Instructions to candidates
 Write your name and admission number in the spaces provided above.
 This paper consists of two sections: Section I and Section II.
 Answer all the questions in Section I and only ﬁve questions from Section II.
 Show all the steps in your calculations, giving your answers at each stage in the spaces provided below each question.
 Marks may be given for correct working even if the answer is wrong.
 Nonprogrammable silent electronic calculators and KNEC Mathematical tables may be used, except where stated otherwise.
QUESTIONS
SECTION 1 (50 marks) Answer all questions in this section
 Use logarithms to evaluate , (4marks)
 Make d the subject of the formula (3marks)
 Simplify the following surds leaving your answer in the form a+ b√c (3marks)

 Expand the binomial expression up to the third term. (1mark)
 Use the expansion above (where x > 1) to estimate the value of (99)^{4} to 3 s.f. (2marks)
 A(3,2) and B(7,4) are points on the circumference of a circle. Given that chord AB passes through the centre of the circle determine the equation of the circle. (4marks)
 Without using logarithms tables of calculator: Evaluate. (3marks)
 Solve for x given that the following is a singular matrix. (3mks)
 The sides of a triangle were measured and recorded as 8.4 cm, 10.5 cm and 15.32 cm . Calculate the percentage error in it’s perimeter 2d.p. (3marks)
 Given that 64, b, 4…. are in continued proportion, find the value of b. (3marks)
 The figure below shows a circle centre O. AB and PQ are chords intersecting externally at a point C. AB=9cm, PQ=5cm and Qc =4cm, find the length BC.(3marks)
 Two variables x and y are such that y varies directly as x^{n} where n is a constant. Given that y=320 when x=16 and y = 2560 when x = 64. Find the value of n. (3marks)
 A man sold a motor cycle at 84000. The rate of depreciation was 5% per annum. Calculate the value of the motor cycle after 3 years to 1d.p. (3marks)
 Vector r has a magnitude of 14 and is parallel to vector s. Given that s = 6i – 2j +3k, express vector r in terms of i, j and k. (3marks)
 Solve for x in the range 0 ≤ x ≤ 360º
If 2sin^{2}x + sin x – 1=0 (4marks)  The prefects body of a certain school consists of 7 boys and 5 girls. Three prefects are to be chosen at random to represent the school at a certain function at Nairobi. Find the probability that the chosen prefects are boys. (2mks)
 A trigonometric function is given as (4mks)
y = 0.5 cos (2x – 40)º
Determine Amplitude
 Period
 Phase angle
SECTION B(50 MARKS)
Answer any five questions from this section in the spaces provided.


 Taking the radius of the earth, R=6370km and π=^{22}/_{7}, calculate the shortest distance between two cities P(600N, 290W) and Q(600N, 310E) along the parallel of latitude. (3marks)
 If it is 1200hrs at P, what is the local time at Q (3marks)
 An aeroplane flew due south from a point A(60ºN, 45ºE) to a point B, the distance covered by the aeroplane was 8000km, determine the position of B. (4marks)

 The diagram below shows a square based pyramid V vertically above the middle of the base. PQ=10cm and VR=13cm. M is the midpoint of VR.
Find
 the length PR. (2marks)
 the height of the pyramid (2marks)

 the angle between VR and the base PQRS (2marks)
 the angle between MR and the base PQRS (2marks)
 the angle between the planes QVR and PQRS. (2marks)

 Complete the following table for the equation
 y = 2x^{3} + 3x^{2} –6x –4 for the values –3≤ x ≤ 2 (2marks)
x
3
2
1
0
1
2
2x^{3}
16
0
2
16
3x^{2}
27
3
0
12
6x
12
0
12
4
4
4
4
4
4
4
y
4
4
12
 On the grid provided draw the graph of y=2x^{3} + 3x^{2} – 6x – 4 (3marks)
 By drawing a suitable straight lines use your graph to solve the equations
 2x^{3} + 3x^{2} – 4x – 2 =0 (2marks)
 2x^{3} + 3x^{2} – 6x – 4 =0 (3marks)
 y = 2x^{3} + 3x^{2} –6x –4 for the values –3≤ x ≤ 2 (2marks)
 The diagram below shows a triangle OPQ in which M and N are points on OQ and PQ respectively such that OM= ^{2}/_{3} OQ and PN = ^{1}/_{4}PQ. Lines PM and ON meets at X.
 Given that OP = p and OQ= q express in term of p and q the vectors.
 PQ (1mark)
 PM (1marks)
 ON (1marks)
 You are further given that OX=KON and PX=hPM.
 Express OX in terms of P and q in two different ways. (2marks)
 Find the value of h and K. (4marks)
 Find the ratio PX:XM (1mark)
 Given that OP = p and OQ= q express in term of p and q the vectors.
 In the figure below , O is the centre of the circle.PQR is a tangent to the circle at Q. Angle PQS=28º, angle UTQ=54º and UT=TQ
Giving reasons, determine the size of Angle STQ (2mks)
 Angle TQU (2mks)
 Angle TQS (2mks)
 Reflex angle UOQ (2mks)
 Mr. Kimutai a teacher from Tuiyotich Secondary School earns K£12000 per annum and lives in a house provided by the employer at a minimum rent of Ksh2000 per month. He gets a family relief of K£1320p.a and is entitled to a relief of 10% of his insurance of K£800p.a.
 Calculate his annual tax bill based on the table below. (6mks)
Income slab in k£p.aRate
1 – 2100 10%
2101 – 4200 15%
4201 – 6300 25%
6301 – 8400 35%
Over 8400 45%  Kimutai other deductions include.
 W.C.P.S = sh600.00pm
 NHIF = sh500.00pm
Calculate Kimutai’s net salary monthly. (4mks)
 Calculate his annual tax bill based on the table below. (6mks)

 Use the midordinate rule with five strips to estimate the area bounded by the curve y = x^{2} +1, the x–axis, lines x=1 and x=6 (4mks)
 Find the exact area of the region in (a) above (3mks)
 Calculate the percentage error in area when midordinate rule is used. (3mks)
 An arithmetic progression AP has the first term a and the common difference d.
 Write down the third, ninth and twenty fifth terms of the AP in terms of a and d. (2mks)
 The AP above is increasing and the third, ninth and twenty fifth terms form the first three consecutive terms of a geometric progression (G.P). The sum of the seventh and twice the sixth term of AP is 78. Calculate
 The first term and common difference of the A.P (5mks)
 The sum of the first 5 terms of the G.P (3mks)
MARKING SCHEME
SECTION 1 (50 marks) Answer all questions in this section
 Use logarithms to evaluate , (4marks)
No Log 24.36 1.3867 0.066547 2.8231 0.2098 1.48^{2} 0.1703 x 2 0.3406 1.8692
39.045 x 10^{1} 1.9564 0.9045  Make d the subject of the formula (3marks)
a^{4} = 1 d^{2}  b
b^{2} 3
1  d^{2} = a^{4} + ^{b}/_{3}
b^{2}
1  d^{2} = a^{4}b^{2} + b^{3}
3
d2 = 1  a^{4}b^{2} + b
3
d = ±√1  a^{4}b^{2} + b
3  Simplify the following surds leaving your answer in the form a+ b√c (3marks)
√5(2√5 + √5) + √2(2√2  √5)
(2√2)^{2}  (√5)^{2}
=2√10 + 5 + 4  √10
8  5
= 9 + √10
3 
 Expand the binomial expression up to the third term. (1mark)
x^{4}  4x^{2} + 6 + .......................  Use the expansion above (where x > 1) to estimate the value of (99)^{4} to 3 s.f. (2marks)
(9.9)^{4} = (10  1/10)^{4}
(x  1/x)^{4} = (10  1/10)^{4}
x = 10
(9.9)^{4} = (10)^{4}  4(10)^{2} + 6
=10000  400 + 6
=9606
 Expand the binomial expression up to the third term. (1mark)
 A(3,2) and B(7,4) are points on the circumference of a circle. Given that chord AB passes through the centre of the circle determine the equation of the circle. (4marks)
 Without using logarithms tables of calculator: Evaluate. (3marks)
 Solve for x given that the following is a singular matrix. (3mks)
(x + 3)  2x^{2} = 0
2x2  x  3 = 0
p = 6
2x  3
1
2  3
2x^{2} + 2x  3x  3 = 0
2x(x + 1)  3(x + 1) = 0
(x  1) (2x  3) = 0
x + 1 = 0or 2x  3 = 0
x = 1 2x = 3
x = 3/2
x = 1 or 1.5  The sides of a triangle were measured and recorded as 8.4 cm, 10.5 cm and 15.32 cm . Calculate the percentage error in it’s perimeter 2d.p. (3marks)
max per = 8.45 + 10.55
+15.325
=34.325
min per = 8.35 + 10.45 + 15.315
=34.115
absolute error = 34.325  34.115
2
= 0.21 = 0.105
2
actual 8.4 + 10.5 + 15.32
=34.22
% error = 0.105 x 100
34.22
= 0.3068%  Given that 64, b, 4…. are in continued proportion, find the value of b. (3marks)
64 = b
b 4
b^{2} = 256
b = 16  The figure below shows a circle centre O. AB and PQ are chords intersecting externally at a point C. AB=9cm, PQ=5cm and Qc =4cm, find the length BC.(3marks)
x(x + 9) = 4 x 9
x^{2} + 9x  36 = 0
BC = 3 cm  Two variables x and y are such that y varies directly as x^{n} where n is a constant. Given that y=320 when x=16 and y = 2560 when x = 64. Find the value of n. (3marks)
 A man sold a motor cycle at 84000. The rate of depreciation was 5% per annum. Calculate the value of the motor cycle after 3 years to 1d.p. (3marks)
A = p  (1  r/100)n
A = ??
p = 84 000
n = 3
r = 5%
A = 84000(1  5/100)^{3}
= 84000(0.95)^{3}
5% = 72019.5  Vector r has a magnitude of 14 and is parallel to vector s. Given that s = 6i – 2j +3k, express vector r in terms of i, j and k. (3marks)
 Solve for x in the range 0 ≤ x ≤ 360º
If 2sin^{2}x + sin x – 1=0 (4marks)  The prefects body of a certain school consists of 7 boys and 5 girls. Three prefects are to be chosen at random to represent the school at a certain function at Nairobi. Find the probability that the chosen prefects are boys. (2mks)
p = 7/12 x 6/11 x 5/10
= 7/22  A trigonometric function is given as (4mks)
y = 0.5 cos (2x – 40)º
Determine Amplitude
0.5  Period
360 = 180
2  Phase angle
40º
 Amplitude
SECTION B(50 MARKS)
Answer any five questions from this section in the spaces provided.


 Taking the radius of the earth, R=6370km and π=^{22}/_{7}, calculate the shortest distance between two cities P(600N, 290W) and Q(600N, 310E) along the parallel of latitude. (3marks)
 If it is 1200hrs at P, what is the local time at Q (3marks)
θ = 60
time diff = 4θ
=4 x 60 = 240
= 4hrs
1200
4
= 0800 hrs
 Taking the radius of the earth, R=6370km and π=^{22}/_{7}, calculate the shortest distance between two cities P(600N, 290W) and Q(600N, 310E) along the parallel of latitude. (3marks)
 An aeroplane flew due south from a point A(60ºN, 45ºE) to a point B, the distance covered by the aeroplane was 8000km, determine the position of B. (4marks)
dist = θ 2πR
360
θ x 2 x 22 x 6370 = 8000
360 7
1001θ = 8000
9
θ = 71.92º
71.92º  60
=11.92º
B(11.92ºS, 45ºE)

 The diagram below shows a square based pyramid V vertically above the middle of the base. PQ=10cm and VR=13cm. M is the midpoint of VR.
Find
 the length PR. (2marks)
PR = √10^{2} + 10^{2}
=√200
=14.14 cm  the height of the pyramid (2marks)
h = √13^{2} + 7.07^{2}
=√161.93
=12.73cm
 the length PR. (2marks)

 the angle between VR and the base PQRS (2marks)
Tan θ = 12.73
5
= 2.546
θ = Tan ^{1}2.546
=68.56º  the angle between MR and the base PQRS (2marks)
cos α = 7.07
13
= 0.5438
α = 57.06º  the angle between the planes QVR and PQRS. (2marks)
Tan θ = 12.73
5
=2.546
θ = Tan ^{1}2.546
=57.06º
 the angle between VR and the base PQRS (2marks)

 Complete the following table for the equation
 y = 2x^{3} + 3x^{2} –6x –4 for the values –3≤ x ≤ 2 (2marks)
x
3
2
1
0
1
2
2x^{3}
54 16
2 0
2
16
3x^{2}
27
3
0
12
6x
18 12
6 0
12
4
4
4
4
4
4
4
y
13 4
3 4
5 12
 On the grid provided draw the graph of y=2x^{3} + 3x^{2} – 6x – 4 (3marks)
 By drawing a suitable straight lines use your graph to solve the equations
 2x^{3} + 3x^{2} – 4x – 2 =0 (2marks)
x = 2.2 or  0.5 or 1.1  2x^{3} + 3x^{2} – 6x – 4 =0 (3marks)
y = 2x^{3} + 3x^{2} – 6x – 4
0 = 2x^{3} + 3x^{2} – 6x – 4
y = 0
x = 2.35 or 1.3 0r  0.6
 2x^{3} + 3x^{2} – 4x – 2 =0 (2marks)
 y = 2x^{3} + 3x^{2} –6x –4 for the values –3≤ x ≤ 2 (2marks)
 The diagram below shows a triangle OPQ in which M and N are points on OQ and PQ respectively such that OM= ^{2}/_{3} OQ and PN = ^{1}/_{4}PQ. Lines PM and ON meets at X.
 Given that OP = p and OQ= q express in term of p and q the vectors.
 PQ (1mark)
= q  p  PM (1marks)
2/3q  p  ON (1marks)
= 3/4p + 1/4q
 PQ (1mark)
 You are further given that OX=KON and PX=hPM.
 Express OX in terms of P and q in two different ways. (2marks)
ox = (1  h)p + 2/3hq  Find the value of h and K. (4marks)
k = 8/9  Find the ratio PX:XM (1mark)
=1:2
 Express OX in terms of P and q in two different ways. (2marks)
 Given that OP = p and OQ= q express in term of p and q the vectors.
 In the figure below , O is the centre of the circle.PQR is a tangent to the circle at Q. Angle PQS=28º, angle UTQ=54º and UT=TQ
Giving reasons, determine the size of Angle STQ (2mks)
28º angles in alternate segment are equal  Angle TQU (2mks)
63º base angles of isosceless triangle are equal  Angle TQS (2mks)
35º angles on a staright line add up to 180º  Reflex angle UOQ (2mks)
252º angles at a point add up to 360º
 Angle STQ (2mks)
 Mr. Kimutai a teacher from Tuiyotich Secondary School earns K£12000 per annum and lives in a house provided by the employer at a minimum rent of Ksh2000 per month. He gets a family relief of K£1320p.a and is entitled to a relief of 10% of his insurance of K£800p.a.
 Calculate his annual tax bill based on the table below. (6mks)
Income slab in k£p.aRate
1 – 2100 10%
2101 – 4200 15%
4201 – 6300 25%
6301 – 8400 35%
Over 8400 45%  Kimutai other deductions include.
 W.C.P.S = sh600.00pm
 NHIF = sh500.00pm
Calculate Kimutai’s net salary monthly. (4mks)
gross income
12000 x 20 = 20000
12
total deductions
3791.67 + 600 + 500 = 4891.67
net income
= 20 000  4891.67
= 15108.33
 Calculate his annual tax bill based on the table below. (6mks)

 Use the midordinate rule with five strips to estimate the area bounded by the curve y = x^{2} +1, the x–axis, lines x=1 and x=6 (4mks)
 Find the exact area of the region in (a) above (3mks)
= 76^{2}/_{3}  Calculate the percentage error in area when midordinate rule is used. (3mks)
error =76^{2}/_{3 } 76.25
= 0.417
0.417 x 100
76^{2}/_{3}
= 0.543
 Use the midordinate rule with five strips to estimate the area bounded by the curve y = x^{2} +1, the x–axis, lines x=1 and x=6 (4mks)
 An arithmetic progression AP has the first term a and the common difference d.
 Write down the third, ninth and twenty fifth terms of the AP in terms of a and d. (2mks)
T_{3} = a + 2d
T_{a} = a + 8d
T_{25} = a + 24d  The AP above is increasing and the third, ninth and twenty fifth terms form the first three consecutive terms of a geometric progression (G.P). The sum of the seventh and twice the sixth term of AP is 78. Calculate
 The first term and common difference of the A.P (5mks)
 The sum of the first 5 terms of the G.P (3mks)
 The first term and common difference of the A.P (5mks)
 Write down the third, ninth and twenty fifth terms of the AP in terms of a and d. (2mks)
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