5
x and v being constant, show that
i) at maximum height H, 8v2 = 9gH,
ii) when the particles collide 9x = 20H.
b) A stone thrown upwards at an angle to the
horizontal with speed,ums-1, just clears a vertical wall
4m high and 10m from the point of projection when
travelling horizontally. Find the angle of projection.
(Ans: 38.66°)
- 390 -
1998 NOV/DEC. PAPER ONE
16
A
2cm
2cm D
C 2cm
8cm
J
8cm
E
F
G
4cm
4cm
I
B
6cm
H
The figure ABCDEFGHIJ shows a symmetrical
composite lamina made up of a semi-circle, radius 3cm,
a rectangle CDEF 2cm × 8cm and another rectangle
GHIJ 6cm × 4cm. Find the distance of the centre of
gravity of this lamina from IH. If the lamina is
suspended from H, by means of a peg through a hole,
calculate the angle of inclination of HG to the vertical.
(Ans: 6.716 cm from IH, 24.07 to the vertical)
SECTION A
1. Solve cos 3 sin 2 for 00 ≤ θ ≤ 3600
(Ans: θ = 600)
2. The gradient of a certain curve is given by kx. If the
curve passes through the point (2, 3) and the tangent
that this angle makes is an angle of tan-1 6 with the
positive direction of the x-axis, find the equation of the
3
curve. (Ans: y x 2 3 )
2
3. Given that the roots of the equation x2 − 2x + 10 = 0
are α and β, determine the equation whose roots are
1
(2 )
2
1
and
(2 ) 2
(Ans: 324x2 + 1 = 0)
4. By row reducing the appropriate matrix to echelon
form, solve the system of equations
x + 2y − 2z = 0
2x + y − 4z = -1
4x − 3y + z = 11
(Ans: x = 3, y = 1 and z = 2)
5. Find in the simplest form the derivative of
1 x2
2
cos1
(Ans:
)
1 x 2
1
x2
6. Show that tan n xdx
tan n 1
tan n 2 xdx . Hence or
n 1
otherwise evaluate tan 4 xdx .
7. Calculate the area of triangle with vertices
(–1, 3), (5, 2) and (4, –1) (Ans: 7.6811 sq. units)
8. PQRS is a quadrilateral with vertices P(2, -1), Q(4, -1)
and S(2, 1). Show that the quadrilateral is a rhombus.
SECTION B
9. a) Given that Z1 = -i + 1, Z2 = 2 + i and
Z3 = 1 + 5i, represent Z2Z3, Z2 − Z1 and
1
on the
Z1
Argand diagram. Also show the representation of
Z 2 Z3
1
Z 2 Z1 Z1
b) Prove that for positive integer n,
(cos i sin )n cos n i sin n . Deduce that this
formula is also true for negative values of n.
10. a) Solve 4x − 2x + 1 − 15 = 0 (Ans: x = 2.3219)
b) Five million shillings are invested each year at a
rate of 15% interest. In how many years will it
accumulate to more than sh. 50 million?
(Ans: 6 years)
d2y
dy
x 1
, show that 2 y
2
dx
dx
11(a) If y tan
b) Find the equation of the tangent to the curve x2 +
y2 − 2xy = 4x at (1, -1) (Ans: y = -1)
- 391 -
12. The vector equations of lines P and Q are given as
rp = t(4i + 3j) and rq = 2i + 12j + 5(i − j).
a) Use the dot product to find the angle between P
and Q (Ans: 8.13)
b) If the lines P and Q meet at M, find the coordinates
of M. Find also the equation of the line through M
perpendicular to the line Q (Ans: M(7, 7))
8
13. Sketch the curve y x 2 for x > 0, showing any
x
asymptotes. Find the area enclosed by the x-axis, the
8
line x = 4 and the curve y x 2 . (Ans: 10 sq units)
x
If this area is now rotated about the x-axis through
3600, determine the volume of the solid generated,
correct to 3 significant figures.
(Ans: 42.1497 cubic units)
14. From the top of a tower 12.6m high, the angles of
depression of ships A and B are 120 and 180
respectively. The bearing of ship A and ship B from
the tower are 1480 and 209½0 respectively. Calculate:
i ) how far apart the ships are from each other
(Ans: 53.1412 m)
ii) the bearing of ship A from ship B. (Ans: 108.11°)
2 tan x
15. a)Prove that sin 2 x
1 tan 2 x
b) Solve for x in:
(i) tan x + 3 cot x = 4
(Ans: Hence x = (45o, 71.565o, 225o, 251.565o)
(ii) 4 cos x − 3 sin x = 2; 0 ≤ x ≤ 3600
(Ans: x = 29.50, 256.70)
16. A rumour spreads through a town at a rate which is
proportional to the product of the number of people
who have heard it and that of those who have not
heard it. Given that x is a fraction of the population of
the town who have heard the rumour after time, t.
i) Form a differential equation connecting x, t and a
dx
k 1 x x. )
constant, k. (Ans:
dt
ii) If initially a fraction C of the population had heard the
C
rumour. Deduce that x
kt
C (1 C )e
iii) Given that 15% had heard the rumour at 9:00 a.m
and another 15% by noon, find what fraction of the
population would have heard the rumour by 3:00
p.m (Ans: 21%)
1998 NOV/DEC PAPER
TWO
SECTION A
1. The probability that two independent events occur
C or both
together is 152 . The probability that either
events occur is 53 . Find the individual probabilities of
the two events. (Ans: 1/3, 2/5)
2. Find an expression for the power exerted by a force r
= 4t2i + 4tj + 7t2k acting on a particle to give it a
velocity of v = ti − 3t2j + 2tk. Find also the
acceleration of the particle. (Ans: 6t3, i 6tj 2k )
D
3.
2 2N
6N
A
B
4N
Three forces of magnitude 6N, 4N, and 2 2 N act along
AD, AB and AC respectively as shown above. ABCD is
the square. Determine the resultant of the three forces
and the angle, which it makes with AB.
(Ans: 10N, 53.13°)
4. Given X = 2.2255, Y = 0.449 correct to the given
number of decimal places. State the maximum
possible errors in the value of X and Y. hence
determine the
i) absolute error (Ans: 5.63 × 10-3)
ii) limits with in which the value of quotient
X
Y
lies giving your answers to 2 decimal places
(Ans: 4.95 and 4.96)
5. The probability that Bob wins a tennis game is 2/3. He
plays 8 games. What is the probability that he wins
i) at least 7 games, (Ans: 0.1951)
ii) exactly 5 games?
(Ans: 0.2731)
6. An elastic string of natural length 1.2 m and modulus
of elasticity 8 N is stretched until the extending force
is 16N. Find the extension and the work done.
(Ans: 0.6m, 27 joules)
7. By using the Newton Raphson formula and x0
2
as
the initial approximation to the root of the equation
10cos x – x = 0, show that the next approximation is
5
11
8. The table below shows the distribution of marks
gained by a group of students in a mathematics test
marked out of 50.
Marks
1 − 10
Frequency
15
- 392 -
11 − 20
21 − 30
31 − 40
41 − 50
20
32
26
7
Plot an ogive for the data and use it to estimate the
median mark and semi – interquartile range.
(Ans: 25.5, 8.5)
SECTION B
9 (a) Show that the iterative formula for finding the 4th
root of a number N is given by:
x n 1
3
N
xn
4
3x n 3
,
n = 0,1,2,3.
(b) Draw a flowchart that
(i) Reads the number N and the initial
approximation xn
(ii) Computes and prints N and its fourth root after 4
iterations and give the root correct to 3dps
(c) Perform a dry run for N = 39.0 and xn = 2.0
(Ans: 39.0)
10. A particle is projected with a speed of 102g ms-1
from the top of a cliff, 50m high. The particle hits
the sea at a distance 100m from the vertical
through the point of projection. Show that there are
two possible directions of projection, which are
perpendicular. Determine the time taken from the
point of projection in each case.
(Ans: 76.72°, -13.28°, t = 2.321 s)
11. A probability density function is given as
kx(4 x 2 ) ; 0 x 2
f ( x)
; Elsewhere
0
when a box is chosen at random. If a sample of 16
boxes is drawn, find the probability that the mean is
between
i) 4.6 and 4.7 kg, (Ans: 0.0761)
ii) 4.3 and 4.7 kg. (Ans: 0.3108)
15. The ages of people in a town were as follows
Age(years)
0-<5
5-<15
Number
(thousands)
4.4
8.1
15-<30 30-<50 50-<70 70-<90
10.5
14.6
9.8
4.7
a. Draw a histogram for this data
b. State the modal age interval (Ans: 0 - < 5)
c. Estimate the:
i. average age of the town (Ans: 36.0125)
ii. number of people under 18 years
(Ans: 14600)
iii. median age (Ans: 34.1781)
16. (a). Ship A is sailing with speed of Ukmh–1 in a
direction N 300E. a second ship B is sailing with a
speed v kmh-1 in a direction N ° E. The velocity of
ship A relative to B is due North East. Show that
u = v (3 + 1) (cos − sin)
b) Ship A changes its course to N60°E, while it continue
with the same speed. Ship B continues with the same
velocity. The velocity of ship A relative to ship B is
now due East. Find tan. (Leave your answer in surd
form). (Ans:
3 1
3 1
)
Find the
i) value of k (Ans: ¼)
ii) median (Ans: 1.0824)
iii) mean (Ans: 1.0667)
iv) standard deviation (Ans: 0.4422)
12. Show that the root of the equation
f(x) = ex + x3 – 4x = 0 lies between 1 and 2. By using
the Newton Raphson method. Find the root to 2decial
places. (Ans: 1.12)
13. To one end of a light inelastic string is attached a
mass of 1kg which rests on a smooth wedge of
inclination 300. The string passes over a smooth fixed
pulley at the edge of the wedge, under a second
smooth moveable pulley of mass 2kg and over a third
smooth fixed pulley ,and has a mass of 2kg attached to
the other end . Find the accelerations of the masses
and the moveable pulley and the tension in the string.
(Assume the portions of the string lie in the vertical
1g 1
plane). (Ans: a1
, g , 0, T = 9.8N).
2 2
14. Boxes made in a factory have weights which are
normally distributed with a mean of 4.5 kg and a
standard deviation of 2.0kg. Find the probability of
there being a box with a weight of more than 5.4 kg
- 393 -
1999 PAPER ONE
SECTION A
1. Given that the equation 2x2 + 5x – 8 = 0, has roots α
1
and β, find the equation whose roots are
( 2) 2
and
1
( 2) 2
(Ans:100x2 – 49x + 4 = 0)
2. The vector equations of two lines are
4 2
5
3
r1 and r2 .
1 3
6
1
Determine the point where r1 meets r2. (Ans: (8, 5))
3. Solve cos(θ + 350) = sin(θ + 250) for 0 ≤ θ ≤ 3600 .
(Ans: Hence θ = 150, 1950 for 00 ≤ θ ≤ 3600)
4. The population of a country increases by 2.75% per
annum. How long will it take for the population to
triple?
(Ans: 40.5 years)
5. Differentiate with respect to x:
i) 3x ln x2 ii) cot 2x
(Ans: 2cosec2 2x )
tan 1 ( x)
dx.
6. Evaluate
2
0 1 x
1
(Ans: impossible)
7. A curve is defined by the parametric equations
x = t2 – t,
y = 3t + 4.
Find the equation of the tangent to the curve at (2, 10).
(Ans: y = x + 8)
8. A cliff which is 100m high runs in the S.E - N.W.
direction along the coast. From the top of the cliff, the
angle of depression of a ship moving at steady speed of
24 kmh-1 towards the cost is 080. Calculate the distance
of the ship from the coast at the instant. What is the
angle of elevation of the cliff from the ship one minute
later? (Ans: 17.796°)
SECTION B (60 marks)
9. The locus of P is such that the distance OP is half the
distance PR, where O is the origin and R is the point (3, 6).
i) Show that the locus of P describes a circle in the x –
y plane. (Ans: x2 + y2 − 2x + 4y − 15 = 0)
ii) Determine the radius and centre of the circle.
(Ans: 2.5495 Units (4 d.p.))
iii) Where does P cut the line x = 3?
(Ans (3, -6) and (3, 2))
10.a) Solve the equation 2(32x) – 5(3x) + 2 = 0
(Ans: x = –0.6309 or 0.6309)
b) The equations of three planes P1, P2 and P3 are 2x –
y + 3z = 3, 3x + y +2z = 7 and
x + 7y – 5z = 13 respectively. Determine where the
three planes intersect. (Ans: (-2, 5, 4))
11. If z is a complex number, describe and illustrate on
the Argand diagram the locus given by each of the
following:
zi
3,
z2
i)
(Ans: x2 y 2 14 y 92 x 35
0;
8
94 , 18 ; r = 0.8385)
x 3
. (Ands: y
3)
3
6
1
12a). Solve sin 3 x 2 cos2 x for 0 ≤ x ≤ 3600.
2
ii) Arg(z + 3) =
(Ans: Hence x = 300, 600, 1200, 1500, 2400 and 3000)
b). Given that in any triangle ABC,
bc
tan B 2C
cot A2 , Solve the triangle with
bc
two sides 5 and 7 and the included angle 45°.
(Ans: a = 4.95, A = 450, B = 89.40 and C = 45.60)
13. A research to investigate the effect of a certain
chemical on a virus infection crops revealed that the
rate at which the virus population is destroyed is
directly proportional to the population at that time.
Initially, the population was P0 at t months later, it
was found to be P.
a) Form a differential equation connecting P and t
dP
KP )
(Ans:
dt
b) Given that the virus population reduced to one
third of the initial population in 4 months, solve
t
the equation in (a) above. (Ans: P Po 3 4 )
c) Find:
i) how long it will take for only 5% of the original
population to remain. (Ans: 10.907 months)
ii) what percentage of the original virus population
will be left after 2 12 months? (Ans: 503268%)
14.
i) Find
x2
( x4 1) dx .
x 1
1
(Ans: ln
tan ( x) C )
4 x 1 2
1
1
ii) Evaluate
0
1
x
dx . (Ans: 0.3905)
1 x
15. A hemispherical bowl of radius a cm is initially full
of water. The water runs out through a small hole at
the bottom of the bowl at a constant rate such that it
empties the bowl in 24s. Given that when the depth
of the bowl is x cm and the volume of the water is
1
3
πx2 (3a – x) cm3, show that the Depth of the water at
that instant is decreasing at a rate a3[36x(2a – x)]-1 cm
s-1. Find how long it will take for the depth of the
water to be at 13 a cm and the rate at which the depth
is decreasing at that instant. (Ans: 20.4 s,
1
3
a cm)
16. a) Find in Cartesian form the equation of the line
passing through the points A(1, 2, 5), B(1, 0, 4) and
C(5, 2, 1).
- 394 -
b) Find the angle between the line x 4 y 2 z 1
8
2
and the plane 4x + 3y – 3z + 1 = 0 (Ans: 69.30)
1999 PAPER TWO
4
SECTION A
1. Given that A and B are mutually exclusive events and
2
1
and P(B) = , find:
3
2
i) P(A B), (Ans: 9/10)
ii) P(A B1) (Ans: 2/5)
iii) P(A1 B1). (Ans: 1/10)
P(A) =
2. Four forces ai + (a – 1)j, 3i + 2aj, 5i – 6j and –i – 2j act
on a particle. The resultant of the forces makes an angle
of 450 with the horizontal. Find the value of a. Hence
determine the magnitude of the resultant force.
(Ans: a = 8, 21.2N)
3. Show by means of a graph that the equation x + logex
= 0.5 has only one real root that lies between 12 and 1.
4. An overloaded taxi travelling at a constant speed of 90
kmh-1 overtakes a stationary traffic police car. Two
seconds later, the police car sets off in pursuit of the
taxi accelerating at a rate of 6 ms-2. How long does the
traffic car travel before catching up with the taxi?
(Ans: 300m)
5. The table below shows the variation of temperature
with time in a certain experiment.
Time(s)
0 120 240 360 480 600
Temperature(0C) 100 80 75 65 56 48
use linear interpolation to find the
i) value of °C corresponding to 400 s, (Ans: 62°C)
ii) time at which the temperature is 770C
(Ans: 192J)
6. A box of mass 4.5 kg rests on a rough horizontal plane
inclined at an angle of 60° to the horizontal. If the
coefficient of friction between the box and the plane is
0.35, determine the force acting parallel to the plane
which will move the box up the plane. (Ans: 50 N)
7. At a bus park, 60% of the buses are of Isuzu make,
25% are styer type and the rest are of Tata make. Of
the Isuzu type, 50% have radios while for the Styer
and Tata make types only 5% and 1% have radios,
respectively. If a bus is selected at random from the
park, determine the probability that:
i) it has a radio. (Ans: 0.0315)
ii) a styer type is selected given that it has a radio
(Ans: 0.0398)
8. Given the variables x and y below,
x
y
80 75 86 60 75 92 86 50 64 75
62 58 60 45 68 68 81 48 50 70
Obtain a rank correlation coefficient between the
variable x and y. comment on you result (Ans: 0.7151)
SECTION B
9.a) The area A of a parallelogram formed by vectors a
and b is given by A = |a| |b| sinθ, where θ is the angle
between the vectors. Find the percentage error made in
the area if |a| and |b| are measured with errors ± 0.05
and the angle with an error of ± 0.50, given that |a| = 2.5
cm, |b | = 3.4 cm and θ = 300. (Ans: 5%)
- 395 -
b) Use the trapezium rule with six sub intervals to
estimate
x sin xdx correct to 3 decimal places.
0
Determine the error in your estimation and suggest
how this error may be reduced. (Ans: 0.073)
10.a) A man buys 10 tickets from a total of 200 tickets in
a lottery. There is only one prize ticket of sh.
10,000.
i) Find the probability that one of the tickets is a prize
ticket. (Ans: 0.0478)
ii) If the price of each ticket is sh.100 and assuming
that all tickets were sold, find the expected loss.
(Ans: 522)
b). A man lives at a point which is 20 minutes’ walk
from the taxi stage. Taxis arrive at the stage
punctually. If the probability density function for
getting a taxi is given by
1 , 0 x 20,
f ( x) 20
0, Else where
Determine the:
i) expected time it takes to wait for a taxi (Ans: 10)
ii) variance of the time it takes to wait for the taxi
(Ans: 33.33)
11. A particle is describing simple Harmonic motion in
a straight line directed towards a fixed point O. When
its distance from O is 3m, its velocity is 25 ms-1 and
its acceleration 75 ms-2. Determine the
i) period and amplitude of oscillation,
2
(Ans:
s and 5.83m)
5
)
10
iii) velocity of the particle as it passes through O
(Ans: 29.15ms–1)
12 (i) Show that the iterative formula for approximating
the root of f(x) = 0 by the Newton Raphson
process for the equation xex + 5x – 10 = 0 is:
ii) time taken by the particle to reach O, (Ans:
x n 1
b) Find the 95% confidence limits for the mean of length
of type A bars of soap. (Ans: 109.12 < μ < 120.88)
14. A rod AB of length 0.6 m long and mass 10 kg is
hinged at A. Its centre of mass is 0.5 m from A. a light
inextensible string attached at B passes over a fixed
smooth pulley 0.8 m above A and supports a mass M
hanging freely. If a mass of 5 kg is attached at B so as
to keep the rod in a horizontal, find the:
i) value of M (Ans: M = 16.7kg)
ii) reaction at the hinge. (Ans: 99.309 N)
15 When a biased tetrahedron is tossed, the probability
that any of its face shows up is proportional to the
number of square of the number on the face that
shows up.
i) Find the probability with which each of the
numbers 1, 2, 3, and 4 on the face of the
tetrahedron appear. (Ans: 8/9)
ii) If three independent tosses of the tetrahedron are
made, what is the probability that the sum of the
numbers on the faces that show up is a 3 or a 5?
(Ans: 0.0028)
D
A
16
1 cm
6 cm
B
2 cm
1 cm
1 cm
8 cm
1 cm
1 cm
C
ABCD is uniform rectangular sheet of cardboard of
length 8 cm and width 6 cm. A square and a circular
hole are cut off from the cardboard as shown above.
Calculate the position of the centre of gravity of the
remaining sheet.
(Ans: 3.944cm from AB and 2.825cm BC)
x n 2 e xn 10
x n e xn e xn 5
(ii) Show that the root of the equation in (i) above lies
between l and 2. Hence find the root of the equation
correct your answer to 2decimal places. (Ans: 1.20)
13. A factory produces two types of bars of soap, A and
B. Their lengths are normally distributed with type A
having average length of 115 cm and standard
deviation 3 cm. Type B has an average length 190 cm
and standard deviation 5 cm
a) Determine the percentage of type
i) A bars of soap that have a length of more than
120 cm (Ans 4.78%)
ii) B bars of soap that have a length of more than
180 cm. (Ans: 97.72%)
- 396 -
b) A point Q is given parametrically by x = 2t,
y 2t 1 . Determine the Cartesian equation of Q
2000 PAPER ONE
SECTION A
4x
)
x
12.a) Show that the equation of the plane through points A
with position vector -2i + 4k perpendicular to the
vector i + 3j – 2k is x + 3y – 2z +10 = 0
b)(i) Show that the vector 2i – 5j + 3.5k is perpendicular
to the line r = 2i – j + λ(4i + 3j + 2k).
and sketch it. (Ans: y
1. Solve the simultaneous equations:
x – 2y + 3z = 6
3x + 4y – z = 3
4x + 6y – 5z = 0 (Ans: x = 2, y = -1/2, z = 1)
2. Solve cos 3 sin 2 using the t-formulae,
(Ans: 2 n
3
)
3. Differentiate x10sinx with respect to x.
(Ans: x10sin x 1x cos x log e 10 )
2
4. Show that log8 x log 4 x . Hence without using
3
tables or calculator, evaluate log8 6 correct to three
decimal places, if log43 = 0.7925.
ii) Calculate the angle between the vector 3i – 2j + k
and the line in b (i) above. (Ans: 66.6°)
13. a) Solve cot2𝜃 = 5(cosec 𝜃 + 1) for 0° ≤ 𝜃 ≤ 360°
(Ans: θ = 9.6°, 170.4°, and 270°)
b) Use tan 2 t to solve 5cos 𝜃 – 2sin = 2;
0 ≤ 𝜃 ≤ 360°. (Ans: θ = 46°, 270°for 0° < θ < 360°)
6x
14. Express f ( x)
into partial fractions.
( x 2)( x 4) 2
cos x
0 1 sin 2 x .
2
5. Evaluate
(Ans: 0.7854)
Hence evaluate
f ( x)dx .
(Ans: ln x 2
1
x 4
6. Show that the line x – 2y + 10 = 0 is a tangent to the
3
4
c)
x4
15. Show that the tangent to the curve 4 – 2x – 2x2 at
x2 y 2
ellipse
1.
64 9
points (-1, 4) and
7. In a culture of bacteria, the rate of growth is
proportional to the population present at time, t. The
population doubles every day. Given that the initial
population P0, is one million, determine the day when
the population will be 100 million. (Ans: 7th day)
8. Show that the equation of the line through the points
(1, 2, 1) and (4, -2, 2) is given as
x 1 y 2
z 1
4
3
SECTION B
9. a) The nth term of a series is Un = a3n + bn + c. given
that U1 = 4, U2 = 13 and U3 = 46, find the values of
a, b and c.
(Ans: a = 2, b = -3, c = 1)
b) If and β are the roots of the equation x2 – px + q =
3
3
0, find the equation whose roots are 1 and 1 .
(Ans: qx2 – (p2q – 2q2 – p)x + (q3 – p3 + 3pq) + 1 = 0
10. a) Prove by induction that 2n + 32n – 3 is always
divisible by 7 for n 2.
the point
1
4
12 ,
2 12 respectively passes through
, 5 12 . Calculate the area of the curve
enclosed between the curve and the x-axis.
(Ans: 9 sq units)
16. An inverted cone with vertical angle of 600 is
collecting water leaking from a tap at a rate of 0.2cm3s-1.
If the height of water collected in the cone is 10cm, find
the rate at which the surface area of water is increasing.
(Ans: 0.12 cm2s-1)
b) Given that y e
tan x
, show that
d2y
dy
(2 tan x sec2 x)
0.
2
dx
dx
1
x 2
b) Expand 1 as far as the term in x2. Hence
3
evaluate 8 , correct to three decimal places
(Ans: ; 2.829)
11.a) A point P is twice as far from the line x + y = 5 as
from the point (3, 0). Find the locus of P.
(Ans: 7x2 + 7y2 – 38x + 10y + 47 = 0)
- 397 -
2000 PAPER TWO
SECTION A
1. A family plans to have 3 children.
i) Write down the possible sample space and construct
its probability distribution table.
(Ans: {BBB, BBG, BGB, BGG, GBB, GBG, GGB}
ii) Given that X is the number of boys in family, find
the expected number of boys. (Ans: 1.5)
b) If the error in each of the values of ex and e-x is
0.0005, find the maximum and minimum values of the
quotient ex/e-x, when x = 0.04, giving your answer
correct to 3 decimal places. (Ans: 0.694)
10. An interval bisection Algorithm that computes and
prints the approximate value of the root, r of the
equation f(x) = 0, in the interval [a, b], correct to 3
decimal places is given in the flow chart below:
2. By the method of linear interpolation, use the table
below to find the value of
i) ln(1.66) (correct to 3 decimal places) (Ans: 0.606)
ii) x corresponding to ln(x) = 0.4000. (Ans: 1.492)
x
1.4
1.5
1.6
1.7
lnx 0.3365 0.4055 0.4700 0.5306
3. Two balls are randomly drawn without replacement
from a bag containing 10 white and 6 red balls.
Find the probability that the second ball drawn is
i) red given that the first one was white. (Ans: 0.4)
ii) white
(Ans: 0.375)
4. A boat travelling at 5ms –1 in the direction 0300 in
still water is blown by wind moving at 8ms-1 from
the bearing of 1500 . Calculate the speed and the
course the boat will be steered. (Ans: 7 ms-1)
1
5. Estimate the value of
dx
1 x
2
0
8. A particle executing simple harmonic motion about
point O has a velocity of 33ms-1 and 3ms-1 when at
distances of 1m and 0.268m respectively, from the end
point. Find the amplitude of the motion. (Ans: 2m)
SECTION B
9. Given that x = 2.5, y = 14.2 and z = 8.1, all the values
given correct to one decimal place, find the maximum
value of
x y
,
z2
ii)
Read a, b
r
1 1 1
x y
iii)
, correct to 3
x y z
z
decimal places. (Ans: 0.259; -1.441; 0.356)
a+b
2
is
a - b < 0 .5 × 1 0 -3
YES
?
NO
b=r
by the trapezium rule
using five sub-intervals. (Give answer correct to 3
decimal places).
(Ans: 0.784)
6. On a certain farm, 20% of the cows are infected by a
tick disease. If a random sample of 50 cows is selected
from the farm, find the probability that not more that
10% of the cows are infected. (Ans: 0.0558)
7. A force acting on a particle of mass 15 kg moves it
along a straight line with a velocity of 10ms-1. The rate
at which work is done by the force is 50 watts. If the
particle starts from rest, determine the time it takes to
move a distance of 100m.
(Ans: 10 6 )
i)
START
Print r
STOP
YES Is
f(r)< 0is r < 0 ?
NO
a=r
By determining f(x) and locating the approximate
interval [a, b], perform a dry run for the flow chart to
determine 3
1
, correct to 3 decimal places. Tabulate
3
values of a, b and r at each stage. (Ans: 0.694)
11. A particle moving with an acceleration given by a =
4e-3ti + 12sintj – 7costk is located at the point (5, -6, 2)
and has velocity, v = 11i – 8j + 3k at time t = 0. Find
the
i. Magnitude of the acceleration when t = 0.
(Ans: 8.0623)
ii. Velocity at any time, t,
(Ans: 13 (37 4e3t )i (4 12cos t ) j (3 7sin t )k )
iii. Displacement at any time, t.
ue 3t
41 37
t
i (6 4t 12sin t ) j
(Ans: r
3
a
a
(5 3t 7cos t) k )
12. At 7:30am daily, a bus leaves Kampala for Masaka.
The time (min) taken to cover the journey were
recorded over a certain period of time and were
grouped as shown in the table below:
Time(min) 80-84 85-89 90-94 95-99 100-104
Freq.(f)
10
15
35
40
28
- 398 -
105-109
110-114
115-119
120-124
15
4
2
1
a. Calculate the mean time of travel from Kampala to
Masaka by the bus. (Ans: 96.6 minutes)
b. Draw a cumulative frequency curve for the data. Use
your curve to estimate the :
i. Median time for the journey, (Ans: 96.5)
ii. Number of times the bus arrived in Masaka
between 9:00 – 9: 25 am, (Ans: 119)
iii. Semi-interquartile range of time of travel from
Kampala to Masaka. (Ans: 5 minutes)
13. The diagram below shows a uniform rod AB of weight
W and length l resting at an angle against a smooth
vertical wall at A. The other end B rests at a smooth
horizontal table. The rod is prevented from slipping by
an inelastic string OC, C being a point on AB such that
AC is perpendicular to AB and O on the point of
intersection of the wall and the table. Angle AOB is 900
A
θ
C
W
B
O
W
tan 2 )
2
ii) reactions at A and B in terms of and W
W tan 2
)
(Ans:
4
14. a) A random variable X takes on the values of the
interval 0 < x < 2 and has a probability density
function given by
Find the i): tension in the string (Ans:
f(x)
1
Determine the expression for f(x). Hence obtain the
i. expression for the cumulative probability density
function
1
1 2 x, 0 x 2
(Ans: f ( x)
, elsewhere
0
ii. mean and variance of X
, x0
0
1 2
(Ans: f ( x) x 5 x , 0 x 2 )
1
,x 2
15. A sugar factory sells sugar in bags of mean weight
50kg and standard deviation 2.5kg. Given that the
weight of the bags are normally distributed, find the
i) Probability that the weight of any bag of sugar
randomly selected lies between 51.5 and 53kg.
(Ans: 0.1592)
ii) Percentage of bags whose weight exceeds 54kg.
(Ans: 5.48%)
iii) Number of bags that will be rejected out of 1000
bags purchased for weighing below 45.0kg.
(Ans: 23)
16. Six forces, 9N, 5N, 7N, 3N, 1N and 4N act along the
sides PQ, QR, RS, ST, TU and UP of a rectangular
hexagon of side 2m, their direction being indicated by
the order of letters. Taking PQ as the reference axis,
express each of the forces in vector form. Hence find
the
i) magnitude and direction of the resultant of the
forces.
(Ans: 8.9 N, 43° with PQ)
ii) distance from P, where the line of action of the
resultant cuts PQ. (Ans: 7.43m from P)
y
T
0 x 1
a
a
f ( x) 2 x 1 12 x 2
2
ele where
0
3N
1
2
16
Find
i) value of a (Ans:
)
25
ii) P(x <16) (Ans: 0.9744)
b) the probability density function f(x) of the random
variable X takes on the form shown in the diagram
below
x
2
1
S
1N
U
7N
600
600
R
5N
6N
600
600
P
9N
Q
x
- 399 -
2001 PAPER ONE
SECTION A
1. Solve the simultaneous equations:
x2 − 10x + y2 = 25
y−x=1
(Ans: x = 2, y = 7; x = -2, y = -1)
y
2. If y = √x, show that
x
deduce
1
x x
. Hence
c. Solve (z + 2z*)z = 5 + 2z where z* is the complex
conjugate of z. (Ans: Z* = 1 + 2i )
10. It can be proved by induction that for all positive n,
13 23 33 .... n3
From this result, deduce that
n 13 n 23 .... 2n 3
3. Given that sin2θ = cos3θ, find the value of
sinθ, 0 ≤ θ ≤ π. (Ans: 0.309)
b) Use Maclaurines theorem to expand ln(1 + ax),
where a is a constant. Hence or otherwise expand
4. Find the points of intersection of the line
x y 2 z 1
with the plane
5
2
4
3x + 4y + 2z − 25 = 0 (Ans: (5, 0, 5))
(1 x)
ln
(1 2 x)
5. A cylinder is inscribed in a semi-hemisphere of radius
r as shown in the figure below
r
Find the maximum volume of the cylinder in terms of
2 r 3
)
r
(Ans:
3 3
6. Expand (1+ x)-2 in descending powers of x including
the term x−4. If x = 9, Find the percentage error in
using the first two terms of the expression.
(Ans: 3.978%)
7. Find the locus of the point which is equidistant from
the line x = 2 and the circle x2 + y2 = 1. Illustrate this
with a sketch (Ans: y2 + 6x – 9 = 0)
y
, given x =
2x 1
4
when y = 6. Hence determine the value of x when
y = 10
(Ans: x = 12, y2 = 8x + 4)
SECTION B
9.a) Use De Moivres’ theorem or otherwise to simplify
(cos i sin )(cos 2 i sin 2 )
cos / 2 i sin / 2
(Ans: cos( 52 ) i sin( 52 ) )
b. Express
i
in modulus-argument form
4 6i
(Ans: Z = 0.1387[cos(0.187π) + isin(0.87π)])
up to the term in x3. For what values
of x is the expansion valid?
x2 5x2
... ; x 12 or | x | 12 )
(Ans: 2 x
2
3
12. a)(i) Find the equation of the chord through the points
(at12, 2at1) and (at22, 2at2) of the parabola y2 = 4ax
(Ans: (t1 + t2)y – 2x – 2at1t2 = 0 )
ii) Show that the chord cuts the directrix when
y
dx
1 2
n 3n 1 5n 3
4
b) A man deposits sh. 800, 000 into his savings account
on which interest is 15% per annum. If he makes no
withdrawals, after how many years will his balance
exceed sh. 8 million? (Ans: 16.5 years)
11 a) Using calculus of small increments or otherwise, find
98 correct to one decimal place. (Ans: 9.9)
x
dy
.
dx
8. Solve the differential equation dy
1 2
2
n n 1 .
4
2a (t2t1 1)
t1 t2
b) Find the equation of the normal to the parabola
y2 = 4ax at (at2, 2at) and determine its point of
intersection with the directrix (Ans: (a1at(t2 + 3))
x y
x y
2sin y
13a) Show that tan
tan 2 cos x cos y
2
b) Find in radians the solution of the equation
cos x + sin 2x = cos 3x, for 0 ≤ x ≤ 2π
3
(Ans: x 0, , , , 2 )
2
2
14(a) Find the Cartesian equation of the plane through
A(0, 3, -4), B(2, -1, 2) and C(7, 4, -1). Show that
Q(10, 13, -10) lies in the same plane.
b) Express the equation of the plane in (a) in the scalar
3
product form. (Ans: r 6 2 )
5
c) Find the area of ABC in (a)(Ans: 25.0998 sq. unts)
15. i) Find the Cartesian equation of the curve given
parametrically by:
x
( x 1) 2
1 t
, y
1 t
x 1
(Ans: y
( x 1) 2
)
x 1
(ii) Sketch the curve
- 400 -
(iii) Find the area enclosed between the curve and
the line y = 1. (Ans: 1.9548 sq units)
2x
16) Integrate
with respect to x
x2 4
(Ans: 2( x2 4) c )
b) Evaluate
6
sin x sin 3x.dx (Ans: 0.1083)
0
c) Using the substitution x = 3 sin θ, evaluate
3
0
3 x
3 x dx (Ans: 7.7124)
2001 PAPER TWO
SECTION A
1. The events A and B are neither independent nor
mutually exclusive. Given that P(B) = 1/3, P(A)= ½
and P(A B1) = 1/3, Find:
i)
P(A1 B1),
(Ans: 5/6)
ii) P(A1/B1),
(Ans: ½)
2. In an experiment to measure the rate of cooling of an
object, the following temperatures, (θ0 C) against
time T were recorded
Temperature, θ0C
80 70.2 65.8 61.9 54.2
Time, T(s)
0
10
15
20
30
Use linear interpolation to find
(i) The value of θ when T = 18s. (Ans: 63.5°C)
(ii) T when θ = 600
(Ans: 22.5 seconds)
3. If x = 4.5, y = 2.54 and z = 26.4 all measured to the
nearest number of decimal places of x, y and z
respectively. Find the range within which the exact
value of the expression x
y
lies. (Ans:
xz
(4.42830, 4.52894))
4. A coin is biased so that a head is twice as likely to
occur as a tail. If the coin is tossed 15 times, determine
the,
i) expected number of heads (Ans: 10)
ii) probability of getting at most 2 tails (Ans: 0.0793)
5. A particle of mass 5 kg is placed on a smooth plane
1
to the horizontal. Find the
inclined at tan 1
3
magnitude of the force acting horizontally, required to
keep the particle in equilibrium and the normal
reaction to the plane. (Ans: 56.5803N)
6. A physics student measured the time required in
seconds for A trolley to run down slopes of varying
gradients and obtained the following results: 32.5,
34.5, 33.5, 29.3, 30.9, 31.8. Calculate the mean time
and standard deviation. (Ans: 32.5333, 2.064)
7. A, B and C are points on a straight road such that AB =
BC = 20 m. A cyclist moving with uniform
acceleration passes A and then notices that it takes him
10 seconds and 15 seconds to travel between A and B ,
and A and C respectively.
Find:
i) his acceleration (Ans: 0.27 ms-2)
ii) the velocity with which he passes A.
(Ans: 0.65 ms-1)
8. An inextensible string attached to two scale pans A
and B, each of weight 20gm, passes over a smooth
fixed pulley. Particles of weight 3.8N and 5.8N are
placed in pans A and B respectively. Find the reaction
- 401 -
of the scale pan holding the 3.8N weight, if the system
is released from rest. [Take g = 10 ms–2].
a) Calculate the approximate mean and modal weights of
the patients. (Ans: 38.8333 kg)
(Ans: R = 4.56 N)
b) Plot an ogive for the above data. Use the ogive to
estimate the
i) median and semi interquartile range for the
weights of the patients, (Ans: 41.5 kg, 14.75 kg)
ii) the probability that patients weighing between
13kg and 52.5kg visited the health unit.
(Ans: 43/75)
13. An object of mass 5kg is initially at rest at appoint
whose position vector is -2i + j. If it is acted upon by a
force, F = 2i + 3j − 4k, find
i) the acceleration. (Ans: 15 (2i + 3j – 4k))
SECTION B
9. a) i) Round off 6.00213 (Ans: 6.00)
ii) Truncate 5415000, (Ans: 5410000)
to 3 significant figures. (2 marks)
b) Use the trapezium rule with eight su-intervals to
4
estimate
10
2 x 1 dx Correct to 4 decimal places.
2
Calculate the percentage error in your result. How
may this error be reduced?
(Ans: 2.9418, 0.098%)
10. Bag A contains 2 green and 2 blue balls, while bag B
contains 2 green and 3 blue balls. A bag is selected at
random and 2 balls drawn from it without
replacement. Find the probability that the balls are of
different colours. (Ans: 19/30)
b) A fair die is drawn 6 times. Calculate the
probability that
i) a 2 or 4 appears on the first throw, (Ans: 1/3)
ii) four 5s will appear in the 6 throws.
(Ans: 0.008)
11. Given two iterative formulae I and II (shown below)
for calculating the positive root of the quadratic
equation f(x) = 0
I
xn 1
II
1 2
xn 1
2
xn 1
1 x 2 1
n
2 xn 1
For n = 1, 2, 3...
Taking x0 = 2.5, use each formula thrice to two
decimal places to decide which is the more suitable
formula. Give a reason for your answer.
1 x 2 1
(Ans: xn 1 n
)
2 xn 1
b) If α is an approximate root of the equation x2 = n, show
that the iterative formula for finding the root reduces to
2
, Hence, taking α = 4, estimate 17 correct to 3
2
ii) the velocity after 3 s. (Ans: 15 (6i + 9j – 12k))
iii) its distance from the origin after 3 s.
(Ans: 5.166 m)
14. (a) Amass oscillates with S.H.M of period one second.
The amplitude of the oscillation is 5cm. Given that the
particle begins from the centre of the motion, state the
relationship between the displacement x of the mass at
any time t. Hence find the first times when the mass is
3cm from its end position. (Ans: (0.066, 0.434))
b) A particle of mass m is attached by means of light
strings AP and BP of the same natural length a m and
moduli of elasticity mg and 2mg N respectively, to the
points A and B on a smooth horizontal table. The
particle is released from the mid –point of
AB ,
where AB = 3am.
4 2 a 12
T
3g
15. A continuous random variable X is defined by the
p.d.f.
1
k x , 0 x 3
f ( x)
a
0, Else where
Given that P(x >1) = 0.8, find the:
i) value of a and k (Ans: a = -1, k = 2/15)
ii) probability that X lies between 0.5 and 2.5
(Ans: 0.6667 (4 dp))
iii) mean of X. (Ans: 1.8)
decimal places (Ans: 4.123)
12. The table below shows the weights to the nearest kg 16. (a) Prove that the centre of mass of a solid cone is ¼ of
the vertical height from the base.
of 150 patients who visited a certain health unit
(b)
The figure ABCDE below shows a solid cone of radius
during a certain week:
r,
height (h), joined to a solid cylinder of the same
Weight(kg)
No. of patients
material with the same radius and height H.
0 − 19
30
(Ans: 32.01°)
20 – 29
16
30 – 39
24
40 – 49
32
50 – 59
28
60 – 69
12
70 - 79
8
- 402 -
difference and the first term. Hence find the sum of the
first 60 terms. (Ans: d = 3, a = 2, 5430)
b) A cable 10 m long is divided into ten pieces whose
lengths are in a geometric progression. The length of
the longest piece is 8 times the length of the shortest
piece. Calculate to the nearest centimeter the length of
the third piece. (Ans: 45 cm)
E
h
D
r
r
A
H
C
B
If the centre of mass of the whole solid lies in the
plane of the cone where the two solids are joined,
find H. If instead H = h and r = ½h, find the angle
AB makes with the horizontal, if the body is hanged
from A.
2002 PAPER ONE
SECTION A
1. Solve the equation 2cos θ – cosec θ = 0; 00 < θ < 2700.
(Ans: θ = 45°, 225°)
2. The vertices of a triangle are P(2, -1, 5), Q(7, 1, -3)
0
and R(12, -2, 0). Show that PQR 90 . Find the
coordinates of S if PQRS is a rectangle.
(Ans: (8, -4, 8))
3. Show that 2 + i is a root of the equation 2z3 - 9z2 +
14z – 5 = 0. Hence find the other roots. (Ans: 2 – i, ½)
4. The point R(2, 0) and P(3, 0) lie on the x-axis and
Q(0, -y) lies on the y-axis. The perpendicular from the
origin to RQ meets PQ at point S(X, -Y). Determine
the locus of S in terms of X and Y.
(Ans: 2x2 + 3y2 – 6x = 0)
2
5. If y = √(5x2 + 3), show that y
d 2 y dy
5.
dx 2 dx
6. Given that log3 x = p and log18 x = q, show that
q
.
pq
log6 3 =
1
1
1 2
7. Given y ln1 , 2u x , show that
x
u
x 1 .
dy
dx x 1x 2 1
1
8. Evaluate
x3
0 x 2 1dx. (Ans: 0.15345)
SECTION B
9.(a) The tenth term of an arithmetic progression (A.P) is 29
and the fifteenth term is 44. Find the value of the common
10. P is a variable point given by the parametric
b
equations x a t 1 ; y t 1 . Show that the
t
t
2
2
x2 y2
locus of P is 2 2 1 .
a
b
State the asymptotes. Determine the coordinates of the
point where the tangent from P meets the asymptotes.
bx
bx
(Ans: y
; y
; (at, bt) and a t , b t )
a
a
11(a) Find the equation of the perpendicular line from
2
1
2
point A 1 onto the line r 0 1 . What is
4
2
2
the distance from A to r?
2
4 9
(Ans: p 1 14 9 ; 1.795 units)
4
8 9
b). Find the angle contained between the line OR and the
2
x – y plane, where OR 1 (Ans: 41.81°)
2
sin 3 sin 6 sin sin 2
tan 5 .
sin 3 cos 6 sin cos 2
(b) Express 4cosθ – 5sinθ in the form R cos (θ + β),
where R is a constant and β an acute angle.
i) Determine the maximum value of the expression
and the value of θ for which it occurs
(Ans: 6.403, when θ = -51.3° or 308.7°)
ii) Solve the equation 4 cos θ – 5 sin θ = 2.2, for
00 < θ <3600. When θ = –51.3° OR 308.7°
(Ans: θ = 18.6° or 238.3°)
13.a)Find the equation whose roots are –1 ± i, where
12(a) Show that
(Ans: Z2 + 2Z + 2 = 0)
i = 1 .
b) Find the sum of the first 10 terms of the series 1 + 2i –
4 – 8i + 16 +…., in the form a + bi where a and b are
constants and i = 1 . (Ans: 205 + 410i)
c) Prove by induction that
(cos θ + i sin θ)n = cos nθ + i sin nθ.
- 403 -
d
0 3 cos .
b) Integrate the following with respect to x:
i)
ln x
ii)
x2 sin 2x
(Ans: 0.6755); x(lnx – 1) + c);
x2
x
1
cos 2 x sin 2 x cos 2 x c )
2
2
4
x( x 3)
15. Given the curve y
,
( x 1)( x 4)
14. (a)Use t tan to evaluate
2
2
i) Show that the curve does not have turning points
ii) Find the equations of the asymptotes. Hence
sketch the curve. (Ans: y = 1, x = 1, x = 4)
16(i). The volume of a water reservoir is generated by
rotating the curve y = kx2 about the y–axis. Show that
when the central depth of the water in the reservoir is h
meters, the surface area, A is proportional to h and the
volume v is proportional to h2.
ii) If the rate of loss of water from the reservoir due to
evaporation is λA m2 per day, obtain a differential
equation for h after t days. Hence deduce that the depth
of water decreases at a constant rate.
iii) Given that λ = ½, determine how long it will take for
the depth of water to decrease from 20 m to 2 m.
(Ans: 36 days)
2002 PAPER TWO
SECTION A
1. On a certain day, fresh fish from lakes: Kyoga,
Victoria, Albert and George were supplied to one of the
central markets of Kampala in the ratios 30%, 40%, 20%
and 10% respectively. Each lake had an estimated ratio
of poisoned fish of 2%, 3% and 1% respectively. If a
health inspector picked a fish at random,
i. What is the probability that the fish was poisoned?
(Ans: 0.025)
ii. Given that the fish was poisoned, what is the
probability that it was from Lake Albert?
(Ans: 0.24)
2. The table below shows how y varies with x in an
experiment at different points
x
y
-1.0
-1.0
-0.5
-2.2
-1.4
-3.7
Use linear interpolation or extrapolation to find
a) y when x = 0.5 (Ans: y = -4.6)
b) x when y = -4.5 (Ans: x = 0.456)
3. A driver of a car travelling at 72 kmh-1 notices a tree
which has fallen across the road, 800 m ahead and
suddenly reduces the speed to 36 kmh-1 by applying the
brakes. For how long did the driver apply the brakes?
(Ans: 53.33 s (2 dp))
4. The chance that a person picked from a Kampala street
is 30 in every 48. The probability that that person is a
university graduate given that he is employed is 0.6.
Find the
a) probability that the person picked at random from
the street is a university graduate and is employed
(Ans: 0.375)
b) number of people that are not university graduates
and are employed from a group of 120 people.
(Ans: 30)
5.
Use a suitable table of values to show that the
function x x 3
8
has two real roots in the interval
x
(-3, 3), Hence use linear interpolation to determine the
approximate value of the negative real roots of the
function giving your answer correct to 1 decimal
place. (Ans: -1.7)
6. The resistance to the motion of a lorry of mass m kg is
1/200 of its weight. When travelling at 108kmh1 on a
level road and ascends a hill its engine fails to work.
Find how far up the hill (in km) the lorry moves before it
comes to rest. Give your answer close to one decimal
90000
)
place (Ans: S
g (1 200sin )
7. The table below shows the cumulative distribution of
the age (in years) of 400 students of a girls’ school
Age(in years)
Cumulative frequency
<12
0
<13
27
<14
85
<15
215
<16
320
<17
370
<18
395
<19
400
Plot an ogive for the data and use it to estimate the:
a)
median age (Ans: 14.9)
b)
20th and 80th percentile age range
(Ans: 2.1)
8. A particle moves in the x – y plane such that its
position vector at any time t is given by
r = (3t2 − 1)i + (4t3 + t − 1)j
Find:
i) its speed, (Ans:
6t 2 (12t 2 1) 2 ; = 48.3735)
ii) the magnitude of acceleration after time t = 2
- 404 -
SECTION B
9. Given below are points of a flow chart not arranged in
order.
READ:A
C
START
IS
B≤0?
C=B+2
B=A−5
STOP
PRINT:C
IS
DIVISIBLE
BY 2
Time(min)
No. of
students
A← A + 1
Rearrange them and draw a complete logical flow chart.
(a) State the purpose of the flow chart
(b) Perform a dry run of your rearranged flow chart
by copying and completing the table below
10. A pair of dice is tossed 180 times. Determine the
probability that a sum of 7 appears
i) Exactly 40 times (Ans: 0.0108)
ii) Between 25 and 35 inclusive times
(Ans: 0.7286)
11. A random variable X has the probability distribution
function
1
,a x b
f X b a
,
0, Else where.
Show that the variance of X is
angles of 500 and 600 with the rod. Calculate the tensions
in the strings. (Ans: 46.92N and 60.34 N)
13. Show graphically that there is only one positive real
root of the equation x3 +2x−2 = 0.
Using the Newton Raphson formula thrice estimate
the root of the equation, give your answer correct to 2
decimal points. (Ans: 0.77 (2 dp))
14. (a) The times taken by a group of students to solve a
mathematical problem are given below
(b a ) 2
.
12
b) During rush hours, it was observed that the number of
vehicles departing for Entebbe from Kampala old taxi
park take on a random variable X with a uniform
distribution over the interval [x1, x2]. If in 1 hour, the
expected number of vehicles leaving the stage is 12,
with variance of 3, calculate the:
i) values of x1 and x2, (Ans: x1 = 9, x2 = 15)
ii) probability that at least 11 vehicles leave the stage
(Ans: 2/3)
12(a) A particle of mass 3Kg is attached to the lower end
B of an inextensible string. The upper end A of the
string is fixed to a point on the ceiling of a roof. A
horizontal force of 22N and an upward vertical force of
4.9 N act upon the particle making it to be in
equilibrium, with the string making an angle with the
vertical. Find the value of and the tension in the string.
(Ans: 33N, α = 45°)
(b) A non – uniform rod of mass 9Kg rests horizontally
in equilibrium supported by two light inextensible
strings tied to the ends of the rod. The strings make
5-9
5
10-14 15-19 20-24 25-29 30-34
14
30
17
11
3
a) Draw a histogram for the data. Use it to estimate the
modal time for solving a problem
b) Calculate a mean time and standard deviation of
solving a problem
(Ans: 18.5 minutes; 5.9896 minutes)
15 (a) The velocities of two ships P and Q are i + 6j and
–i + 3j kmh-1 respectively.
At a certain instant the displacement between the two
ships is 7i +4j km
Find the:
(i) relative velocity of ship P to Q. (Ans: 2i + 3j)
(ii) magnitude of displacement between ships P and Q
after 2hours. (Ans: 14.87 km)
(b) The position vector of two particles are:r1 = (4i −2j)t + (3i + j)t2 and
r2 = 10i + 4j + (5i − 2j)t respectively .
Show that the two particles will collide. Find their
speeds at the time of collision. (Ans: 5.3852 units)
16. A particle is projected from level ground towards a
vertical pole, 4m high and 30m away from the point
of projection. It just passes the pole in one second.
Find:
a) its initial speed and angle of projection.
(Ans: 31.2933; α = 16.5°)
b) the distance beyond the pole where the particle will
fall. (Ans: 24.42m)
- 405 -
2003 PAPER ONE
SECTION A
1. Show that z = 1 is a root of the equation
z3 – 5z2 + 9z – 5 = 0. Hence solve the equation for the
other roots.
2. Given the position vectors OA = (3, -2, 5), and OB = (9,
1, -1), find the position vector of point C such that C
intersects at O. taking O as the origin, use the dot
product to prove that AO is perpendicular to BC
b) Prove that ABC 90 given that A is (0, 5, -3),
B(2, 3, -4) and C(1, -1, 2). Find the coordinates of D
if ABCD is a rectangle. (Ans: D(-1, 1, 3))
12(a) . Use De Moivres theorem to express tan 5θ in
terms of tan θ
3
5
)
(Ans: tan 5 5 tan 102tan tan
4
1 10 tan 5 tan
divides AB internally in the ratio 5: -3
(Ans: 18i
11
2
j 10k )
3. Solve the equation
cos 2θ + cos 3θ + cos θ = 0; 00 ≤ θ ≤ 1800.
(Ans: θ = 45°, 120°, and 135°)
2
2
x
x
c)
ln x
2
4
5. Solve for x in the equation log4(6 − x) = log2x
(Ans: x = 2)
4. Find
x ln xdx.
(Ans:
6. If y et cos(t ) , show that
d2y
dx
2
2dy
2y 0
dx
7. The points A (2, 1), P (α, β) and point B(1, 2) lie in
the same plane. PA, meets the x-axis at the point (h,
0) and PB meets the y-axis at the point (0, k). Find h
and k in terms of α and β.
(Ans: n = 2 , k = 2 )
1
1
d
x
ln
8. Determine
dx
(1 x 2 )
when x = 2. (Ans: 1/10)
9. (a) Show that cot A + tan 2A = cot A sec 2A
3t t 3
1 3t 2
1i 3
)
2
13. Determine the nature of the turning points of the
(Ans: Z = -1, Z
curve y
1
2
i 3
2
, Z
x2 6x 5
. Sketch the graph of the curve
(2 x 1)
for x = -2 to x = 7. State any asymptotes.
(Ans: (-1, -4) maximum; (12, -1) minimum)
14. A conic section is given by x = 4 cos θ; y = 3 sin θ.
Show that the conic section is an ellipse and
determine its eccentricity.
b) Given that the line y = mx + c is a tangent to the
ellipse
x2
a2
y2
b2
1 , show that c2 = a2m2 + b2. Hence
determine the equations of the tangents at the point
(-3, 3) to the ellipse
x2 y 2
1.
16 9
(Ans: y = 3 and y
15(a). Find
x e
3 x
4
dx. (Ans:
1
4
18
7
x
75
7
)
4
ex c )
b) Use the substitution t = tan x to find
1
1
tan 1 ( 2 tan x) c )
dx (Ans:
2
1 sin x
2
16.a) Solve the differential equation
, where t = tan θ
10(a) . Given the inequalities y > x − 5 and 0 y
6
,
x
illustrate graphically by shading out the unwanted
regions.
b) Solve the simultaneous equations
xy + 2x = 5
9x = y + 6.
Illustrate your solutions on a graph.
5
9
dR
e2t t , given
dt
that R(0) = 3 (Ans: R 12 e2t 12 x 2
2 3
Hence or otherwise show that tan 12
(Ans: x = 1, x =
b) Solve the equation z + 1 = 0
SECTION B
(b) Show that tan 3θ =
3
5
2
)
(b) The acceleration of a particle after time t seconds
is given by a = 5 + cos½t. If initially the particle is
moving at 1ms-1, find its velocity after 2π seconds
and the distance it would have covered by then.
(Ans: v = 10𝜋 + 1; d = 10π2 + 2π + 4)
; y = -1, y = 3)
11(a). In a triangle ABC, the altitudes from B and C meet
the opposite sides at E and F respectively. BE and CF
- 406 -
SECTION B
2003 PAPER TWO
9. (i) Show that the equation x = 1n(8 – x ) has a root
between 1 and 2.
(ii) Use Newton Raphson method to find the
approximate root of x = 1n (8 – x). Correct to 3
decimal places. (Ans: 1.821 (3 dp))
SECTION A
1. Events A and B are such that P(A) = ½,
P(B) = 3/8 and P(A/B) = 7/12, find
i) P(A∩B) (Ans:7/32 )
10. Given the cumulative distribution function,
ii) P(B/ A ) (Ans: 5/16)
2. Two decimal numbers x and y are recorded off to give
X and Y with the errors E1 and E2 respectively.
Show that the maximum relative error recorded in
approximating x2y by X2Y is given by: 2
E1
E
2
X
Y
3. ABCD is a square of side a. Forces of magnitude 2N,
1N, 2 N and 4N act along AB, BC, AC and DA
respectively. The directions being in the order of
letters. Find the magnitude and direction of the
resultant force. (Ans: 3.6056N, 33.69°)
4. In an examination, scaling is done such that candidate
A who had originally scored 35% gets 50% and
candidate B with 40% gets 65%. Determine the
original mark for candidate C whose new mark is 80%
(Ans: 45%)
1
5. Find the approximate value of
1
x 2 1dx
using five
x2 1
x,1 x 2;
2
F x
x2
3x , 2 x 3
2
x 3;
1,
a) Find:
x 1 : 1 x 2
i) the p.d.f. (Ans: f ( x) 3 x : 2 x 3 )
0
: elsewhere
ii) P(1.2 < x < 2.4 (Ans: 0.8)
iii) the mean of x (Ans: 2)
b) Sketch f(x)
11. Blocks A and B of masses 2 and 3 kg respectively
are connected by a light inextensible string passing
over a smooth pulley as shown below
0
sub-intervals. (Ans: 0.784)
A
6. A spring AB of natural length 1.5m and modulus N is
fixed at A and hangs in a vertical position. The other
end is joined to a second spring BC of natural length
o
B
30
1m and modulus 2N. A particle of weight 15 N is
then attached to end C of the second spring. When the
Block A is resting on a rough plane inclined at at 30 to
system is hanging freely in equilibrium the distance
the horizontal while block B hangs freely. When the
25
)
AC is 4m. Find the value of . (Ans:
system is released from rest, block B travels a distance
3
of 0.75m before it attains a speed of 2.25ms-1.
7. The table below shows the marks scored in a
mathematics examination by students in a certain
Calculate the
school
(i) acceleration of the blocks (Ans: 3.375 ms-1)
Marks
Number students
(ii) coefficient of friction between the plane and block A,
30 – 39
12
(Ans: 0.16)
40 – 49
16
(iii) reaction of the pulley on the string. (Ans: 33.4N)
50 – 59
14
(Ans: 33.4N)
60 – 69
10
12. i) Determine the iterative formula for finding the fourth
70 – 79
8
root of a given number N.
80 – 89
4
(Ans: 3 xn N 3 for n = 0, 1, 2, … )
Draw a histogram and use it to estimate the mode.
4
3xn
Calculate the mean score. (Ans: 46; mean, 54.1875)
ii) Draw a floor chart that reads N and the initial
8. A train starts from station A with a uniform
approximation, X0, computes and prints the fourth
acceleration of 0.2 ms–2 for 2 minutes and attains a
root of N correct to 3decimal places and N.
maximum speed and moves uniformly for 15 minutes.
(Ans: x = 3.500)
It is then brought to rest at constant retardation of 5/3
iii)
Perform
a
dry
run
for
N
=150.10
and
x
0 =3.200.
ms–2 at station B. Find the distance between stations A
13. In a school of 800 students their average weight is
and B. (Ans:23212.8 m )
54.5 kg and standard deviation 6.8 kg. If the weights
of all the students in the school assume a normal
distribution, find the
- 407 -
i)
probability that a student picked at random
weighs 52.8 or less kg, (Ans: 0.4013)
number of students who weigh over 75 kg
(Ans: 0.8)
weight range of the middle 56% of the students of
the school (Ans: 49.2504 < x < 59.7496 kg)
ii)
iii)
14. A particle of weight 24N is suspended by a light
inextensible string from a light ring. The ring can slide
along a rough horizontal rod. The coefficient of
friction between the rod and the ring is ⅓. A force of P
Newtons acting upwards on a particle at 45 to the
horizontal, keeps the system in equilibrium with the
ring at a point of sliding. Find the:
i)
value of P
(Ans: 6 2 N)
ii) tension in the string (Ans: 6 10 N)
15. (a) The table below shows the percentage sand, y in
the soil at different depths, x, (in cm);
Soil depth(x) in cm 35 65 55 25
% of sand, y
86 70 84 92
45 75 20 90 51 60
79 68 96 58 86 77
a) i) Plot a scatter diagram for the data.
Comment on the relationship between the depth of
the soil and the percentage of sand in the soil
ii) Draw a line of best fit through the points of the
scatter diagram. Use your result to estimate the
- percentage of the sand in the soil at depth of
31 cm.
- depth of the soil with 54 % sand
(Ans: 92%; 96 cm)
b) Calculate a rank correlation coefficient between the
percentage of sand in soil and depth of the soil
(Ans: -0.9485)
16 Two particles P and Q initially at positions
3i + 2j
and 13i + 2j respectively begin moving. Particle P
moves with a constant velocity of
2i + 6j while
particle Q moves with a constant velocity of 5j, the
units being in meters and metres per second
respectively.
a) Find the:
i) time the two particles are nearest to each other
(Ans: 43)
ii) bearing of particle P from Q when they are
nearest to each other. (Ans: 333.43)
b) Given that after half the time, the two particles are
moving closest to each other, particle P reduces its
speed to half its original speed, in the direction to
approach particle Q and the velocity of Q remains
unchanged , find the direction of particle P
(Ans: N50.8°E)
2004 PAPER ONE
SECTION A
1. Solve the inequality (0.8)-3x > 4.0, correct to 2 decimal
places.
(Ans: x > 2.07)
2. A right circular cone of radius r cm has a maximum
volume, the sum of its vertical height h, and the
circumference is 15 cm. If the radius varies, show that
the maximum volume of the cone is
625
cm 3 .
3. Solve cos θ + sin 2θ = 0 for 00 ≤ θ ≤ 3600.
(Ans: θ = 90°, 210°, 270°, 330°)
1 sin x
, show that
1 sin x
4. Given that: y
dy
1
.
dx 1 sin x
5. A is the point (1, 3) and B the point (4, 6). P is a
variable point which moves in such a way that
AP PB
2
2
34 . Show that the locus of P
describes a circle. Find the centre and radius of the
circle. (Ans: centre 5 2 , 9 2 , r = 5 2 2 units)
6. Find the equation of the plane through the point (1, 2,
3) and perpendicular to the vector r = 4i + 5j + k.
(Ans: 4x + 5y + z = 17)
n
7. Prove by induction that:
3r 1
r 1
3n 1
, where n is
2
a whole number.
8. Use the substitution t tan
x
to evaluate
2
dx
3 5 cos x . (Ans: 0.2747)
0
SECTION B
9. Find n if P4 30n C5 , (Ans: 8)
n
b) How many arrangements can be made from the letters
of the name MISSISSIPPI,
When all the letters are taken at a time
(Ans: 34650 ways)
If the two letters P begin every word?
(Ans: 630 ways)
c) Find the number of ways in which a senior six
mathematics student can choose one or more of the
four girls in the mathematics class to join a discussion
group.
(Ans: 15 ways)
10 a) Find all the values of θ, 00 ≤ θ ≤ 3600, which
satisfy the equation sin2 θ – sin 2θ – 3 cos2 θ = 0.
- 408 -
b) Show that
(Ans: θ = 71.6°, 135°, 251.6°, 315°)
cos A
cot A 45 0 . Hence or
2
1 sin A
otherwise solve
cos A
1
; 00≤ A ≤ 3600
1 sin A 2
11. a) Find the equation of the line through A(2, 2, 5) and
x 2 1
B(1, 2, 3). (Ans: y 2 t 0 )
z 5 2
b) If the line in (a) above meets the line
x 1 y 2 z 1
at P, find the:
1
0
3
i) coordinates of P (Ans: (3, 2, 7))
ii) angle between the two lines. (12 marks)
(Ans: θ = 8.1° or 171.9°)
x 4
in partial fractions.
( x 1) 2 ( x 5)
2
12. Express f ( x)
7
Hence evaluate
f ( x)dx correct to 4 decimal
6
places.
(Ans: 04689)
13. a) Show that the line 5y = 4x + 25 is a tangent to the
2
2
ellipse x y 1
25
9
b) Find the equation of the normal to the ellipse at the
point of contact. (Ans: y
5
4
x 165 )
c) Determine the eccentricity of the ellipse.
(Ans: 54 )
14. a) Differentiate the following with respect to x
i) (sin x)x (Ans: (sin x)x(x cot x + ln sin x))
( x 1) 2
Giving your answers in their simplest
( x 4) 3
ii)
forms. (Ans:
(5 x)( x 1)
)
( x 4)4
b) The distance of a particle moving in a straight line
from a fixed point after time t is given by
x = e-tsin t.
Show that the particle is instantaneously at rest at time
t
seconds. Find its acceleration at t
4
4
seconds. (Ans: -0.6447s)
15. a)Without using table or calculators, simplify
cos 17 i sin 17
9
cos 17 i sin 17
8
b) Given that x and y are real, find the values of x and y
which satisfy the equation:
2 y 4i
y
0.
2x y x i
(Ans: x = -1, y = -2; x = 1, y = 2)
16. Solve the differential equation
tan x
dy
y sin 2 x . (Ans: y = sin2x + c sin x)
dx
b) An athlete runs at a speed proportional to the square
root of the distance he still has to cover. If the athlete
starts running at 10 ms-1 and has a distance of 1600 m
to cover, find how long he will take to cover the
distance. (Ans: 320 s)
2004 PAPER TWO
SECTION A
1. A particle is performing simple Harmonic motion
with center O, amplitude 6m and period 2π. Points B
and C lie between O and A with OB = 1m, OC =
3m and OA = 6m. Find the least time taken while
traveling from
a) A to B
(Ans: 0.4467 π s)
b) A to C.
(Ans: 0.3333π)
2. The probability of two independent events P and Q
occurring together is 1/8. The probability that either
or both events occur is 5/8. Find
a) Prob(P) (Ans: ½, ¼)
b) Prob(Q)
(Ans: ¼, ½)
3. In the table below is an extract of part of log x to base
10, log10 x
x
80.00 80.20 80.50 80.80
log10 x 1.9031 1.9042 1.9058 1.9074
Use linear interpolation to estimate:
a) log10 80.759 (Ans: 1.9072)
b) the number whose logarithm is 1.90388.
(Ans: 80.14)
4. A particle is projected at an angle of 600 to the
horizontal with a velocity of 20 ms-1. Calculate the
greatest height the particle attains. [Use g = 10 ms-2]
(Ans: 15m)
5. Twenty percent of eggs supplied by a certain farm
have cracks on them. Determine the probability that a
sample of 900 eggs supplied by the farm will have
more than 200 eggs with cracks. (Ans: 0.0439)
6. Two forces of magnitude 12N and 9N act on a particle
producing an acceleration of 3.65 ms-2. The forces act
at an angle of 600 to each other. Find the mass of the
particle.
(Ans: 5)
(Ans: -1)
- 409 -
8. Given the numbers; x = 2.678 and y = 0.8765,
measured to the nearest number of decimal places
indicated,
a) State the maximum possible error in x and y.
(Ans: 0.0005); 0.00005)
b) Determine the absolute error in xy
(Ans: 0.00057215)
c) Find the limit within which the product xy lies,
correct to 4 decimal places. (Ans: (2.3467, 2.3478))
SECTION B
9. a) Abel, Bob and Charles applied for the same job in a
certain company. The probability that Abel will take
the job is ¾. The probability that Bob will take it is ½
while the probability that Charles won’t take the job is
1/3. What is the probability that:
i) None of them will take the job? (Ans: 1/224)
ii) One of them will take the job? (Ans: ¼)
b) Two events A and B are independent. Given that
P(A B1) = ¼ and P(A1/B) = 1/6, use a Venn
diagram to find the probabilities
i)
P(A) (Ans: 5/6)
ii)
P(B) (Ans: 7/10)
iii)
P(A B) (Ans: 7/12)
iv)
P(A B)1. (Ans: 7/20)
10(a). Use the trapezium rule to estimate the area of y =
52x between the x-axis, x = 0 and x = 1, using five sub
intervals. Give your answer correct to 3 decimal places
(Ans: 7.712)
1
b) Find the exact value of: 52 x dx . (Ans: 7.456)
0
c) Determine the percentage error in the two calculations
in (a) and (b) above (Ans: 3.43%)
11. The probability density function of a random
variable is given by
k ( x 2) ; 1 x 0,
2k
; 1 x 1,
f ( x) k
2 (5 x ) ; 1 x 3,
0
; Else where
Sketch the function f(x).
Find the:
(i) value of k, (Ans: k = 13)
(ii) mean of X, (Ans: 12/13)
(iii) P(0 < x < 1/x > 0). (Ans: 7/13)
12(a). Use a graphical method to find the approximation
to the real root of
x3 – 3x + 4 = 0.
(b) Use the Newton-Raphson method to find the root of
the equation correct to 2 decimal places.
(Ans: -2.20)
13 A car of mass m kg has an engine which works at a
constant rate of 2H kW. The car has a constant speed of
V ms-1 along a horizontal road
a) Find in terms of m, H, V g and θ the acceleration of
the car when traveling:
i) up a road of inclination θ with a speed of 3/4V ms-1,
2000 H 3mvg sin
a
3mv
ii) down the same road with a speed of 3/5 Vms-1, the
resistance to the motion of the car apart from the
gravitational force, being constant.
4000 H 3mug sin
(Ans: a1
)
3mu
b) If an acceleration in (a) (ii) above is 3 times that of (a)
(i) above, find the angle of inclination θ of the road.
2000 H
(Ans: sin 1
12mug
)
c) If the car continues directly up the road, in case (a) (i)
above, show that its maximum speed is 12/13 V ms-1.
12v
)
(Ans: v1
13
14. The heights (in cm) of senior six candidates in a
certain school were recorded as in the frequency table
below.
Height(cm) Frequency(f)
149 – 152
5
153 – 156
17
157 -160
20
161 – 164
25
165 – 168
15
169 – 172
6
173 - 176
2
a) Estimate the mean height and standard deviation
of the candidates. (Ans: 160.9 cm, 5.5873)
b) Plot a cumulative frequency curve (Ogive)
c) Use your Ogive in (b) above to estimate the:
i. Median height (Ans: 161)
ii. Range of the height of the middle 60% of the
candidates.
(Ans: 54)
15. a) A non-uniform ladder AB, 10m long and mass 8
kg, lies in limiting equilibrium with its lower end A
resting on a rough horizontal ground and the upper
end B resting against a smooth vertical wall. If the
centre of gravity of the ladder is 3 m from the foot of
the ladder, and the ladder makes an angle of 300 with
the horizontal, find the:
i)
coefficient of friction between the ladder and the
ground. (Ans: 3 3 )
10
- 410 -
ii)
5
b) The diagram below shows a cross section ABCD of a
uniform rectangular block of base 8 cm and height, 10
cm resting on a rough horizontal table
C
D
2005 PAPER ONE
(Ans: 12 g 3 )
reaction at the wall.
SECTION A
1. Given the complex number
z
(3i 1)(i 2)2
, determine:
i 3
i)
z in the form a + bi, where a and b are
constants, (Ans: -4 – 3i)
Arg(z) (Ans: -143.13°)
10cm
ii)
2
D
D
8 cm
B
An increasing force F parallel to the table is applied on
the upper edge. If the coefficient of friction between
the block and the table is 0.7, show that the table will
tilt before sliding.
16. Two planes A and B are both flying above the Pacific
Ocean. Plane A is flying on a course of 0100 at a speed
of 300kmh-1 Plane B is flying on a course of 3400 at
200 kmh-1. At a certain time, plane B is 40 km from
plane A. Plane A is then on a bearing 0600. After what
time will they come closest together and what will be
their minimum distance apart? Give your answers
correct to 1 decimal place) (Ans: 14.4 minutes; 8.1km)
2. Find
2 x cos( x)
2
dx . (Ans: 1)
0
3. Solve the inequality (0.6)–2x < 3.6, correct to two
decimal places.
(Ans: x < 1.252)
4. Given that the vectors ai – 2j + k and 2ai + aj – 4k
are perpendicular, find the values of a.
(Ans: a = 2, or -1)
5. A spherical balloon is inflated such that the rate at which
its radius is increasing is 0.5 cms-1. Find the rate at which
i)
the volume is increasing at the instant
(Ans: 157.08 cm3 s-1)
ii) the surface area is increasing when r = 8.5 cm.
(Ans: 106.814 cm2 s-1 (3 dp))
2
6. Solve the equation 2 sin 3 cos 0 ,
00 ≤ θ ≤ 3600.
(Ans: θ = 120°, 240°)
7. Sketch the parabola y2 = 12(x – 4). State the focus and
equation of the directrix.
x
8. Determine d
, when x = 2.(Ans: 1/10)
ln
2
dx
1 x
SECTION B (60 marks)
9. (a) Given that X, Y and Z are angles of a triangle XYZ.
Prove that tan( X Y )
x y
z
cot . Hence solve
2
x y
the triangle if x = 9 cm, y = 5.7 cm and Z = 570.
(07 marks)
b) Use the substitution t tan
2 to solve the equation
3cosθ – 5sinθ = -1 for 00 < θ <3600.
(Ans: θ = 40.84°, 201.08°)
10. i) Determine the coordinates of the point of
intersection of the line x 1 y 3 z 2 and the
2
1
5
plane x + y + z = 12. (Ans: (3, 13, -4))
ii) Find the angle between the line
x 1 y 3 z 1 and the plane x + y + z = 12.
2
5
1
(Ans: 39.2315°)
11. a) Determine the Binomial expansion of 1
x 4
2
4
Hence evaluate (2.1) correct to 2 decimal places.
2
3
4
(Ans: 1 2 x 3 x x x ; = 19.45)
2
2
16
- 411 -
b) A geometric progression (G.P) has a common ratio r
< 1, u1 = 15, and S∞ = 22.5, where S∞ is its sum to
infinity and u1, the first term.
Find the:
i)
value of r,
(Ans: 1/3)
ii)
ratio of u1 : u2. (Ans: 3:1)
12. a) Find the equation of a circle which passes through
the points (5, 7), (1, 3) and (2, 2).
(Ans: x2 + y2 – 7x – 9y + 24 = 0)
b) i) If x = 0 and y = 0 are tangents to the circle, x2 + y2
+ 2gx + 2fy + c = 0, show that c = g2 = f 2.
ii) Given that the line 3x – 4y + 6 = 0 is also a tangent to
the circle in (b) (i) above, determine the equation of
the circle lying in the first quadrant.
(Ans: x2 + y2 – 2x – 2y + 1 = 0)
13. Given the curve y = sin 3x, find the
a) i) value of
dy
at the point
dx
2 , 0 . (Ans: -3)
ii) equation of the tangent to the curve at this point
(Ans: y = -3x + π)
b) i) Sketch the curve y = sin 3x.
ii) Calculate the area bounded by the tangent in a (i)
above, the curve and the y - axis.
(Ans: 0.9783 sq. units)
14. a) Solve the equation 4 x 3 y 2 y x z 4 y and
4
3
2
17
16
18
, y
, z
)
6x + 6y + 2z = 6. (Ans: x
15
15
5
b) Given the polynomial f(x) = Q(x) g(x) + R(x), where
Q(x) is the quotient, g(x) = (x – α)(x − β) and R(x) the
remainder, show that
R( x)
( x ) f ( ) ( x) f ( )
when f(x) is
divided by g(x).
Hence find the remainder when f(x) is divided by x2 –
9, given that f(x) divided by x – 3 is 2 and when
divided by x + 3 is -3. (07 marks)
15. Express
3x 2 x 1
( x 2)( x 1)3
into partial fractions. Hence
3x 2 x 1
dx. Give your answer
evaluate
( x 2)( x 1) 3
3
4
correct to 3 decimal places. (Ans: 0.317)
16. a) Solve the differential equation
1 dy
sin x sec2 3 y (Ans: sin x – x cos x + c)
x dx
b) A hot body at a temperature of 1000C is placed in
a room of temperature of 200C. Ten minutes later,
its temperature is 600C.
i) Write down a differential equation to
represent the rate of change of temperature,
θ of the body with time, t.
dt
k ( R ) )
d
ii) Determine the temperature of the body after
another 10 minutes. (Ans: 40°)
(Ans:
2005 PAPER TWO
SECTION A
1. A particle moves in a straight line with S.H.M. of
period 5 seconds. The greatest speed is 4ms-1. Find the
10
m)
i) Amplitude (Ans:
2
ii) speed when it is 6 m from the centre.
(Ans: 3.2 ms-1)
2. In the table below is part of an extract from sec x0.
x = 600 0′
12′
24′
36′
48′
Sec x 2.0000 2.0122 2.0245 2.0371 2.0498
Use linear interpolation to estimate the
i) value of sec 60015′, (Ans: 2.03065)
iii) angle whose secant is 2.0436. (Ans: 42'')
3. A good football striker is nursing his injury in the leg.
The probability that his team will win the next match
when he is playing is 4/5, otherwise it is 2/3. The
probability that he will have recovered by the time of
the match is ¼. Find the probability that his team will
win the match.
(Ans: 7/10)
4. On the average 15% of all boiled eggs sold in a
restaurant have cracks. Find the probability that a
sample of 300 boiled eggs will have more than 50
cracked eggs.
(Ans: 0.215)
5. A particle is projected with a velocity of 40 ms-1 at an
angle of 600 to the horizontal from the foot of a plane
inclined at an angle of 300 from the horizontal. Find
the time at which the particle hits the plane. [Use g =
10 ms-1] (Ans: 4.6188 s)
6. The table below shows the marks scored by ten
students in Mathematics and Fine Art tests.
A B C D E F G H I J
Mathematics
40 48 79 26 55 35 37 70 60 40
Fine Art
59 62 68 47 46 39 63 29 55 67
Calculate the rank correlation coefficient for the
students performance in the two subjects. Comment
on your result.
(Ans: 0.1121)
7. The forces 3N, 4N, 5N and 6N act along the sides AB,
BC, CD and DA of a rectangle. Their direction is in
the order of the letters. BC is the horizontal. Find the
resultant force and the couple at the centre of a
rectangle of sides 2 m and 4 m.
(Ans: 2 2 , 26m anticlockwise)
8. Given the numbers a = 23.037 and b = 8.4658,
measured to their nearest number of decimal places
indicated,
i) state the maximum possible errors in a and b
(Ans: eA = 0.0005)
- 412 -
ii) determine the absolute errors in
a
b
(Ans: 0.00007513)
iii) find the limits within which
a
lies. Correct to 4
b
decimal places. (Ans: 2.7211 and 2.7213)
SECTION B
9. a) A and B are intersecting sets as shown in the Venn
diagram below.
A
B
x
y
z
0.15
Given that P(A) = 0.6, P A1 / B
5
, and P(A B)
7
= 0.85, find
i) the value of x, y and z
(Ans: x = 0.5, y = 0.1. z = 0.25)
ii) P A / B
b) A bag contains 4 white balls and 1 black ball. A
second bag contains 1 white ball and 4 black balls. A
ball is drawn at random from the first bag and put
into the second bag, then a ball is taken from the
second bag and put into the first bag. Find the
probability that a white ball will be picked when a
ball is selected from the first bag.
(Ans: 7/10)
10. i) Use the trapezium rule to estimate the area of y =
3x between the x – axis, x = 1 and x = 2, using five
sub-intervals. Give your answer correct to four
significant figures. (Ans: 5.48338)
2
ii) Find the exact value of
3
x
dx. (Ans: 5.4614)
1
iii) Find the percentage error in calculations (i) and
(ii) above. (Ans: 0.4028%)
11. The probability density function of a random
; 0 x 1
2kx
variable x is given by f ( x) k (3 x) ; 1 x 2
0
; Else where
a) Sketch the function f(x)
b) Find the
i) value of k (Ans: k = 2/5)
ii) Mean of x (Ans: 1.1333)
iii) P (1 x
2
0) .
x
12. Use a graphical method to show that the equation ex
+ x – 4 = 0 has only one real root. Use the NewtonRaphson method to find the root of the equation
correct to 3 significant figures. (Ans: 1.07)
13. a) A pump raises 2 m3 of water through a vertical
distance of 10 meters in one and a half minutes,
discharging it at a speed of 2.5 ms-1. Show that the
power it develops is approximately 2.25 kW (to 3
significant figures).
b) A car of mass 1000 kg has a maximum speed of
150 km h-1 on a level rough road and the engine is
working at 60 kW.
i) Calculate the coefficient of friction between the car
and the road if all the resistance is due to friction.
(Ans: 0.1469)
ii) Given that the tractive force remains unaltered and
the non-gravitational resistance in both cases
varies as the square of the speed, find the greatest
slope on which a speed of 120 km h-1 could be
maintained.
(Ans: 3.0322°)
14. A particle P moving with a constant velocity 2i + 3j
+ 8k, passes through a point with position vector 6i –
11j + 4k. At the same instant, a particle Q passes
through a point with position vector i – 2j + 5k,
moving with constant velocity 3i + 4j – 7k. Find the
i) position and velocity of Q relative to P at that
5 1
instant (Ans: 9 1 t )
1 15
ii) shortest distance between P and Q in the
subsequent motion. (Ans: 10.3183 units)
iii) time that elapses before the particle are nearest to
11
S)
one another.
(Ans: 227
15. The weights of senior five science class in a certain
school were recorded as in the frequency table below.
Weight (kg)
Frequency
50 – 53
3
54 – 57
8
58 – 61
12
62 − 65
18
66 – 69
11
70 – 73
5
74 – 77
2
78 − 81
1
a. Estimate the mean and standard deviation of the
students’ weights (Ans: 63.1, 6)
b. Plot an ogive
c. Use your ogive to estimate the:
i) median weight (Ans: 63.1)
ii) number of students who weigh between 58.9
kg and 66.7 kg.
(Ans: 29 students)
16. A uniform ladder of length 2l and weight W rests in
a vertical plane with one end against a rough vertical
wall and the other against a rough horizontal
surface, the angles of friction at each end being
1
1
tan 1 and tan 1 respectively.
3
2
a) If the ladder is in limiting equilibrium at either end,
find θ, the angle of inclination of the ladder to the
horizontal (Ans: 39.8°)
- 413 -
b) A man of weight 10 times that of the ladder begins
to ascend it, how far will he climb before the ladder
slips?
(Ans: ½ of the ladder)
2006 PAPER ONE
SECTION A
1. Prove that tan tan 2 tan 2
4
2. Differentiate
3x 4
4
2 x 2 3x 2
with respect to x
7 x 24
(Ans:
2(2 x 2 3 x 2) 2
3. Show that when the quadratic expressions
x2 + bx + c = 0 and x2 + px + q = 0 have a common
root, then, (c – q)2 = (b – p)(cp – bq)
4. Prove that y = -3x + 6 is a tangent to a rectangular
hyperbola whose parametric co- coordinates are of
3
)
3
the form 3t ,
t
5.Find the point of intersection of the plane
3
1
r 1 2
5
2
11x – 3y + 7z = 8 and the line
where λ is a scalar. (Ans: (-4, -1, 7))
6. A group of nine has to be selected from ten boys and
eight girls. It can consist of either five boys and four
girls or four boys and five girls. How many different
groups can be chosen? (Ans: 29400)
7. Solve the differential equation
dy
3 y e2 x given
dx
that when x = 0, y = 1. (Ans: y 15 (e 2 x 4e 3 x ) )
2
8. Evaluate
8x
x2 4 x 12dx correct to 2 decimal places.
0
(Ans: -1.05 )
SECTION B
9. Express the complex numbers Z1 = 4i and Z2 = 2 − 2i in
the trigonometric form r(cos θ + isin θ).
Hence or otherwise evaluate
Z1
.
Z2
(Ans: Z1 = 4(cos90° + i sin90°);
Z2 = 2 2 ( cos(-45°) + sin(-45°);
Find the values of x and y in
Z1 1
(1 0i) )
Z2 2
6 2i
x
y
2 3i 3 2i 1 8i
(Ans: x = 2.8, y = 0.4)
x
10(a). Differentiate from first principles y 2
with
x 1
dy
1 x2
2
)
respect to x (Ans:
dx ( x 1) 2
(b) (i) Determine the turning points of the curve
- 414 -
y = x2(x − 4)
(Ans: (0, 0)
8 3 , 27256 )
2006 PAPER TWO
(ii) Sketch the curve in (i) above for -1≤ x ≤ 5.
(iii) Find the area enclosed by the curve above and the
x – axis. (Ans: 643 sq. units)
11. a) Given the vectors a = 3i − 2j + k and
b = i − 2j + 2k, Find:
i) the acute angle between the vectors (Ans: 36.7°)
ii) vector c such that it is perpendicular to both
2
vectors a and b. (Ans: 5 )
4
b) Given that OA = a and OB = b, point R is on OB
such that OR : RB 4 :1 . Point P is on BA such that
BP : PA 2 : 3 and when RP and OA are both
produced they meet at point Q.
Find:
i) OR and OP in terms of a and b.
(Ans: 54 b , OP 15 (2a 3b) )
ii) OQ in terms of a. (Ans: 85 a )
12. a) Solve the equation 3 cos x + 4 sin x = 2 for
0 ≤ x ≤ 3600. (06 marks)
b) If A, B, C are angles of the triangle, show that cos
2A + cos 2B + cos 2C = -1 − 4 cosA cos B cos C.
(Ans: 119.6°, 346.7°)
13. a) Form the equation of a circle that passes through the
points A (-1, 4), B (2, 5) and C (0, 1)
(Ans: x2 + y2 – 2x – 6y + 5 = 0)
b) The line x + y = c is a tangent to the circle
x2 + y2 −4y + 2 = 0. Find the coordinates of the point of
contact of the tangent for each value of c. (Ans: (1, 3))
14. a) Find the first three terms of the expansion of
SECTION A
1. A and B are two independent events with A twice as
likely to occur as B. If P A
i)
P A B
ii)
p A B / A
1
, find:
2
2. A cylindrical pipe has a radius of 2.5 cm measured to
the nearest unit. If the relative absolute error made in
calculating its volume is 0.125, find the relative
absolute error made in measuring its height.
(05 marks)
3. Joan played 12 chess games. The probability that she
wins a game is
3
. Find the probability that she will
4
win:
i) exactly 8 games
ii) more than 10 games
4. The resultant of the forces F1 = 3i + (a − c)j , F2 = (2a
+ 3c)i + 5j, F3 = 4i + 6j acting on a particle is 10i +
12j.
Find the:
i)
values of a and c
ii) magnitude of force F2.
5. The figure below shows a uniform lamina OPQR in the
form of a trapezium. OP a m , RQ 3a m . The
vertical height of P from RQ = a m. Calculate the
centre of mass of OPQR.
1
1 x
, using Maclaurin’s theorem. (Ans: 1 – x + x2)
b) Use Maclaurin’s theorem to expand tan x in
ascending powers of x up to the term in x3.
(Ans: x
x3
)
3
15. (a) Expand (a + b)4. Hence find (1.996)4 correct to 3
decimal places. (Ans: 15.872)
(b)A credit society gives out a compound interest of
4.5% per annum. Muggaga deposits Shs 300,000 at
the beginning of each year. How much money will he
have at the beginning of four yeas If there are no
withdrawals during this period? (Ans: 1341212917)
16. Find:
ln x dx (Ans: 2x(lnx – 1) + c)
dx
b) x
(Ans: ln(1 – e-x) + c)
e 1
a)
2
6. A vehicle of mass 2.5 metric tonnes is drawn up on a
slope of 1 in 10 from rest with an acceleration of 1.2 ms2 against a constant frictional resistance of
1
of the
100
weight of the vehicle, using a cable. Find the tension in
the cable.
7. The probability distribution for the number of heads
that show up when a coin is tossed 3 times is given by
1 3
P( X x) ,
k x
Find:
i)
ii)
x = 0, 1, 2, 3.
the value of k
E(X).
- 415 -
of the string and moves with speed u in a circle
whose centre is the mid-point of AB.
Show that the tensions in the upper and lower strings
are: (15mu2 + 40 mga)/48a and (15mu2 − 40
mga)/48a respectively.
Hence deduce that the motion is possible if 15u2 ≥
40ga.
(07 marks)
8. Use the trapezium rule with 7 ordinates to estimate
3
1
1 x dx , correct to 3 decimal places
0
SECTION B
9. a) Among the spectators watching a football match,
80% were the home team’s supporters while the rest
were the visitor team’s supporters. If 2500 of the
spectators are selected at random, what is the
probability that there were more than 540 visitors in
this sample? (06 marks)
b) The times a factory takes to make a unit of a product
are approximately normally distributed. A sample of
49 units of the product were taken and found to take
an average of 54 minutes with a standard deviation of
2 minutes.
Calculate the 99% confidence limits of the mean time
of making all the units of the product.
10. a) Study the flow chart below and answer the
questions that follow:
START
n=0
A=1
n = n+1
12. The table below is the distribution of weights of a
group of animals.
Mass(kg)
Frequency
21 – 25
10
26 – 30
20
31 – 35
15
36 – 40
10
41 – 50
30
51 – 65
45
66 – 74
5
(a) Draw a cumulative frequency curve to estimate the
semi-inter-quartile range.
b) Find the:
i) mode
ii) standard deviation of the weights. (12 marks)
13. A light elastic string of natural length l has one end
fastened to a fixed point O. The other end of the string
is attached to a particle of mass m. When the particle
hangs in equilibrium so that it moves in equilibrium,
A = A n
PRINT n, A
is
b) A particle is placed on the lowest point of a smooth
spherical shell of radius 3a m and is given a horizontal
velocity of 13ag m/s. How high above the point of
projection does the particle rise?
(05
marks)
the length of the string is
YES
n 6?
NO
7l
. The particle is displaced
2
from equilibrium so that it moves vertically with simple
harmonic motion when the string is taut.
3l
.
g
a) Show that its period is
b) At t = 0, the particle is released from rest at appoint A,
STOP
i) Perform a dry run for the flow chart
ii) State the purpose of the flow chart
iii) Write down the relationship between n and A
b) Draw a flow chart that reads and prints the mean of
the first twenty numbers and perform a dry run of your
chart.
(07 marks)
11. a) A light inextensible string of length 5a meters has
one end attached to end A and the other end to point
B which is vertically below A and 3a meters from it.
A particle P of mass m kg is fastened to the mid-point
at a distance
3l
vertically below O. Find the:
2
i) depth below O of the lowest point L,
ii) time taken to move from A to L
1
3
3l
g
iii) Depth below O of the particle at time t
.
14.
Show that the Newton-Raphson formula for
approximating the Kth root of a number N is given
by:
- 416 -
1
N
K 1 xn k 1
K
xn
Use your formula to find the positive square root of
67 correct to four significant figures.
b) Show that one of the roots of the equation x2 = 3x − 1
lies between 2 and 3. By use of linear interpolation,
find the root to two decimal places
2007 PAPER ONE
xn1
SECTION A
1. The 5th term of an Arithmetic Progression (A. P) is 12
and the sum of the first 5 terms is 80. Determine the first
term and the common difference. (Ans: 20, -2)
a
2. Given that
x
2
2 x 6 dx 0 , find the value of a
0
15. A continuous random variable X has a probability
density function given by:
2 x3
,
f x x 2 , 3 x 4
0,
Otherwise
i) Sketch f(x)
ii) Find the value of β, hence f(x).
iii) Median, m.
iv) P(2.5 < x < 3.5)
(12. marks)
(Ans: a = -6)
3. Solve the equation log2 x − logx8 = 2.
(Ans: x = 8 or x = ½)
sin 2sin 2 sin 3
tan 2
4. Show that
sin 2sin 2 sin 3
2
5. A point P has co-ordinates (1, -2, 3) and a certain
plane has the equation x + 2y + 2z = 8. The line
through
P
parallel
to
the
line
x y 1
z 1 meets the plane at a point Q.
3
1
16 (a) A particle is projected at an angle of elevation of
Find the co-ordinates of Q. (Ans: (6, 311 , 143 ) )
0
30 with a speed of 21 m/s. If the point of projection
is 5m above the horizontal ground, find the
6. A hemispherical bowl of internal radius, r is fixed with
horizontal distance that the particle travels before
its rim horizontal and contains a liquid to a depth, h.
striking the ground. (Take g = 10 ms-s).
Show by integration that the volume of the liquid in
(b) A boy throws a ball at an initial speed of 40 m/s at an
the bowl is 1 h 2 3r h .
angle of elevation, α. Show , taking g to be 10 m/s2,
3
that the times of flight corresponding to a horizontal
7.
Find
the
locus
of the point P(x, y) which moves such
range of 80m are positive roots of the equation T4that
its
distance
from the point S( -3, 0) is equal to
64T2 + 256 = 0
its distance from a fixed line x = 3. (Ans: y2 = -12x)
8. Differentiate: log e 11 xx
1
2
Simplify your answer.
(Ans:
dy
1
)
dx 1 x 2
SECTION B
9 (a) The function f(x) = x3 + px2 – 5x+q has a factor
(x – 2) and has a value of 5 when x = –3. Find p and q.
(Ans: p = 3, q = -10)
(b) The roots of the equation ax2 + bx + c = 0 are
and . Form the equation whose roots are
(Ans: acx2 – (b2 – 2ac)x + ac = 0)
and
(c) Simplify:
3 2
in the form p + q
2 33
3 where p,
q are rational numbers. (Ans: p = 4, q = -7/3)
10. Sketch the curve: y 4( x 3) .
x( x 2)
11. (i) Show that the equation of the tangent to the
hyperbola (a sec , b tan ) is bx – ay sin – ab
cos 0 .
x2 y2
1,
4
9
at the points where = 450 and where = –135°.
(ii) Find the equations of the tangents to
- 417 -
(Ans: y
2 x 3; y
3
2
3
2
(iii) Find the asymptotes. (Ans: y 32 x )
12. (a) Solve 2 sin 2x = 3 cos x, for –1800 x 180°.
(Ans: x = (-90°, 48.6°, 90°, 131.4°)
(b) Solve sinx – sin 4x = sin2x – sin3x for x
(Ans: x = 5 , 2 , 35 ,0, 5 , 2 , 35 , 35 )
13. (a)
x2
1 x
2
(Ans: 13 (1 x2 ) 2 ( x2 2) c )
1
1
2
dx
1
(b) Use the substitution t tan x to evaluate
2
2
dx
0 1 sin x cos x (Ans: ln 2)
14. (a) What is the smallest number of terms of the
Geometric Progression (G.P.) 5, 10, 20, ... that can
give
a
sum
greater
than 500,000?
(Ans: 17)
n
(b) Prove by induction
n
1
1
r
r
n
1
r 1
(c) Solve simultaneously a3 + b3 = 26 and a + b = 2.
(Ans: a = -1, or 3; b = 3 or -1)
15. Given that the position vectors of A, B and C are
3
1
7
OA = 2 , 0B = 2 and OC = 2 ,
1
2
2
(i) Prove that A, B and C are collinear.
(ii) Find the acute angle between OA and OB.
(Ans: 106.1°)
(iii) If OABD is a parallelogram, find the position
vectors of E and F such that E divides DA in the ratio 1:
2 and F divides it externally in the ratio 1: 2.
3
53
(Ans: 2 , OF = 10 )
8
4
3
16. (a) Given x = r cos and y = r sin , show that
d2y
dx 2
1
cosec3 .
r
(b) Solve:
dy
+ 2y tan x = cos2x given that y = 2,
dx
when x = 0. (Ans: y = cos2x(2 + x))
2007 PAPER TWO
2 x 3)
SECTION A
1. A die is tossed 40 times and the probability of
getting a six on any one toss is 0.122. Estimate the
probability of getting between 6 to 10 sixes.
(Ans: 0.2048)
2. Find the position vector of the centre of gravity of a
uniform lamina in the form of a triangle whose
vertices are; (2, 2), (4, 6 ) and (0,3). (Ans: (2, 11/3))
3. Use the Trapezium rule with 7 ordinates to find the
value of
(1 sin x)
1
2
dx , correct to two decimal
0
places. (Ans: 3.98)
4. A particle of mass 2 kg moving with speed 10 ms–1
collides with a stationary particle of mass 7 kg.
Immediately after impact the particles move with the
same speed but in opposite directions. Find the loss
in Kinetic Energy during collision. (Ans: 283)
5. The table below shows the likelihood of where A
and B spend a Saturday evening.
A
B
Goes to dance
1
Visits a neighbour
1
2
2
1
3
3
6
1
1
6
6
(i) Find the probability that they both go out.
(Ans: 25/36)
(ii) If we know that they both go out, what is the
probability that they both went to dance?
(Ans: 12/25)
6. Show that the equation f(x) = x3 + 3x – 9 has a root
between x = l and x = 2. Using the Newton Raphson
formula once, estimate the root of the equation,
rounded off to two significant figures. (Ans: 1.6)
Stays at home
7. The heights, in centimetres, of children in a SeniorOne class were:
Heights 151160154-156 157-159
(cm)
153
162
Frequency 2
14
13
13
163- 166165 168
2
1
Calculate the:
(i) mean height, (Ans: 158.133)
(ii) standard deviation. (Ans: 3.222)
8. The initial velocity of a particle moving with
constant acceleration is (3i − 5j) ms-1. After 2
seconds the velocity of the particle is of
magnitude 6 ms−1 and parallel to (i + j). Find the
acceleration of the particle.
(Ans: a
3
2
2 32 i
3
2
2 32 j )
SECTION B
- 418 -
9. The departure time, T of pupils from a certain day
primary school can be modeled as in the diagram
below, where t is the time in minutes after the final
bell at 5.00 pm.
START
n=0
f (t)
k
Read: N, x0, TOL
0
5
10
t
Determine the:
(a) value of k, (Ans: k = 15 )
(b) equations of the p.d.f.
251 t
,0 t 5
(Ans: f (t ) 1 (10 t ),5 t 10 )
25
0
, otherwise
(c) E(T). (Ans: 5)
(d)
probability that a pupil leaves between 4 and 7
minutes after the bell. (Ans: 0.5)
10. A car started from rest, accelerated uniformly for 2
minutes and then maintained a speed of 50 kmh-1.
Another car started 2 minutes later from the same spot,
and this car too accelerated uniformly for 2 minutes
and it then maintained a speed of 75 kmh-1.
(i) Draw a Velocity – Time graph and find when and
where the second car overtook the first.
(ii) The first car maintained the speed of 50kmh-1 for 10
minutes. It then decelerated uniformly for a further
2½ minutes before coming to rest. How far has the
car travelled from the start? (Ans: 10.2 km)
11. (a) The table below gives the values of x and their
corresponding values of f(x):
x
2
3
4
5
f(x) 3.88
5.11
8.14
11.94
Use linear interpolation to determine the value
of:
(i)
f(x) when x = 2.15, (Ans: 4.06)
(ii)
x when f(x) = 10.72. (Ans: 4.68)
(b) Study the flow chart below:
n = n+ 1
Is
?
NO
YES
PRINT: xn + 1, N
STOP
(i) Perform a dry run for x0 = 2 and N= 65, TOL =
0.0005.
(ii) State the purpose of the flow chart.
12. Below are marks scored by 8 students A, B, C, D, E,
F, G and H in Mathematics, Economics and
Geography in the end of term examinations.
A
B
C
D
E
F
G
H
Maths 52 75 41 60 81 31 65 52
Econ
50 60 35 65 66 45 69 48
Geog
35 40 60 54 63 40 55 72
Calculate the Rank Correlation Coefficients between the
performances of the students in:
(i) Mathematics and Economics. (Ans: 0.8512)
(ii) Geography and Mathematics. (Ans: 0.1905)
Comment on the significance of Mathematics in the
performance of Economics and Geography.
[Spearman, = 0.86, Kendall’s, 0.79 based on 8
observations at 1% level of significance.]
(Ans: Not significant at 1% level)
13. (a) A rod AB, 1m long, has a weight of 20 N and it’s
centre of gravity is 60 cm from A. It rests
horizontally with A against a rough vertical wall. A
string BC is fastened to the wall at C, 75 cm
vertically above A.
Find the:
(i) normal and frictional forces at A. If friction is
limiting, find the coefficient of friction.
(Ans: 0.5)
- 419 -
(b) A car is working at 5kW and is travelling at a
constant speed of 72kmh1. Find the resistance
to motion. (Ans: 250 N)
(ii) tension in the string.
(Ans: 20N)
(b) A uniform rod LM of weight W rests with L on a
smooth plane PO of inclination 25° as shown in the
diagram below.
M
2008 PAPER ONE
P
SECTION A
450
L
250
O
The angle between LM and the plane is 45°. What force
parallel to PU applied at M will keep the rod in
equilibrium?
(Give your answer in terms of W.)? (Ans: 0.66W)
14 The numbers A and B are rounded off to a and b
with errors e1 and e2, respectively.
(i) Show that the maximum relative error made in the
approximation of
(ii)
e
e
a
A
by
is 1 2 .
a
b
b
b
If also the number C is rounded off to c with
error e3, deduce the expression for the maximum
relative error in taking the approximation of
as
A
BC
a
in terms of e1, e2, e3, a, b and c.
bc
| e | | e3 |
e
e
(Ans: 1 2
3 )
a
bc
bc
(iii) Given that a = 42.326, b = 27.26 and C = -12.93 are
rounded off to the given decimal places, find the
range within which the exact value of the
expression, B lies.
A
2.9558 )
(Ans: ±0.0021 = 2.9516
BC
15 (a) A box contains 7 red balls and 6 blue balls. Three
balls are selected at random without replacement.
Find the probability that:
(i) they are of the same colour. (Ans: 0.1923)
(ii) at most two are blue.
(Ans: 0.9301)
(b) Two boxes P and Q contain white and brown cards.
P contains 6 white cards and 4 brown cards. Q
contains
2
white
cards
and
3 brown cards. A box is selected at random and a
card is selected.
Find the probability that:
(i) a brown card is selected. (Ans: 0.5)
(ii) box Q is selected given that the card is white.
(Ans: 0.4)
16 (a) A particle of mass, m kg is projected with a
velocity of 10 ms-1 up a rough plane of inclination 300
to the horizontal. If the coefficient of friction between
the particle and the plane is 1 calculate how far up
4
the plane the particle travels. (Ans: 7.121 m)
1. Find the fourth root of 4 + 3i
(Ans: ±(1.4760 + 0.23971i); ±(0.2397 – 1.4760i))
2. Without using tables or calculators, show that
tan 150 = 2 -
3
3. Evaluate
02 sin 2 Cos d
(Ans:
2
3
)`
4. Given the vectors a = i - 3j + 3k and b = -i – 3j + 2k
find the:
(i) acute angle between vectors a and b
(Ans: 30.86°)
(ii) equation of the plane containing a and b
(Ans: -3x + 5y + 6z = 0)
5. Given the points O(0, 0) and P(4, 2), A is the locus
of the points such that OA : AP = 1 : 2. Q is the
midpoint of AP. Find the locus of Q in its simplest
form. (Ans: 3x2 + 3y2 – 8x – 4y = 0)
6. Given that and are the roots of the equation x2
+ px + q = 0, express ( − ) and ( − ) in
2
2
3
3
terms of p and q (Ans: p 2 4q ; p(3q – p2));
7.
x2
3
Differentiate tan 2 2 x with respect to x
-1
4 x(1 6 x)
)
4 ( x 2 4 x3 ) 2
8. Find the volume of the solid of revolution formed by
rotating the area enclosed by the curve y = x (1+x),
the x – axis, the lines x = 2 and x = 3 through four
right angles about the x= axis
(Ans: 81.033π cubic units)
(Ans:
SECTION B:
9. A circle cuts the y – axis at two points A and B. It
touches the x- axis at a distance 4 units from the origin
and distance AB is 6 units. A is the point (0,1)
Find the;
(a) equation of the circle
(Ans: x2 y 2 23
x 8 y 7 0 , centre ( 238 , 4) )
4
(b) equations of the tangents to the circle at A and B
(Ans: 24y = 23x + 168; 24y = -23x + 24)
10. (a) Solve the equation Cos x + Cos 2x = 1 for values
of x from 00 to 3600 inclusive
(Ans: x = 38.67°, 321.33°)
(b) (i) Prove that
cos A cos B
sin A sin B
cot
A B
2
- 420 -
(ii) Deduce that
cos A cos B
tan
sin A sin B
C
where A, B and
2
C are angles of a triangle.
1 sin x
If the number of accidents increased to 60 at the
beginning of 2002, estimate the number that was
expected at the beginning of 2005. (Ans: 79)
2
11. (a) Given that y =
dy
dx
3sin 2 x
(cos x 1)
2
Hence, find
dy
dx
2
cos x 1
2
(Ans:
, show that
3sin 2 x
, = -1.6628 )
(cos2 x 1)2
when x
2
3
(b) A curve is represented by the parametric
equations
x = 3t and y
4
2
t 1
. Find the
general equation of the tangent to the curve in
terms of x, y and t. hence determine the equation
of the tangent at the point (3, 2).
(Ans: 3y(t2 + 1)2 + 8 + x = 24t2 + 12(t2 + 1); 3y + 2x = 12)
12. The position vectors of points A and B are OA = 2i –
4j – k and OB = 5i – 2j + 3k respectively. The line
AB is produced to meet the plane 2x + 6y – 3z = -5 at
a point C.
Find the;
(a) coordinates of C
(Ans: (8, 0, 7))
(b) angle between AB and the plane (Ans: (80.8414°)
6
dx
13. (a) Use partial fractions to evaluate 2
4 x 2x 3
(Ans: 0.1905)
(b) Evaluate
2
0
x sin 2 2 x dx
(Ans:
2
)
16
14. On the same axes sketch the curves
f(x) = x2 (x+2) and g(x) =
1
f ( x)
Show the asymptotes and turning points.
2
15. (a) Find the binomial expansion of 1 x . Use your
2
expansion to estimate (0.875)5 to four decimal places.
(Ans: 0.5129)
(b) A financial credit society gives a 2%compound
interest per annum to its members. If Ochola deposits
shs 100,000 at the beginning of every year starting
with 2004, how much would he collect at the end of
2008 if there are no withdraws within this period?
(Ans: 530, 810.0563)
16. (a) Solve the differential equation:
x
2
2
dy
y x 3e x (Ans: y 2x (e x A) )
dx
(b) The number of car accidents x in a year on a high
way was found to approximate the differential equation
dx
kx, where t is the time in years and K a constant.
dt
At the beginning of 2000 the number of recorded
accidents was 50.
- 421 -
2008 PAPER TWO
SECTION A
1. The probability that Anne reads the New Vision is
0.75 and the probability that she reads the New
Vision and not the Daily Monitor is 0.65. The
probability that she reads neither of the papers is
0.15. Find the probability that she reads the Daily
Monitor.
(Ans: 0.2)
less than 500g of sugar, calculate the mean weight
of the sugar in the packets. (Ans: 508.216 g)
5. Use the trapezium rule with 6 ordinates to evaluate
1
ex
2
correct to 2 decimal places. (Ans: 0.74)
0
6. The engine of a train exerts a force of 35, 000 N on
a train of mass 240 tonnes and draws it up a slope of
1 in 120 against resistance totaling to 60N/ tonne.
Find the acceleration of the train.
(Ans: 0.004167 ms-2)
7. A discrete random variable X has the following
probability distribution.
x
0
1
2
3
4
P(X=x) 0.09 0.15 0.40 0.25 0.11
2. Study the flow chart below
START
Find the mean and standard deviation of the
distribution. (Ans: 2.14; 1.0865)
8. Find the coordinates of the centre of mass of the
lamina shown below. Take A as the origin and AD,
AB as x – and y – axes respectively.
(Ans: (4.227, 2.12))
Read: X, Y
P=0
Q=X
C
10 cm
6 cm
3 cm
YES
is Q < Y ?
A
3 cm E
8 cm
D
SECTION B
Write: P, Q
NO
P=P+1
STOP
Q =Q – Y
Using the flow chart, perform a dry run and complete the
table below for X = 22 and Y = 7
P
Q
O
22
………………… ……………………
………………… ……………………
………………… ……………………
9. The table below shows the amount of money (in
thousands of shillings) that was paid out as
allowance to participants during a certain workshop.
Amount (sh’000s)
No. of participants
110 – 114
13
115 – 119
20
120 – 129
32
130 – 134
17
135 – 144
16
145 – 159
12
(a) Draw a histogram and use it to estimate the modal
allowance. (Ans: 11800)
(b) Calculate: (i) median allowances (Ans: 126375)
(ii) mean allowances (Ans: 128,000)
10. (a) The numbers X and Y were estimated with
maximum possible errors of X and Y
respectively. Show that the percentage relative error
P = …………………
Q = …………………
X Y
in XY is
100.
What is the purpose of the flow chart? (05 marks)
Y
X
3. The force A of magnitude 5N acts in the direction with
unit vector 15 (3i 4 j ) And force B of magnitude 13N acts (b) Obtain the range of values within which the exact
value of 3.551 × 2.71635 lies.
in direction with unit vector 15 i 12
j Find the resultant of
13
(Ans: (9.6444, 9.6471).)
forces A and B. (Ans: 8 2 N, 315°
(c) Locate each of the three roots of the equation
x3 – 5x2 + 5 = 0 (Ans: (-1, 0), (1, 2), (4, 5))
4. Sugar packed in 500g packets is observed to be
approximately normally distributed with a standard
deviation of 4g. If only 2% of the packets contained
Thus record:
- 422 -
11. (a) Derive the equation of the path of a projectile
projected from origin O at angle to the horizontal
with initial speed Ums-1.
(05 marks)
(b) A particle projected from a point on a horizontal
ground moves freely under gravity and hits the
ground again at A. Taking O as the origin, the
equation of the path of the particle is 60y = 20
3 x x 2 , where x and y are measured in metres.
Determine the:
(i) initial direction and speed of projection
(Ans: 20 ms-1, 30°)
(ii) distance OA (Take g as 10ms-2). (Ans: 20 3 )
12. A continuous random variable X has the probability
density function
0 x 2 ,
(1 cos x);
f ( x) sin x; 2 x
0
elsewhere.
(a) Find:
(i) the value of (Ans: 2 )
3
(ii) P X
4
3
(Ans: 0.6982)
(b) The times a machine takes to print each of the 10
documents were recorded in minutes as given
below:
16.5, 18.3, 18.5, 16.6, 19.4, 16.8, 18.6, 16.0, 20.1,
18.2
If the times of printing the documents are
approximately normally distributed with variance
of 2.56 minutes, find the 80% confidence interval
for the mean time of printing the documents
(Ans: (17.25, 28.55))
16. A uniform beam AC of mass 8kg and length 8m is
hinged at end A and maintained in equilibrium by
two strings attached to it at points C and D as shown
below.
The tension in BC is twice that in BD;
3
AC
4
AB 4m, AD
B
4
m
(b) Show that the mean, of the distribution is
1
4
13. A particle of mass 1.5 kg lies on a smooth horizontal
table and is attached to two light elastic strings fixed
at points P and Q 12m apart. The strings are of
natural length 4m and 5m and their moduli λ and 2.5λ
respectively.
(a) Show that the particle stays in equilibrium at a point
R midway between P and Q.
(b) If the particle is held at some point S in the line PQ
with PS = 4.8m and then released, show that the
particle performs simple harmonic motion and find
the:
(i)
period of oscillation
(ii)
velocity when the particle is 5.5m from P
14. (a) Show graphically that there is only one positive
real root of the equation
ex – 2x–1=0, between 1 and 2. (05 marks)
(b) Use the Newton Raphson method to calculate
the root of the equation in (a), correct to 2 decimal
places.
(Ans: 1.26)
15. (a) Sixty students sat for a mathematics contest whose
pass mark was 40 marks. Their scores in the contest
were approximately normally distributed. 9 students
scored less than 20 marks while 3 scored more than 70
marks. Find the:
(i) Mean score and standard deviation of the contest,
(Ans: μ = 39.32, σ = 18.65)
(ii) Probability that a student chosen at random passed
the contest
A
D
C
Find the:
(i) tension in string BC, (Ans: 59.8339 N)
(ii) magnitude and direction of the resultant force at the
hinge (Ans: 85.8937N, 24.084°)
2009 PAPER ONE
SECTION A
1. Solve the simultaneous equations
p + 2q – r = -1
3p – q + 2r =16
2p +3q +r = 3
2. Given that sin(𝜃 – 45°) = 3cos(𝜃 + 45°), show that
tan 𝜃 = 1. Hence find if 0° ≤ 𝜃 ≤ 360°.
3. Differentiate
4. If y
e ax
2
3 2x
, find the range of possible values of y
4 x2
for real x
5. The points P(2,3), Q(−11, 8) and R(-4, -5) are vertices
of a parallelogram PQRS which has PR as a diagonal.
Find the co-ordinates of vertex S.
6. Find
dx
1 cos x
- 423 -
7. Find the equation of a line through the point (1, 3, 2) and perpendicular to the plane whose equation is
4x + 3y – 2z – 16 = 0
dy
8. Solve the differential equation x 1 y y 0 ,
dx
given that y = 1 when x = e
SECTION B
15. a) Find the equation of the tangent and normal to the
ellipse
x2
4
y2
1
1 at the point P(2cos θ, sin θ)
b) If the tangent in (a) cuts the y – axis at point A and the
x− axis at point B, and the normal cuts the x-axis at
point C, find the co-ordinates of the points A, B and
C.
9. (a) By using the binomial theorem, expand
8 24x
2
as far as the 4th term. Hence evaluate
3
2
4 3 to one decimal place.
b) Find the coefficient of x in the expansion of
10
2
x 2
x
10 a) Differentiate 1 2x 2
1
2
with respect to x
x x x 1
with respect to x.
x3 x
11. a) Use the factor formula to show that
sin A 2 B sin A
tan A B
cos A 2 B cos A
b) Express y = 8cosx + 6sin x in the form R cos (x –
) where R is positive and is acute .
Hence find the maximum and minimum values of
4
3
2
b) Integrate
16. In a certain process the rate of production of yeast is kx
grammes per minute, where x grammes is the amount
produced and k = 0.003.
a) Show that the amount of yeast is doubled in about
230 minutes.
b) If in addition yeast is removed at a constant rate of
m grammes per minute, find the
i) amount of yeast at time t minutes, given that
when t = 0, x = p grammes
ii) Value of m if p = 20,000 grammes and the
supply of yeast is exhausted in 100 minutes.
1
8 cos x 6 sin x 15
12. a) Given that
ix
3x i 4
, find the values of x
1 iy x 3 y
and y
b) If Z = x + iy, find the equation of the locus
Z 3
4
Z 1
13. a) Find the angle between the planes
x – 2y +z = 0 and x – y =1.
b) Two lines are given by the parametric equations;
−i + 2j + k + t(i − 2j + 3k) and
−3i + pj + 7k + s(i − j + 2k)
If the lines intersect, find the
i) values of t, s and p.
ii) coordinates of the points of intersection
14. a) Use Maclaurin’s theorem to expand
to the term in x
b) Given that
d2y
dx 2
1
1 x
up
3
e x = tan 2y, show that
e x e3 x
2 1 e2 x
2
- 424 -
2009 PAPER TWO
SECTION A
1. If A and B are independent events;
(i) show that events A and B' are also independent
(ii) find P(B) given that P(A) = 0.4 and P(A B) =
0.8
(Ans: 0.66)
2. A car moves from Kampala to Jinja and then back. Its
average speed on the return journey is 4km−1 greater
than that on the outward journey and it takes 12
minutes less. Given that Kampala and Jinja are 80km
apart, find the average speed on the outward journey.
(Ans: 38.05 ms-1)
3. The table below shows the distance in kilometers
(km) a truck can run with a given amount of fuel in
litres (l).
Distance(km)
20
28
33
42
Fuel (l)
10
13
21
24
Estimate:
(a) how far the truck can move on 27.5l of fuel,
(Ans: (Ans: 52.5 km)
(b) the amount of fuel required to cover 29.8km
(Ans: 18.88 l)
4. The random variable X has a probability function
k 22 ; x 0, 1, 2,3
f ( x)
0 ; elsewhere
Find:
(a) the value of the constant k (Ans: k = 15 )
34
(b) E(X) (Ans: 15
)
5. A body of mass 8kg rests on a rough plane inclined at
θ to the horizontal. If the coefficient of friction is ,
find the least horizontal force in terms of , θ and g
which will hold the body in equilibrium. (Ans:
P
8 g (sin cos )
)
(cos sin )
6. Use the trapezium rule with 6 ordinates to estimate
2
ln x
1 x dx . Give your answer correct to three decimal
places (Ans: 0.237)
7. The following information relates to three products
sold by a company in the year 2001 and 2004
Product
2001
2004
Quantity in Selling Quantity in Selling
thousands price per thousands price per
unit
unit
A
76
0.60
72
0.18
B
52
0.75
60
1.00
C
28
1.10
50
1.32
Calculate:
(a) percentage increase in sales over the period
(Ans: 10.26%)
(b) corresponding percentage increase in income of the
period
(Ans: 8.98%)
8. The velocity of a particle at any time t is given by an
equation;
v(t ) a sin t b cos t
(a) find the expression for the displacement x at any
time given that x = 0 when time
t = 0 (Ans: x = 9cos ωt + bω sin ωt)
(b) show that the motion of the particle is Simple
Harmonic
SECTION B
9. The dimensions of a rectangle are 6.2cm and
5.36cm
(i) State the maximum possible error in each
dimension (Ans: 0.05, 0.005)
(ii) Find the range within which the area of the
triangle lies (Ans: (32.93325, 33.53125))
b) The numbers a = 26.23, b = 13.18 and c = 5.1 are
calculated with percentage errors of 4, 3 and 2
respectively. Find the limits to two decimal places
within which the exact error of the expression
b
ab lies. (Ans: (319.21, 367.87))
c
10. A pile driver of mass 1200kg falls freely from a
height of 3.6m and strikes without rebounding, a
pile of mass 800kg. The blow drives the pile a
distance of 36cm into the ground. Find the
(a) resistance of the ground (Ans: 90160N)
(b) time for which the pile is in motion (Ans: 0.14 s)
[Assume the resistance of the ground to be uniform]
11. The table below shows the income of 40 factory
workers in millions of shillings per annum.
1.0 1.1
1.0 1.2
5.4
1.6
2.0
2.5
2.1 2.2
1.3 1.7
1.8
2.4
3.0
2.2
2.7 3.5
4.0 4.4
3.9
5.0
5.4
5.3
4.4 3.7
3.6 3.9
5.2
5.1
5.7
1.5
1.6 1.9
3.4 4.3
2.6
3.8
5.3
4.0
(a) Form a frequency distribution table with class
intervals of 0.5 million shillings starting with the
lowest limit of 1 million shillings.
(b) Calculate the
i. mean income (Ans: 3175,000)
ii. standard deviation (Ans:
1413992.574)
(c) Draw a histogram to represent the above data. Use it
to estimate the modal income.
12. Forces of magnitude 3N, 4N, 4N, 3N and 5N act
along the lines AB, BC, CD, DA and AC
respectively of the square ABCD whose side has a
length of a units. The direction of the forces are
indicated by the order of the letters.
(a) find the magnitude and direction of the resultant
force
- 425 -
(Ans: 5.196 N, 60.8°)
(b) if the line of action of the resultant force cuts AB
produced at E, find the length AE (Ans: 1.76a)
13. (a) A box contains two types of balls, red and black.
When a ball is picked from the box, the probability
that it is red is 7 12 . Two balls are selected at
random from the box without replacement. Find the
probability that
(i) the second ball is black (Ans: 52 )
(ii) the first ball is red, given that the second one is
black
(Ans: 117 )
b) An interview involves written, oral and practical
tests. The probability that an interviewee passes a
written test is 0.8, the oral test is 0.6 and the practical
test is 0.7. What is the probability that the
interviewee will pass
(i) the entire interview? (Ans: 0.336)
(ii) exactly two of the interview tests? (Ans: 0.452)
14. (a) Show that the root of the equation
2 x 3cos( x 2) 0 lies between 1 and 2
(b) Use Newton Raphson’s method to find the root of
the equation in (a) above. Give your answer
correct to two decimal places. (Ans: 1.23 (2 dp))
15. The masses of soap powder in certain packets is
normally distributed with mean 842 grams and
variance 225 (grams)2.
Find the probability that a random sample of 120
packets has sample mean with mass
(i) between 844 grams and 846 grams (Ans:
0.0702)
(ii) less than 843 grams (Ans: 0.7673)
b) A random sample of size 76 electrical components
produced by a certain manufacturer has
resistances r1, r2, … r76 ohms where ri = 740 and
(b) distance of P from the origin at the time when
the distance between P and Q is least (Ans:
28.8 m)
(c) least distance between P and Q. (Ans: 24.14 m)
2011 PAPER ONE
SECTION A
1. Solve the equation log25 4x2 = log5 (3 – x2).
(Ans: x = -3, x = 1)
2. Find the equation of a line through the point (2, 3) and
perpendicular to the line x + 2y + 5 = 0.
(Ans: y = 2x + 1)
3
3x 2 4 x 1
3. Evaluate 3
dx (Ans: ln 12)
x 2x2 x
1
4. A committee of 4 men and 3 women is to be formed
from 10 men and 8 women. In how many ways can
the committee be formed? (Ans: 11760 ways)
1
1
1
5. Show that tan ( 12 ) tan ( 15 ) tan ( 97 )
2
6. Given that R q (1000 q ) , find
dR
1000 2q 2
, (Ans:
)
dq
1000 q 2
(b) the value of q when R is maximum. (Ans: 500 )
7. Show that the points A, B and C with position vectors
3i + 3j + k , 8i + 7j + 4k and lli + 4j + 5k respectively,
are vertices of a triangle.
(a)
8. (a) Form a differential equation by eliminating the
constants a and b from x = a cos t + b sint. (Ans: -x)
(b) State the order of the differential equation formed in
(a) above. (Ans: second order)
SECTION B
9. (a) The first term of an Arithmetic Progression (A.P.) is
½. The sixth term of the A.P is four times the fourth
term. Find the common difference of the A.P.
(Ans: 3 )
ri 2 = 8,216
Calculate the
(i) unbiased estimate for the population variance
(Ans: 13.476)
(ii) 91.86% confidence interval for the mean
resistance of the electrical components
produced. [Give answers correct to 3 decimal
places]
(Ans: (9.003, 10.470))
16. The particles P and Q move with constant velocities
of (4i + j − 2k)ms−1 and (6i + 3k)ms−1 respectively.
Initially P is at the point with position vector (−i +
20j + 21k)m and Q is at the point with position
vector (i + 3k)m.
Find the
(a) time for which the distance between P and Q is
least
(Ans: 2.2 s)
14
(b) The roots of a quadratic equation x2 + px + q = 0
are α and β. Show that the quadratic equation
whose roots are α2 – qα and β 2 – q is given by x2 –
(p2 + pq – 2q) x + q2(q + P + 1) = 0.
10. (a) Form a quadratic equation having -3 + 4i as one
of its roots. (Z2 + 6Z + 25) = 0
(b) Given that Z1 = –1 + i 3 and Z2 = –1 – i 3
(i) express Z1 in the form a + i b ,where a
Z2
and b are real numbers. (Ans:
1
2
3
2
i)
(ii) represent Z1 on an Argand diagram.
Z2
(iii) find
Z1
Z2
(Ans: 1)
- 426 -
11. In the diagram below, the curve y = 6 – x2 meets the
line y = 2 at A and B, and the x - axis at C and D.
y
y = 6 – x2
A
B
C
D
x
2011 PAPER TWO
Find the (a) coordinates of A, B, C and D
(Ans: A(-2, 2), B(2, 2), C ( 6, 0), D ( 6, 0))
(b) area of the shaded region, correct to one
decimal place. (Ans: 8.9293 sq. units)
12. (a) Find the angle between the lines
x y 1 z 2
y 1 z 2
x
and
.
2
3
4
2
3
(Ans: 8.53°)
(b) Find in vector form the equation of the line of
intersection of the two planes 2x + 3y − z = 4
19 5
1
6
and x − y + 2z = 5. (Ans: r
1 )
5
0
1
13. (a) Find the equation of the tangent to the parabola
x
y2
at the point t 2 , 4t .
16
(Ans: x – 8ty + t2 = 0)
(b) If the tangents to the parabola in (a) above at the
points P p 2 ,
p
4
and Q q 2 ,
q
4
meet on the
line y = 2,
(i) show that p + q = 16,
(ii) deduce that the mid-point of PQ lies on the
line y = 2.
14. (a) Solve 3 sin x + 4 cos x = 2 for
-180° ≤ x ≤ 180°. (Ans: x = -29.55°, 103.29°)
(b) Show that in any triangle ABC,
a 2 b 2 sin( A B )
c2
sin( A B )
15.(a) Differentiate the following with respect to x:
1
(5 x 6)( x 2)
2
)
(i) x 1 2 x 2 (Ans:
1
2( x 1) 2
2 x 2 3x
x 2 3 x) )
(Ans: (4 x 3( x 4) 2(2
2
3
( x 4)
( x 4)
(b) The base radius of a right circular cone increases
and the volume changes by 2%. If the height of
the cone remains constant, find the percentage
increase in the circumference of the base.
(Ans: 1%)
dy sin 2 x
16.(a) Solve the differential equation
,
dx
y2
given that y = 1 when x = 0.
(Ans: 4y3 = 6x – 3 sin 2x + 4)
(ii)
(b) It is observed that the rate at which a body cools
is proportional to the amount by which its
temperature exceeds that of its surroundings. A
body at 78°C is placed in a room at 20°C and
after 5 minutes the body has cooled to 65°C.
What will be its temperature after a further 5
minutes?
(Ans: θ = 54.9°)
SECTION A
1. The data below represents the lengths of the leaves in
centimeters.
4.4, 6.2, 9.4, 12.6, 10.0, 8.8, 3.8 and 13.6
Find the: (a) mean length, (Ans: 8.575 cm)
(b) variance. (Ans: 11.224 cm)
2. A particle of mass 2 kg moves under the action of
three forces, F1, F2, and F3, At a time, t,
F1 = ( 4 t – 1)i + (t – 3)j N,
1
F2 = ( 12 t + 2)i + ( 12 t – 4)j N and
F3 = ( 4 t – 4)i + ( 2 t + 1)j N
Find the acceleration of the particle when t = 2
seconds. (Ans: a = -½ ms-2-)
1
3
3. The table below shows delivery charges by a courier
company.
Mass (gm)
200
400
600
Charges( shs)
700
1200
3000
Using linear interpolation or extrapolation, find the:
(a) delivery charge of a parcel weighing 352gm,
(Ans: 1080)
(b) mass of a parcel whose delivery charge is
shs3,300. (Ans: 633 13 kg)
4. Two events A and B are such that P(A'∩B) = 3x,
P(A∩B') = 2x, P(A'∩B') = x, and P(B) =
Venn diagram, find the values of
(a) x, (Ans: 1/7)
(b) P(A∩B) (Ans: 1/7)
4
7
. Using a
5. A man can row a boat in still water at 6 kmh-1. He
wishes to cross a river to a point directly opposite his
starting point. The river flows at 4 kmh-1 and has a
width of 250 m. Find the time the man would take to
cross the river. (Ans: 3.354 min)
6. Study the flow chart given below.
- 427 -
(c) linear speed of the particle. (Ans: 4 ms-1)
11. (a) Use the trapezium rule with five sub-intervals to
START
3
estimate
x=0
tan x dx correct to three decimal places.
0
(Ans: 0.704 )
y=1
3
(b) (i) Find the value of
0
x=x+1
places. (Ans: 0.693)
(ii) Calculate the percentage error in your
estimation in (a) above. (Ans: 1.587%)
(iii) Suggest how the percentage error may be
reduced.
(Ans: Increasing number of sub-intervals)
y=y×x
Is
x>
6
tan x dx to three decimal
NO
YES
PRINT y
STOP
(a) Perform a dry run.
(b) What is the purpose of the flow chart?
7. Given that X ~ N(2, 2.89), find P(X < 0).
(Ans: 0.1198)
8. Particles of weights 12 N, 8 N and 4 N act at points
(1, -3), (0, 2) and (1, 0) respectively. Find the centre
of gravity of the particles. (Ans: 2 3 , 5 6 )
SECTION B
9. The continuous random variable X has the
probability density function (p.d.f.) given by
1 x 3,
k1 x,
f ( x ) k 2 (4 x) 3 x 4,
0
otherwise
where k1 and k2 are constants.
(a) Show that k2 = 3k1
(b) Find:
(i)
The value of k1 and k2
(Ans: k1 = 2/11, k2 = 6/11)
(ii)
E(X), the expectation of X (Ans: 2.485)
10. An elastic string of length a metres is fixed at one
end P and carries a particle of mass 3 kg at its other
end Q. The particle is describing a horizontal circle
of radius 80 cm with an angular speed of 5 rads-1.
Determine the:
(a) (i) angle the string makes with the horizontal,
(Ans: 26.1°)
(ii) tension in the string. (Ans: 66.816N)
(b) value of a. (Ans: (0.891 - x)
12. The heights and masses of ten students are given in
the table below.
Height (cm)
Mass (kg)
156
62
151
58
152
63
146
58
160
70
157
6O
149
55
142
57
158
68
141
56
(a) (i) Plot the data on a scatter diagram.
(ii) Draw the line of best fit. Hence estimate the
mass corresponding to a height of 155cm.
(Ans: 65 kg)
(b) (i) Calculate the rank correlation coefficient for
the data. (Ans: 0.87 (2 dp))
(ii) Comment on the significance of the heights
on masses of the students. [Spearman’s =
0.79 and Kendall’s τ = 0.64 at 1% level of
significance based on 10 observations.]
(Ans: 1%)
13. A football player projects a ball at a speed of 8ms-1 at
an angle of 30° with the ground. The ball strikes the
ground at a point which is level with the point of
projection. After impact with the ground, the ball
bounces and the horizontal component of the velocity
of the ball remains the same but the vertical
component is reversed in direction and halved in
magnitude. The player running after the ball kicks it
again at a point which is at a horizontal distance of
1.0 m from the point where it bounced, so that the
ball continues in the same direction. Find the:
(a) horizontal distance between the point of
projection and the point at which the ball first
strikes the ground. [Take g = 10 ms-2 ].
(Ans: 5.543m)
(b) (i) the time interval between the ball striking the
ground and the player kicking it again.
(Ans: 0.1443 s)
- 428 -
(ii) the height of the ball above the ground when
it is kicked again. [Take g = 10 ms-2]
(Ans: 0.1845 m)
14. (a) (i) On the same axes, draw graphs of y = x2 and y
= cos x for 0 ≤ x ≤ 2 at intervals of 8 .
(ii) From your graphs, obtain to one decimal
place, an approximate root of the equation
x2 – cosx = 0. (Ans: 0.8 (1 dp))
(b) Using Newton-Raphson method, find the root of
the equation x2 – cos x = 0, taking the
approximate root in (a) as an initial
approximation. Give your answer correct to three
decimal places. (Ans: 0.824 (3 dp))
15. Box A contains 4 red sweets and 3 green sweets. Box
B contains 5 red sweets and 6 green sweets. Box A is
twice as likely to be picked as box B. If a box is
chosen at random and two sweets are removed from
it, one at a time without replacement;
(a) find the probability that the two sweets removed
are of the same colour. (Ans: 0.4372)
(b) (i) construct a probability distribution table for
the number of red sweets removed.
(ii) find the mean number of red sweets removed.
(Ans: 1.065)
16. The diagram below shows a uniform wooden plank
AB of mass 70 kg and length 5m. The end A rests on
a rough horizontal ground. The plank is in contact
with the top of a rough pillar at C. The height of the
pillar is 2.2m and AC = 3.5 m.
B
3.5m
3x − y + z = 4,
x − 2y + 4z = 3,
2x + 3y − z = 4.
2. (a) Prove that
2 tan
1 tan 2
(Ans: x = 1, y = 1, z = 1)
sin 2
(b) Solve sin 2θ = cos θ for 0° ≤ θ < 90°
3x 1
with respect to x
3. Differentiate
x2 1
x3
)
3
( x 2 1) 2
4. A line passes through the points A(4,6,3) and B(1,3,3).
(a) Find the vector equation of the line.
4
3
(Ans: 6 3 )
3
0
(b) Show that the point C(2, 4, 3) lies on the line in
(a) above.
5. The sum of the first n terms of a Geometric
4
Progression (G.P) is (4n 1) . Find its nth term as an
3
integral power of 2. (Ans: 22n)
6. The line y = mx + c is a tangent to the ellipse
(Ans:
x2
a
2
y2
b2
1 when c a2 m2 b2 . Find the
equations of the tangents to the ellipse
Wooden plank
C
Pillar
2.2m
5.0m
2012 PAPER ONE
1. Solve the simultaneous equations
from the point (0,
Given that the coefficient of friction at the ground is
0.6 and the plank is just about to slip, find the:
(a) angle the plank makes with the ground at A.
(Ans: 38.945°)
(b) normal reaction at
(i)
A (Ans: 380.52 N)
(ii)
C (ANs: 381.097 N)
(c) coefficient of friction at C. (Ans: 0.0379)
a2
y2
b2
1
5 ) (Ans: y = x + 5 , y = -x+ 5 )
7. Using a suitable substitution, find
A
x2
sin 1 2 x
(1 4 x 2 )
dx.
2
1
(Ans: sin 2 x c )
2
8. Find the equation of the normal to the curve x2y + 3y2
– 4x – 12 = 0 at the point (1, 2). (Ans: -3x + 2)
9. If z
(2 i )(5 12i )
(1 2i ) 2
(a) Find: (i) modulus of z (Ans: 5.814)
(ii) argument of z (Ans: -86.055°)
(b) represent z on a complex plane.
(c) Write z in the polar form.
(Ans: 5.814(cos0.478π – i sin 0478π))
10. (a) Solve the equation 8cos4x – 10cos2x + 3 = 0 for
x in the range 0° ≤ x ≤ 180°
(Ans: x = 30°, 45, 135°, 150°)
(b) Prove that cos 4A – cos 4B – cos 4C
= 4sin2B sin 2C cos 2A, given that A, B, C are
angles of a triangle.
11.(a) Find the derivatives with respect to x of the
following:
- 429 -
cos 2 x
1 sin 2 x
i)
(Ans:
(ii) ln(sec x + tan x
(b)
2
)
1 sin 2x
(Ans: sec x)
(Ans: a = -6, p = 9.5)
2. Two points A and B are such that P( A)
2
x
2
0
sin x
(Ans: π – 2)
P( B)
12. Triangle OAB has OA = a and OB = b. C is a point
2a
on OA such that OC
. D is the midpoint of
3
AB . When CD is produced, it meets OB produced
at E, such that DE = nCD and BE = kb. Express
DE in terms of:
5n
n
a b)
(a) n, a, and b (Ans:
6
2
(b) k, a, and b. Hence find the values of n and k.
1
2k 1
(Ans: a
b , n = 3 5 , k = 15 )
2
2
13. (a) Find the equation of the locus of a point which
moves such that its distance from D(4, 5) is thrice
its distance from the origin.
(Ans: x2 + y2 + x + 1.25y – 5.125 = 0)
(b) The line y = mx intersects the curve y = 2x2 – x at
the points A and B. Find the equation of the locus
of the point P which divides AB in the ratio 2:5.
(Ans: y = 7x2 – x)
14. (a) On the same axes, sketch the curves
y = x(x + 2) and y = x(4 – x)
(b) Find the area enclosed by the two curves in (a).
(Ans: 1/3 sq. units)
(c) Determine the volume of the solid generated
when the area enclosed by the two curves in (a) is
rotated about the x-axis. (Ans: π cubic units)
15. Solve for x in the following equations:
(a) 9x – 3(x + 1) = 10 (Ans: x = 1.465)
(b) log4x2 – 6logx4 – 1 = 0 (Ans: x = 16, x = 1/8)
16. At 3:00 pm, the temperature of a hot metal was
80°C and that of the surroundings 20°C. At 3:03
pm, the temperature of the metal had dropped to
42°C. The rate of cooling of the metal was directly
proportional to the difference between its
temperature 𝜃 and that of the surroundings.
(a)(i) Write a differential equation to represent the
rate of cooling of the metal
(Ans: θ = 20 + Ae-kt)
(ii) Solve the differential equation using the given
conditions. (Anss: 20 60e 3 22 )
(b) Find the temperature of the metal at 3:05 pm.
(Ans: θ = 31.27°)
t
ln( 60 )
2012 PAPER TWO
0
0
0
1. The forces , , and 0 N act at points (p,
2 4 3
q
1), (2, 3), (4, 5) and (6, 1) respectively. The resultant
is
1
and
5
0 N
3
acting at (1, 1). Find the values of a and p.
1
2
. Find P(A∪B) when A and B are:
(a) independent events (Ans: 0.6)
(b) mutually exclusive events (Ans: 0.7)
3. Use the trapezium rule with four sub-intervals to
estimate
2
1
1 sin x dx (Give your answer correct to
0
three decimal places). (Ans: x = 1.013)
4. (a) Show that the velocity v of a body which starts
with an initial velocity u and moves with uniform
acceleration a consequently covering a distance x,
1
is given by v u 2 2ax .
2
(b) Find the value of x in (a) if v = 30 m/s, u = 10 m/s
and a = 5 m/s2. (Ans: x = 80m)
5. A teacher gave two tests in chemistry. Five students
were graded as follows.
GRADE
Test 1
A
B
C
D
E
Test 2
B
A
C
D
E
Determine the rank correlation coefficient between
the two tests. Comment on your result.
(Ans: 0.9, high positive)
6. A light extensible string passes over a smooth pulley
fixed at the top of a smooth plane inclined at 30° to
the horizontal. A mass, m is attached to the other end
of the string and rests on the inclined plane. If the
system is in equilibrium, find m. (Ans: m = 8 kg)
7. The table below shows the cost y shillings for hiring a
motor cycle for a distance x kilometres.
Distance (x km)
10
20
30
40
Cost (Shs. Y)
2800 3600 4400 5200
Use linear interpolation or extrapolation to
calculate the:
(a) cost of hiring the motor cycle for a distance of
45 km. (Ans: 5600)
(b) Distance Mukasa travelled if he paid Shs 4000
(Ans: 25 km)
8. A random variable X has the following probability,
distribution: P(X=0)
P(X = 3) =
1
3
, P(X=1) = P(X = 2) = and
8
8
1
. Find the:
8
(a) mean of X, (Ans: 1.5)
(b) variance of X (Ans: 0.75)
9. The table below shows the marks obtained in an
examination by 200 candidates.
Marks (%)
Number of candidates
10 – 19
18
- 430 -
20 – 29
34
30 – 39
58
40 – 49
42
50 – 59
24
60 – 69
10
70 – 79
6
80 – 89
8
(a) Calculate the:
(i) mean mark, (Ans: 40.2%)
(ii) modal mark (Ans: 35.5%)
(b) Draw a cumulative frequency curve for the
data.
Hence estimate the lowest mark for a
distinction one if the top 5% of the candidates
qualify for the distinction. (Ans: 75%)
5
1
10. At 11:45 a.m, ship A has position vector km and
moving at 8 kmh-1 in the direction N30°E. At 12
8
noon, another ship B has position vector km
7
-1
and moving at 3 kmh in the direction South East.
(a) Find the position vector of ship A at 12 noon
6
(Ans:
km)
2.73205
(b) If the ships after 12 noon maintain their
courses, find the:
(i) time when they are closest
(Ans: 0.49612s)
(ii) least distance between them.
(Ans: 1.091 km)
11. (a) (i) Show that the equation ex – 2x – 1 = 0 has a
root between x = 1 and x = 1.5
(ii) Use linear interpolation to obtain an
estimation of the root (Ans: 1.18 (2dp))
(b) (i) Solve the equation in (a)(i), using each
formula below twice. Take the approximation
in (a)(ii) as the initial value.
1
2
e xn (en 1) 1
Formula 1: xn1 (e xn 1) (Ans: 1.2642)
Formula 2: xn 1
e xn 2
(Ans: 1.2565)
(ii) Deduce with a reason which of the two
formulae is appropriate for solving the given
equation in (a)(i). Hence write down a better
approximate root, correct to 2 decimal places.
(Ans: 1.26 (2 dp))
12. A continuous random variable X has a probability
density function (p.d.f) f(x) as shown in the graph
below.
f (t)
k
0
1
2
x
(a) Find the:
(i) value of k,
(k = 1)
(ii) expression for the probability density function
(p.d.f) of X
x, 0 x 1
(Ans: f ( x) 2 x, 1 x 2 )
0, otherwise
(b) Calculate the:
(i) mean of X, (Ans: 1)
(ii) P(X < 1.5/X > 0.5)
(Ans: 08751)
13. (a) Show that the centre of gravity of a uniform thin
hemispherical cup of radius, r is at a distance
r
2 from the base.
(b) The figure below is made up of a thin
hemispherical cup of radius 7 cm. It is welded to
a stem of length 7 cm and then to a circular base
of the same material and of radius 7 cm. The mass
of the stem is one quarter than of the cup.
7cm
7cm
7cm
Find the distance from the base, of the centre of
gravity of the figure. (Ans: 6.5)
14.(a) The length, width and height of a water tank were all
rounded off to 3.65m, 2.14m and 2.5m respectively.
Determine in m3 the least and greatest amount of
water the tank can contain (Ans: 19.992 m3)
(b) A shop offered 25% discount on all items in its
store and a second discount of 5% to any
customer who paid cash.
(i) Construct a flow chat which shows the amount
paid for each item.
(ii) Using your flow chart in (i), compute the amount
paid for the following items: (Ans: 225000/-)
Item
Mattress
Television set
Price
125,000
340,000
Mode of payment
Cash
Credit
15.(a) A box of oranges contains 20 good and 4 bad
oranges. If 5 oranges are picked at random,
- 431 -
determine the probability that 4 are good and the
other is bad. (Ans: 0.456)
(b) An examination has 100 questions. A student has
60% chance of getting each question correct. A
student fails the examination for a mark less than 55.
A student gets a distinction for a mark of 68 or
more. Calculate the probability that a student:
(i) fails the examination (Ans: 0.1308)
(ii) gets a distinction (Ans: 0.0629)
16. A gun of mass 3000 kg fires horizontally a shell at an
initial velocity of 300 ms-1. If the recoil of the gun is
brought to rest by a constant opposing force of 9000N
in 2 seconds, find the:
(a) (i) initial velocity of the recoil (Ans: 6ms-1)
(ii) mass of the shell (Ans: 60 kg)
(iii) gain in kinetic energy of the shell just after
firing (Ans: 2700 kJ)
(b) (i) displacement of the gun (Ans: 6m)
(ii) work done against the opposing force
(Ans: 54 kJ)
2013 PAPER ONE
1.
2.
3.
4.
5.
SECTION A
Solve logx5 + 4 log5x = 4 (Ans: x = 5)
In a Geometric Progression (G.P), the difference
between the second and fifth term is 156. The
difference between the seventh and fourth term is
1404. Find the possible values of the common ratio.
(Ans: r = 3, r = -3)
Given that r = 3 cos θ is the equation of a circle,
find its Cartesian form. (Ans: x2 + y2 – 3x = 0)
The position vector of point A is 2i + 3j + k, of B is
5j + 4k and of C is i + 2j + 12k. Show that ABC is a
triangle.
Solve 5cos23θ = 3(1 + sin3θ) for 0° < θ < 90°.
(Ans: θ = 7.859°, 52.141°, 90°)
6. If y = (x – 0.5)e2x, find
dy
.
dx
(Ans: 2 xe2 x ,
(ii) On the same Argand diagram, plot Z and
2Z 3i
(b) What are the greatest and least values of |Z| if
|Z – 4| ≤ 3.
(Ans: 7, 1 respectively)
10. Given the equation x3 + x – 10 = 0,
(a) Show that x = 2 is a root of the equation.
(b) deduce the values of α + β and αβ where α and
β are the roots of the equation. Hence form a
quadratic equation whose roots are α2 and β2.
(Ans: x2 + 6x + 25 = 0)
11. (a) Find the point of intersection of the lines
x 5 y 7 z 3 and x 8 y 4 z 5 .
7
1
3
4
4
5
(Ans: (1, 3, 2)
(b) The equations of a line and a plane are
x2 y 2 z 3
and 2x + y + 4z = 9
1
2
2
respectively. P is a point on the line where x =
3. N is the foot of the perpendicular from point
P to the plane. Find the coordinates of N.
(Ans: N(1, 3, 1)
12. (a) Find the equation of the tangent to the
hyperbola whose points are of the parametric
form (2t, 2/t). (Ans: t2y = -x + 4t)
(b)(i) Find the equation of the tangents in (a) which
are parallel to y + 4x = 0
(Ans: y = -4x – 8, y = -4x + 8)
(ii) Determine the distance between the tangents
in (i). (Ans: 3.881 units)
2
13. A curve has the equation y
.
1 x2
(a) Determine the nature of the turning point on the
curve. (Ans: (0, 2), maximum)
(b) Find the equation of the asymptote. Hence
sketch the curve.
(Ans: y = 0, horizontal asymptote)
tan A tan B
14. (a) Prove that tan( A B)
1 tan A tan B
1
Hence determine
2x
xe dx ,
(Ans: 2.0973)
0
7. The region bounded by the curve y = cos x, the yaxis and x-axis from x = 0 to x
2
is rotated
about the x-axis. Find the volume of the solid
2
formed.
(Ans: 4 cubic units)
8. Solve (1 – x2)
dy
– xy2 = 0, given that y = 1 when
dx
1
)
1
ln(1 x 2 ) 2 1
9. (a) The complex number is the conjugate of Z.
(i) Express Z in the modulus argument form
(Ans: Z = 2(cos 30° + i sin 30°)
Hence show that
1 tan15
1
1 tan15
3
(b) Given that cos A = 3 5 and cos B = 12 3
where A and B are acute, find the value of:
(i) tan(A + B)
(Ans: 3.9375)
(ii) cos(A + B)
(Ans: 1.0317)
x3 5 x 2 6 x 6
into partial fractions.
( x 1)2 ( x 2 2)
dy
.
Hence find y dx and
dx
15. Resolve y
x = 0 (Ans: y
- 432 -
1
2
4
,
2
2
x 1 ( x 1)
x 1
2
4
ln( x 1)
tan 1 ( x2 ) c ,
x 1
2
1
4
8x
( x 1)2 ( x 1)3 ( x 2 2)2
dp
16. (a) The differential equation
= kp(c – p) shows
dt
the rate at which information flows in a student
population c. p represents the number who have
heard the information in t days and k is a constant.
(a) Solve the differential equation
1 p
(Ans: ln
kt a )
c c p
(b) A school has a population of 1000 students.
Initially, 20 students had heard the information.
A day later, 50 students had heard the
information. How many students heard the
information by the tenth day? (Ans: 990)
Ans:
2013 PAPER II
1. A class performed an experiment to estimate the
diameter of a circular object. A sample of five
students had the following results in centimetres;
3.12, 3.16, 2.94 and 3.0. Determine the sample:
(a) mean` (Ans: 3.11)
(b) standard deviation
(Ans: 0.1356)
2. The table below shows the values of a function f(x)
x
1.8
2.0
2.2
2.4
f(x) 0.532 0.484 0.436 0.384
Use linear interpolation to find the value of:
(a) f(2.08). (Ans: 0.465 (3 dp))
(b) x corresponding to f(x) = 0.5 (Ans: 1.933)
3. The speed of a taxi increased from 90 kmh-1 to
18kmh-1 in a distance of 120 metres. Find the speed
of the taxi when it had covered a distance of 50
metres. (Ans: 69.7137 kmh-1)
4. Events A and B are such that P(A∩B) =
P(A/B) =
1
and
2
1
. Find P(B∩A'). (Ans: 1/6)
3
2
5. Find the approximate value of
1
1 x2 dx using thr
0
trapezium rule with 6 ordinates. Give your answer to
3 decimal places. (Ans: 1.105 (3 dp))
6. Forces of 7N and 4N act away from a common point
and make an angle of θ° with each other. Given that
the magnitude of their resultant is 10.75N, find the:
(a) value of 𝜃 (Ans: 25.4578°)
(b) direction of the resultant (Ans: 9.2056°)
7. An industry manufactures iron sheets of mean length
3.0 m and standard deviation of 0.05 m. Given that
the lengths are normally distributed, find the
probability that the length of any iron sheet picked at
random will be between 2.95 m and 3.15m.
(Ans: 0.8400)
8. A particle of mass m kg is released at rest from the
highest point of a solid spherical object of radius a
metres. Find the angle to the vertical at which the
particle leaves the sphere. (Ans: 48.19°)
9. The heights (cm) and ages (years) of a random
sample of 10 farmers are given in the table below.
Height(cm) 156 151 152 160 146 157 149 142 158 140
Age(years) 47 38 44 55 46 49 45 30 45 30
(a) (i) Calculate the rank correlation coefficient
(Ans: 0.7515)
(ii) comment on your result (Ans: high positive)
(b) Plot a scatter diagram for the data.
(c) Use your diagram in (b) to find:
(i) y when x = 147 (Ans: y = 37)
(ii) x when y = 43. (Ans: x = 151)
10. A mass of 12 kg rests on a smooth inclined plane
which is 6 m long and 1 m high. The mass is
connected by a light inextensible string, which
passes over a smooth pulley fixed at the top of the
plane, to a mass of 4 kg which is hanging freely.
With the string taut, the system is released from
rest.
(a) Find the
(i) acceleration of the system.
(Ans: 1.225 ms-2)
(ii)
(Ans: 38.2N)
(b) Determine the:
(i) velocity with which the 4 kg mass hits the
ground (Ans: 1.5652 ms-1)
(ii) time the 4kg mass takes to hit the ground.
(Ans: 1.3 s)
11. The probability density function (pdf) of a
continuus random variable X is given by:
kx(16 x 2 , 0 x 4
f ( x)
, else where
0
Where k is a constant.
Find: (a) value of k (Ans: k = 1/64)
(b) mode of X. (Ans: 2.3099)
(c) mean of X (Ans: 2.1333)
12. (a) Particles of masses 3 kg, 2 kg, 3 kg and 2 kg act
at points with position vectors 3i – j, 2i + 3j, -2i +
5j and –i – 2j respectively. Find the position
vector of their centre of gravity. (Ans: 0.917i + j)
(b) The figure ABCD below shows a metal sheet of
uniform material cut in the shape of a trapezium.
AB = x, CD = y, AF = a, EB = b and h is the
vertical distance between AB and CD.
y
D
h
A
C
h
b
a
F
E
B
x
- 433 -
Show that the centre of gravity of the sheet is at a
h 3y a b
distance
from side AB.
3 x y
13. The numbers x and y are measured with possible
errors ∆x and ∆y respectively.
(a) Show that the maximum absolute error in the
quotient
3. Given that 2A – cos2B = -p and sin 2A – sin2B = q,
prove that sec( A B) 1 p 2 q 2 .
q
e tan x
4. Differentiate log5 2 with respect to x.
sin x
| y || x | | x || y |
x
is given by
y
y2
(b) Find the interval within which the exact value of
2.58
is expected to lie. (Ans: (0.7464, 0.7716)
3.4
14. A particle is projected with a speed of 36ms-1 at an
angle of 40° to the horizontal from a point 0.5m
above the level ground. It just clears a wall which is
70 metres on the horizontal plane from the point of
projection. Find the:
(a) (i) time taken for the particale to reach the wall.
(Ans: 2.5384 s)
(ii) height of the wall. (Ans: 27.664 m)
(b) Maximum height reached by the particle from
the point of projection. (Ans: 27.320)
15. (a)
Show that the iterative formula based on Newton
Raphson’s method for solving the equation
lnx + x – 2 = 0 is given by
x (3 lnxn )
, n = 0, 1, 2,
xn 1 n
1 xn
(b) (i) Construct a flow chart that:
- reads the initial approximation as r
- computes using the iterative formula in (a),
and prints the roots of the equation lnx + x – 2
= 0 when the error is less than 1 × 10-4.
(Ans: root = 1.557 (3 dp))
(ii) Perform a dry run of the flow chart when r = 1.6.
16. A research station supplies three varieties of seeds S1, S2
and S3 in the ratio 4:2:1. The probabilities of
germination of S1, S2, and S3 are 50%, 60% and 80%
respectively.
(a) Find the probability that a seed selected at random
will germinate. (Ans: 0.5714)
(b) Given that 150 seeds are selected at random, find
the probability that less than 90 of the seeds will
germinate. Give your answer to two decimal
places. (Ans: 0.73 (2 dp))
(Ans:
dy
1
(sec2 x 2 cot x) )
dx ln 5
5. Find the equation of a line through S(1, 0, 2) and
T(3, 2, 1) in form r = a + λb. Hence deduce the
Cartesian equation of the line.
1
2
(Ans: r 0 2 , x 1 y z 2 )
2
2
1
2
1
6. Solve the equation 2 x 3 x 1 x 2 . Verify
your answer. (Ans: x = 3)
7. Find
x(1 x
2
1
) 2 dx . (Ans:
3
1
(1 x 2 ) 2
3
9.
(a) Given that the complex number Z and its
conjugate Z satisfy the equation
Z Z + 2iZ = 12 + 6i, find Z.
(Ans: Z = 3 + 3i, Z = 3 – i)
(b) One root of the equation Z3 – 3Z2 – 9Z + 13 = 0 is
2 + 3i. Determine the other roots. (Ans: No root)
10. A circle is described by the equation x2 + y2 – 4x – 8y
+ 16 = 0. A line given by the equation y = 2(x – 1)
cuts the circle at points A and B. A point P(x, y)
moves such that its distance from the midpoint of AB
is equal to its distance from the centre of the circle.
(a) Calculate the coordinates of A and B.
(Ans: A(2, 2), B(3.6, 5.2))
(b) Determine the centre and radius of the circle.
(Ans: centre = (2, 4); radius = 2)
(b) Find the locus of P. (Ans: y = 2x – 1)
11. (a) Differentiate y = cot-1(ln x) with respect to x
2014 PAPER I
2. A focal chord PQ to the parabola y2 = 4x has
gradient m = 1. Find the coordinates of the midpoint
of PQ. (Ans: (3, 2))
c)
8. A cylinder has radius r and height 8r. The radius
increases from 4cm to 4.1cm. Find the approximate
increase in the volume. (use 𝜋 = 3.14).
(Ans: 120.576 cm3)
(Ans:
1. Solve the simultaneous equations:
x – 2y – 2z = 0
2z + 3y + z = 1
3x – y – 3z = 3 (Ans: z = -2, y = 3, z = -4)
1
dy
)
dx x(1 (ln x) 2 )
(b) Evaluate
x sin x dx
(Ans: 2.7992)
3
12. (a) Find the Cartesian equation of the plane through
the points whose position vectors are 2i + 2j + 3k,
3i + j + 2k and -2j + 4k. (Ans: 5x – y + 67 = 26)
(b) Determine the angle between the plane in (a) and
the line
x2
y
z 5 . (Ans: 33.66°)
4
2
- 434 -
13. (a) Find the first three terms of the expansion (2 – x)6
and use it to find (1.998)6 correct to two decimal
places. (Ans: 64 – 192x + 240x2, 63.62 (2 dp))
(b) Expand (1 – 3x + 2x2)5 in ascending powers of x
as far as the x2 term. (Ans: 1 – 15x + 100x2)
14. (a) Find the equation of a normal to a curve whose
parametric equations are x = b sec2θ, y = b tan2θ.
(Ans: y + x = b(tan2θ + sec2θ))
(b) The area enclosed by the curve x2 + y2 = a2, the yaxis and the line y = 1 a is rotated through 90°
2
about the y-axis. Find the volume of the solid
generated. (Ans: V 965 a3 or V 329 a3 )
15. Solve:
(a) 4sin2θ – 12 sin 2θ + 35cos2θ = 0 for 0°≤θ ≤ 90°
(Ans: θ = 68.1986°, θ = 74.0546°)
(b) 3cos θ – 2sin θ = 2, for 0° ≤ θ ≤ 360°
(Ans: θ = 22.6199°, 270°)
16. A substance loses mass at a rate proportional to the
amount M present at time t.
(a) Form a differential equation connecting M, t and
the constant of proportionality k.
(Ans: ln M = -kt + c.)
(b) If initially the mass of the substance is Mo, show
that M = MOe-kt.
(c) Given that half of the substance is lost in 1600
years, determine the number of years 15g of the
substance would take to reduce to 13.6g
(Ans: 226.1694 years)
2014 PAPER II
1. The daily number of patients visiting a certain
hospital is uniformly distributed between 150 and
210.
(a) Write down the probability distribution function
(pdf) of the number of patients.
1
, 50 x 210
(Ans f ( x) 60
)
0, elsewhere
(b) Find the probability that between 170 and 194
patients visit the hospital on a particular day.
(Ans: 0.4)
2. A particle starts from rest at the origin (0, 0). Its
acceleration in ms-2 at time t seconds is given by
a = 6ti – 4j. Find its speed when t = 2 seconds.
(Ans: 14.4222 ms-1)
3. Use the trapezium rule with four sub-intervals to
1.0
estimate 2 x 1 dx, correct to two decimal places.
x
0.2
2
3
5. Forces of 1 N, N, 9 N and 5 N act at the
6
4
1
3
points having position vectors (3i – j)m, (2i + 2j)m,
(-i – j)m and (-3i – 4j)m respectively. Show that the
forces reduce to a couple.
6. Given the table below:
x
0
10
20
30
y
6.6
2.9
-0.1
-2.9
use linear interpolation to find:
(d) y when x = 16 (Ans: y = 1.1)
(e) x when y = -1 (Ans: x = 23.2)
7. The table below shows the scores of students in
Mathematics and English tests.
Maths
72 65 82 54 32 74 40 60
English 58 50 86 35 76 43 40 60
Calculate the rank correlation coefficient for the
students’ performance in the two subjects.
(Ans: 0.143)
8. A bullet of mass 50 grammes travelling horizontally
at 80 ms-1 hits a block of wood of mass 10 kg resting
on a smooth horizontal plane. If the bullet emerges
with a speed of 50 ms-1, find the speed with which
the block moves. (Ans: 0.15 ms-1)
9. (a) A bag contains 30 white (W), 20 blue (B) and 20
red (R) balls. 3 balls are drawn at random one after
the other without replacement. Determine the
probability that the first ball is white and the third
ball is also white. (Ans: 0.18012)
4
(b) Events A and B are such that P(A)= ,
7
1
5
P(A∩B') = , and P( A / B) . Find
3
14
(i) P(B) (Ans: 2/3)
(ii) P(A'∩B') (Ans: 0)
10. The diagram below shows two pulleys of mass 8kg
and 12kg connected by a light inextensible string
hanging over a fixed pulley.
1
(Ans: 2.20 (2 dp))
4. Tom’s chance of passing an examination is 2 3 . If he
sits for four examinations, find the probability that he
passes:
(a) Only two examinations (Ans: 0.2963 (Calc))
(b) More than half of the examinations
(Ans: 0.5926 (Calc))
8 kg
4kg
12kg
12kg 3kg
mkg
The accelerations of 4kg and 12kg masses are
and
g
2
g
2
downwards respectively. The accelerations of
the 3kg and m kg masses are g 3 upwards and g 3
downwards respectively. The hanging portions of the
- 435 -
strings are vertical. Given that the string of the fixed
pulley stationary, find the:
(a) tensions in the strings,
(Ans: 98N, T1 = 58.8 N, T2 = 39.2N)
(b) value of m. (Ans: m = 6 kg)
11. The numbers x and y are approximated with possible
errors ∆x and ∆y respectively.
(a) Show that the maximum absolute error in the
quotient
y x x y
x
is given by
y
y2
(b) Given x = 2.68 and y = 0.9 are rounded to the
given number of decimal places, find the
interval within which the exact value of
x
is
y
expected to lie. (Ans: (2.8158, 3.1588)
12. A lorry starts from point A and moves along a
straight horizontal road with constant acceleration of
2 ms-2. At the same time a car moving with a speed
of 20 ms-1 and constant acceleration of 3 ms-2 is
400m behind the point A and moving in the same
direction as the lorry. Find:
(a) how far from A the car overtakes the lorry.
(Ans: 214.3589 m)
(b) the speed of the lorry when it is being overtaken
(Ans: 29.282 ms-1)
13. The cumulative frequency table below shows the ages
in years of employees of a certain company.
Age(years
Cumulative frequency
< 15
0
< 20
17
< 30
39
< 40
69
< 50
87
< 60
92
< 65
98
(a) (i) Use the data in the table to draw a cumulative
frequency curve (Ogive)
(ii) Use the curve to estimate the semi-interquartile
range. (Ans: 9.5)
(b) Calculate the mean age of the employees.
14. (a) Show that the Newton-Raphson formula for finding
the root of the equation x N
4 xn N
5
1
5
is given by
, n = 0, 1, 2, …
5 xn 4
(b) Construct a flow chart that:
(i) reads N and the first approximation xo,
(ii) computes the root to three decimal places,
(iii) prints the root (xn) and number of iterations
(n)
(c) Taking N = 50, x0 = 2.2, perform a dry run for the
flow chart. Give your root correct to three decimal
places. (Ans: 2.187 (3 dp))
15. (a) A non-uniform plan AB of length 4 metres rests in
horizontal position on vertical supports A and B. The
centre of gravity is at 1.5m from A. The reaction at B is
37.5N. Determine the:
(i) mass of the plank (Ans: 10.2 kg)
(ii) reaction at A.
(Ans: 62.5 N)
xn 1
(b) Find the coordinates of the centre of gravity of a
uniform lamina bound by the curve y2 = 2x and the
line x = 4.
(Ans: (2.4, 0)
16. The marks in an examination were normally distributed
with mean μ and standard deviation σ. 20% of the
students scored less than 40 marks and 10% scored
more than 75 marks. Find the:
(a) values of μ and σ (Ans: μ = 53.875, σ = 16.478)
(b) percentage of candidates who scored more than 50
marks. (Ans: 59.29%)
2015 PAPER ONE
1. The first term of an Arithmetic Progression (A.P) is
equal to the first term of the Geometric Progression
(G.P) whose common ratio is 1 3 and sum to infinity
is 9. If the common difference of the A.P is 2, find
the sum of the first ten terms of the A.P. (Ans: S10 =
150)
2. Find the equation of the line through the point
(5, 3) and perpendicular to the line 2x – y + 4 = 0
(Ans: 2y + x = 11)
3. Solve for x in log a ( x 3)
1
2log a 2
log x a
(Ans: x =1)
4. Given that D(7, 1, 2), E(3, -1, 4) and F(4, -2, 5) are
points on a plane, show that ED is perpendicular to
EF.
5. In a triangle ABC all angles are acute. Angle ABC =
50°, a = 10cm and b = 9 cm. Solve the triangle.
(Ans: A = 58.34°, B – 71.66°, c = 11.15 cm)
6. Differentiate e x x3 sin x with respect to x.
3
(Ans: x2e x (2 x2 sin x 3sin x x cos x) )
7. The region enclosed by the curve y = x2, the x-axis
and the line x = 2 is rotated through one revolution
about the x-axis. Find the volume of the solid
32
20.1062)
generated. (Ans:
5
dy
e x y given that y = 2 when x = 0.
8. Solve
dx
(Ans: ex – 1.135)
2
9. (a) Given that f(x) = (x – α) g(x), show that f '(x) is
divisible by 𝛼.
(b) A polynomial P(x) = x3 + 4ax2 + bx + 3 is
divisible by (x – 1)2. Use the result in (a) above to
find the values of a and b. Hence solve the
equation P(x) = 0 (Ans: a 1 4 , b = -5)
10. Sketch on the same coordinate axes the graphs of the
curve y = 2 + x – x2 and the line y = x + 1. Hence
determine the area of the region enclosed between the
curve and the line. (Ans: 1.333 sq units)
11. (a) Solve Z Z 5iZ 5(9 7i) where Z is the
complex conjugate of Z.
(Ans: Z = 7 – 4i OR Z = 7 – i)
(b)(i) Find the Cartesian equation of the curve
given as |Z + 2 – 3i| = 2|Z – 2 + i|
(Ans: 3x2 + 3y2 – 20x + 14y + 7 = 0)
2
- 436 -
(ii) Show that it represents a circle. Find the
centre and radius of the circle.
(Ans: Centre (10 3 , 7 3 ) ; radius = 3.771)
cos 3 cos 5
12. (a) Simplify
(Ans: cot θ)
sin 5 sin 3
(b) Show that cot 2
1
13. Express
evaluate
x 2 ( x 1)
3
dx
x2 ( x 1)
1 tan 2
2 tan
as
partial
4.
5.
fractions.
Hence
correct to 3 decimal places.
6.
2
(Ans:
1
1 1
2
, = 0.12102)
x x
x 1
3
1
2
2
14. (a) Show that the lines a = 4 1 and b =
5
1
0 1 intersect
2
2
(b) Find the:
(i) point of intersection, P of the lines in (a)
(Ans: (2, -3, 4)
(ii) Cartesian equation of the plane which
contains a and b. (Ans: 2x + z = 8)
15. The tangent at the point P(cp, c/p) and Q(cq, c/q)
on the rectangular hyperbola xy = c2 intersect at R.
Given that R lies on the curve xy
2015 PAPER TWO
Find the magnitude and direction of the resultant
3
of the forces N,
1
2.
4
N and
2
1
N
2
(Ans: 5.83N, 30.96°)
Use the trapezium rule with five subintervals to
4
estimate
5
( x 1)dx . Give your answer correct to
2
3.
8.
c2
, show that
2
the locus of the midpoint of the line PQ is given by
xy = 2c2.
16. The rate of increase of a population of certain birds
is proportional to the number in the population
present at that time. Initially, the number in the
population was 32,000. After 70 years, the
population was48,000. Find the:
(a) number of birds in the population after 82 years.
(Ans: 51455)
(b)time when the population doubles the initial
number. (Ans: 119.67 years)
1.
7.
three decimal places. (Ans: 2.558 3 dp)
The table below shows the mass of boys in in a
certain school.
Mass (kg)
15 20 25 30 35
9.
Number of boys
5
6 10 20 9
Calculate the mean. (Ans: 27.2 kg)
Two cyclists A and B are 36m apart on a straight
road. Cyclist B starts from rest with an
acceleration of 6ms-2 while A is in pursuit of B
with velocity of 20 m/s and acceleration of 4 ms2
. Find the time when A overtakes B.(Ans: 2 s)
Events A and B are independent. P(A) = x, P(B) =
x + 0.2 and P(A∪B) = 0.65. find the value of x.
(Ans: x = 0.3)
The table below shows the values of a function
f(x) for a given value of x.
x
f(x)
9
2.66
10
2.42
11
2.18
12
1.92
Use linear interpolation or extrapolation to find:
(a) f(10.4)
(Ans: 2.324)
(b) the value of x corresponding to f(x) = 1.46
(Ans: 13.769 N)
The marks in an examination were found to be
normally distributed with mean 53.9 and
standard deviation 16.5. 20% of the candidates
who sat this examination failed. Find the pass
mark for the examination. (Ans: 40%)
A fixed hollow hemisphere has centre O and is
fixed so that the plane of the rim is horizontal. A
particle A of weight 30 2 N can move on the
inside surface of the hemisphere. The particle is
acted upon by a horizontal P, whose line of
action is in a vertical plane through O and A. OA
makes an angle of 45° with the vertical. If the
coefficient of friction between the particle and
the hemisphere is 0.5 and the particle is just
about to clip downwards, find the:
(a) normal reaction. (Ans: 40N)
(b) value of P.
(Ans:P = 14.1421 N)
The probability density function (p.d.f) of a
random
variable
Y
is
given
by
( y 1)
,
f ( x) 4
0 ,
0 yk
elsewhere
Find:
(a) the value of k
(Ans: k = 2)
(b) the expectation of Y
(Ans: E(Y) = 7/6)
(c) P(1 ≤ Y ≤ 1.5)
(Ans: 0.28125)
10. The numbers A = 6.341 and B = 2.6 have been
rounded to the given number of decimal places.
(a) Find the maximum possible error in AB.
(Ans: 0.31835)
(b) Determine the interval within which
A2
can
B
be expected to lie.
(Ans: 15.171 ≤ exact < 15.770)
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11. The diagram below shows a 12kg mass on a
horizontal rough plane. The 6kg and 4kg masses
are on rough planes inclined at angles 60° and
30° respectively. The masses are connected to
each other by light inextensible strings passing
over light smooth fixed pulleys A and B.
12kg
C
B
position of the centre of gravity of the lamina from the
side OA.
O
D
10 cm
20 cm
C
4kg
6kg
10 cm
60°
30°
The planes are equally rough with coefficient of
1
. If the system is released from rest, the:
2
(a) acceleration of the system. (Ans: 0.73833 ms-2)
(b) extension in the string. (Ans: 25.3823 N)
The table below gives the points awarded to eight
schools by three judges J1, J2 and J3 during a music
competition. J1 was the chief judge.
J1
J2
J3
72
60
50
50
55
40
50
70
62
55
50
70
35
50
40
38
50
48
82
73
67
72
70
67
(a) Determine the rank correlation coefficient
between the judges of
(i) J1 and J2. (Ans: 0.7440)
(ii) J1 and J3. (Ans: 0.7023)
(b) Who of the two other judges had a better
correlation with the chief judge? Give a reason
(Ans: J2 had a better correlation because it had a
slightly higher value with the chief judge.)
A ball is projected from a point A and falls at a point
B which is in level with A and at a distance of 160m
from A. The greatest height of the ball attained is
40m. Find the:
(a) angle and velocity at which the ball is projected.
(Ans: 45°, 39.6 ms-1)
(b) time taken for the ball to attain its greatest height.
(Ans: 2.857 s)
(a) Show that the iterative formula based on Newton
Raphson’s method for approximating the root of the
equation 2 ln x – 1 + 1 = 0 is given by
2 ln xn 1
xn 1 xn
, n = 0, 1, 2, …
xn 1
(b) Draw a flow chart that:
(i) reads the initial approximation xo of the root.
(ii) computes and prints the root correct to 2
decimal places, using the formula in (a).
(Ans: root = 3.51 (2 dp))
The diagram below represents a lamina formed by
welding together a rectangular metal. Find the
friction
12.
13.
14.
15.
A
B
4 cm
(Ans: 3.3576 cm from OA)
16. A box A contains 4 white and 2 red balls. Another box
contains 3 white and 3 red balls. A box is selected at
random at two balls are picked one after the other
without replacement.
(a) Find the probability that the two balls picked are
red.
(Ans: 0.1333)
(b) Given that two white balls are picked, what is the
probability that they are from box B?(Ans: 0.333)
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Eneku Ronald