Instructions to candidates:
 Write your name, Index number, in the spaces provided above.
 Sign and write the date of examination in the spaces provided above.
 The paper contains two sections: Section I and Section II.
 Answer All the questions in Section I and only five questions from Section II
 All answers and working must be written on the question paper in the spaces provided below each question.
 Show all the steps in your calculations, giving your answers at each stage in the spaces below each question.
 Non – programmable silent electronic calculators and KNEC Mathematical tables may be used, except where stated otherwise.
 Candidates should answer the questions in English.
For Examiner’s Use Only
Section I
1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  Total 
Section II
17  18  19  20  21  22  23  24  Total 








QUESTIONS
SECTION I (50MARKS)
Answer ALL the questions in this section in the spaces provided.
 Use logarithm tables to evaluate (3 marks)
 Without using a calculator or mathematical table evaluate leaving your answer in simplified form. (3 marks)
 Expand up to the term in x in ascending powers of x^{3} .Hence find the value of (1.005)^{10} correct to four decimal places. (3 marks)
 Solve for x in the equation
2log_{10}x +log_{10}5 = 1 + 2log_{10}4 (3marks)  In the figure below OS is the radius of a circle centre O. Chords SQ and TU are extended to meet at P and OR is perpendicular to QS at R. OS = 61 cm, PU = 50 cm, UT = 40 and PQ = 30cm.
Calculate the length of QS (2 marks)
 OR to 2 decimal places (2 mark)
 Simplify as far as possible leaving your answer in surd form (3marks)
 In the figure below angle A=68º, B= 39º, BC= 8.4cm and CN is the bisector of angle ACB. Calculate the length CN to 1decimal place. (3 marks)
 Given that the matrix is a singular matrix, find the values of x. (3marks)
 Make x the subject of the equation (3 marks)
 The equation of the circle is given by x^{2} + y^{2} + 8x 2y 1 = 0 . Determine the radius and the centre of the circle. (4marks)
 A coffee blender mixes 6 parts of type A with 4 parts of type B. if type A cost him sh. 24 per kg and type B cost him sh. 22 per kg, at what price per kg should he sell the mixture in order to make 5% profit. Give your answer to 2 decimal places (3marks)
 Musau invested a sum of money which earned him 10% compound interest in the first year. In the second year, the investment earned him 20% compound interest and in the third year, it earned him 25% compound interest. At the end of the three years, the investment was worth sh. 11,550,000. What sum did he invest. (3marks)
 Line AB is 8cm long. On the same side of line AB draw the locus of point P such that the area of triangle APB is 12cm^{2} and angle APB=90º (3marks)
 In a class of 20 students, there are 12 boys and 8 girls. If two students from the class are chosen at random to go to trip, what is the probability that both of them are boys (3marks)
 After transformation T represented by the matrix (2 1), the triangle ABC was mapped onto triangle A_{1}B_{1}C_{1} where A_{1},B_{1,}C_{1}had coordinates (2,0), (4,0) and (4,6) respectively. Determine the coordinates A, B, and C (3marks)
 The length and breadth of a rectangular floor were measured and found to be 4.1m and 2.2m respectively. If a possible error of 0.01m was made in each of the measurements; find the:
 Maximum and minimum possible area of the floor (2marks)
 Maximum wastage in the carpet ordered to cover the whole floor. (1mark)
SECTION II (50 MARKS)
INSTRUCTIONS: Answer ANY FIVE questions only in this section

 complete the table below, giving the values correct to 2 decimal places (2mks)
X^{0}
0^{0}
15^{0}
30^{0}
45^{0}
60^{0}
75^{0}
90^{0}
105^{0}
120^{0}
135^{0}
150^{0}
165^{0}
180^{0}
Cos 2X^{0}
1.00
0.87
0.00
0.5
1.00
0.5
0.00
0.50
0.87
1.00
Sin (X^{0}+30^{0})
0.50
0.71
0.87
0.97
1.00
0.87
0.71
0.50
0.00
0.50
 Using the grid provided draw on the same axes the graph of y=cos 2Xº and y=sin(Xº+30º) for
0º≤X≤180º. (4mks)  Find the period of the curve y=cos 2xº (1mk)
 Using the graph, estimate the solutions to the equations;
 sin(Xº+30º)=cos 2Xº (1mk)
 Cos 2Xº=0.5 (1mk)
 complete the table below, giving the values correct to 2 decimal places (2mks)
 A Quantity P varies partly as the square of m and partly as n. When p= 3.8, m = 2 and n = 3, When p =  0.2, m = 3 and n= 2.
 Find
 The equation that connects p, m and n (4marks)
 The value of p when m = 10 and n = 4 (1mark)
 Express m in terms of p and n (2marks)
 If P and n are each increased by 10%, find the percentage increase in m correct to 2 decimal place. (3marks)
 Find

 The 5th term of an AP is 16 and the 12^{th} term is 37.
Find; The first term and the common difference ( 3 marks)
 The sum of the first 21 terms (2 marks)
 The second, fourth and the seventh term of an AP are the first 3 consecutive terms of a GP. If the common difference of the AP is 2.
Find:  The common ratio of the GP ( 3 marks)
 The sum of the first 8 terms of the GP (2 marks)
 The 5th term of an AP is 16 and the 12^{th} term is 37.
 The table below shows the rates of taxation in a certain year.
In that period, Juma was earning a basic salary of sh. 21,000 per month. In addition, he was entitled to a house allowance of sh. 9000 p.m. and a personal relief of ksh.1056 p.m He also has an insurance scheme for which he pays a monthly premium of sh. 2000. He is entitled to a relief on premium at 15% of the premium paid. Calculate how much income tax Juma paid per month. (7mks)
 Juma’s other deductions per month were cooperative society contributions of sh. 2000 and a loan repayment of sh. 2500. Calculate his net salary per month. (3mks)
 A cupboard has 7 white cups and 5 brown ones all identical in size and shape. There was a blackout in the town and Mrs. Kamau had to select three cups, one after the other without replacing the previous one.
 Draw a tree diagram for the information. (2mks)
 Calculate the probability that she chooses.
 Two white cups and one brown cup. (2mks)
 Two brown cups and one white cup. (2mks)
 At least one white cup. (2mks)
 Three cups of the same colour. (2mks)
 The For a sample of 100 bulbs, the time taken for each bulb to burn was recorded. The table below shows the result of the measurements.
Time(in hours)
1519
2024
2529
3034
3539
4044
4549
5054
5559
6064
6569
7074
Number of bulbs
6
10
9
5
7
11
15
13
8
7
5
4
 Using an assumed mean of 42, calculate
 the actual mean of distribution (4mks)
 the standard deviation of the distribution (3mks)
 Calculate the quartile deviation (3mks)
 Using an assumed mean of 42, calculate
 The position of town A and B on the earth’s surface are (36ºN, 49ºE) and (36ºN, 131ºW) respectively.
 Find the difference in longitude between town A and town B (2marks)
 Given that the radius of the earth is 6370km, calculate the distance between town A and B along;
 Parallel of longitude (2marks)
 A great circle (3marks)
 Another town C is 840km east of town B and on the same latitude as town A and B. find the longitude of town C (3marks)
 A trader is required to supply two types of shirts, type A and type B. the total number of shirts must not be more than 400. He has to supply more of type A than type B shirts. However the number of type A shirts must not be more than 300 and the number of type B shirts must not be less than 80. Let x be the number of type A shirts and y be the number of type B shirts.
 Write down in terms of x and y all the linear inequalities representing the information above (4marks)
 On the grid provided, draw the inequalities and shade the unwanted regions (4marks)
 The profits were as follows;
Type A: sh. 600 per shirt
Type B: sh. 400 per shirt Use the graph to determine the number shirts of each type that he should make to maximize the profit (1mark)
 Calculate the maximum possible profit (1mark)
 Write down in terms of x and y all the linear inequalities representing the information above (4marks)
MARKING SCHEME
SECTION I (50MARKS)
Answer ALL the questions in this section in the spaces provided.
 Use logarithm tables to evaluate (3 marks)
No standard form log 1.19111
3
=0.63700.4239 0.4239 × 10^{1} ¯1.6272 149.6 1.496 × 10^{2} 2.1750 1.9022 log 6 = 0.7782 7.782 × 10^{1} ¯1.8911 1.9111 4.335 4.335 × 10^{0} 0.6370  Without using a calculator or mathematical table evaluate leaving your answer in simplified form. (3 marks)
 Expand up to the term in x in ascending powers of x^{3} .Hence find the value of (1.005)^{10} correct to four decimal places. (3 marks)
 Solve for x in the equation
5log_{10}x +log_{10}5 = 1 + 2log_{10}4 (3marks)
5log_{10}x +log_{10}5 = 1 + 2log_{10}4
log_{10}5x^{5} = log_{10}160
5x^{5} = 160
x^{5} = 32
x^{5} = 2^{5} =
x = 2  In the figure below OS is the radius of a circle centre O. Chords SQ and TU are extended to meet at P and OR is perpendicular to QS at R. OS = 61 cm, PU = 50 cm, UT = 40 and PQ = 30cm.
Calculate the length of QS (2 marks)
PT.PU = PS.PQ
90 × 50 = (30 × QS) × 30
4500 = 900 + 30QS
30QS = 3600
QS = 120  OR to 2 decimal places (2 mark)
OR = √61^{2}  60^{2}
= √121
= 11.00 cm
 QS (2 marks)
 Simplify as far as possible leaving your answer in surd form (3marks)
 In the figure below angle A=68º, B= 39º, BC= 8.4cm and CN is the bisector of angle ACB. Calculate the length CN to 1decimal place. (3 marks)
∠ACN = 180  (68 + 39)
2
=365º
8.4 = x
sin68º sin39º
x = 5.701 cm  Given that the matrix is a singular matrix, find the values of x. (3marks)
x (x  1)  0 = 0
x = 0
x = 1  Make x the subject of the equation (3 marks)
(t)^{2} = b^{2}
(s) x  4
t^{2}(x  4) = s^{2}b^{2}
t^{2}x  4t = s^{2}b^{2}
t^{2}x = s^{2}b^{2} + 4t
x = s^{2}b^{2} + 4t
t^{2}  The equation of the circle is given by x^{2} + y^{2} + 8x 2y 1 = 0 . Determine the radius and the centre of the circle. (4marks)
x^{2} + 8x + 16 + y^{2}  2y + 1 = 1 + 16 + 1
(x + 4)^{2} + (y  1)^{2} = 18
center x(4,1)
radius = √18 = 4.243 units
or
r = 3√2 units  A coffee blender mixes 6 parts of type A with 4 parts of type B. if type A cost him sh. 24 per kg and type B cost him sh. 22 per kg, at what price per kg should he sell the mixture in order to make 5% profit. Give your answer to 2 decimal places (3marks)
avearage cost = total cost
total
=(24 × 6) + (22 × 4)
6 + 4
144 + 88 = 222 = 23.2
10 10
105 × 23.2 = 24.35
100  Musau invested a sum of money which earned him 10% compound interest in the first year. In the second year, the investment earned him 20% compound interest and in the third year, it earned him 25% compound interest. At the end of the three years, the investment was worth sh. 11,550,000. What sum did he invest. (3marks)
1st year
A = P(1 + r )^{n}
100
A = P(1 + 10 )^{1}
100
A = 1.1P
2nd year
A = 1.1P(1 + 20 )^{1}
100
= 1.1P(1 + 0.2)
A = 1.1P × 1.2
A = 1.32P
3rd year
A = 1.32P(1 + 25 )^{1}
100
= 1.65P = 1150000
P = sh 7 000 000  Line AB is 8cm long. On the same side of line AB draw the locus of point P such that the area of triangle APB is 12cm^{2} and angle APB=90º (3marks)
p is a point substended by chord AB to the circumference of a semicircle or circle  In a class of 20 students, there are 12 boys and 8 girls. If two students from the class are chosen at random to go to trip, what is the probability that both of them are boys (3marks)
( 12 × 11) = 132 = 33
20 19 380 95  After transformation T represented by the matrix (2 1), the triangle ABC was mapped onto triangle A_{1}B_{1}C_{1} where A_{1},B_{1,}C_{1}had coordinates (2,0), (4,0) and (4,6) respectively. Determine the coordinates A, B, and C (3marks)
A B C A B C
(2 1) ( x1 x2 x3) = (2 4 4)
(0 1) (y1 y2 y3) (0 0 6)
2x1 + y1 = 2
0 + y1 = 0
y1 = 0
x1 = 1
2x2 + y2 = 4
0 + y2 = 0
y2 = 0
x2 = 2
2x3 + y3 = 4
0 + y3 = 6
y3 = 6
2x3 = 2
x3 = 1
A(1,0)
B(2,0)
C(1,6)  The length and breadth of a rectangular floor were measured and found to be 4.1m and 2.2m respectively. If a possible error of 0.01m was made in each of the measurements; find the:
 Maximum and minimum possible area of the floor (2marks)
maximum area = 4.11 m × 2.21m = 9.0831m^{2}
minimum area = 4.09m × 2.19m = 8.9571m^{2}  Maximum wastage in the carpet ordered to cover the whole floor. (1mark)
actual area = 4.1 ×2.2
=9.02^{2}
wastage = (9.02  8.9571) + (9.0831  9.02)
2
= 0.063
 Maximum and minimum possible area of the floor (2marks)
SECTION II (50 MARKS)
INSTRUCTIONS: Answer ANY FIVE questions only in this section

 complete the table below, giving the values correct to 2 decimal places (2mks)
X^{0}
0^{0}
15^{0}
30^{0}
45^{0}
60^{0}
75^{0}
90^{0}
105^{0}
120^{0}
135^{0}
150^{0}
165^{0}
180^{0}
Cos 2X^{0}
1.00
0.87
0.50
0.00
0.5
0.87
1.00
0.87
0.5
0.00
0.50
0.87
1.00
Sin (X^{0}+30^{0})
0.50
0.71
0.87
0.97
1.00
0.97
0.87
0.71
0.50
0.26
0.00
0.26
0.50
 Using the grid provided draw on the same axes the graph of y=cos 2Xº and y=sin(Xº+30º) for
0º≤X≤180º. (4mks)  Find the period of the curve y=cos 2xº (1mk)
360
b
= 360 = 180º
2  Using the graph, estimate the solutions to the equations;
 sin(Xº+30º)=cos 2Xº (1mk)
x = 18.5 ± 2º
x = 139º ± 2º  Cos 2Xº=0.5 (1mk)
x= 30º
 sin(Xº+30º)=cos 2Xº (1mk)
 complete the table below, giving the values correct to 2 decimal places (2mks)
 A Quantity P varies partly as the square of m and partly as n. When p= 3.8, m = 2 and n = 3, When p =  0.2, m = 3 and n= 2.
 Find
 The equation that connects p, m and n (4marks)
p = xm^{2} + yn
3.8 = 4x  3y
0.2 = 9x + 2y
7.6 = 8x  6y
 0.6 = 27x + 6y +
7 35x
7 = x
35
x = 1/5 = 0.2
3.8 = 0.8  3y
1 = y
p = 0.2m^{2}  n  The value of p when m = 10 and n = 4 (1mark)
p = 0.2m^{2}  n
p = 20  4
p = 16
 The equation that connects p, m and n (4marks)
 Express m in terms of p and n (2marks)
p = 0.2m^{2}  n
0.2m^{2} = p + n
m^{2} = p + n
0.2
m = √p + n
0.2
m = ± √p + n
0.2  If P and n are each increased by 10%, find the percentage increase in m correct to 2 decimal place. (3marks)
m0 = √p + n
0.2
m1 = √1.1(p + n)
0.2
m1 = √1.1p + 1.1n = √5.5(p + n)
0.2
= 2.3452√(p + n)
m0 = √p + n = √5(p + n)
0.2
= 2.2361√(p + n)
% change in m = (m1  m0) × 100%
m0
= 2.3452√(p + n)  2.2361√(p + n)
2.2361√(p + n)
= 2.3452 2.2361 × 100%
2.2361
= 0.1001 × 100%
2.2361
= 0.04879 × 100%
= 4.88% 2dp
 Find

 The 5th term of an AP is 16 and the 12^{th} term is 37.
Find; The first term and the common difference ( 3 marks)
Tn = a + (n  1)d
T5 = a + 4d = 16
T2 = a + 11d = 37
7d = 21
d = 3
a + 4(3) = 16
a + 12 = 16
a = 4  The sum of the first 21 terms (2 marks)
Sn = n/2(2a + (n  1)d)
=21/2((2 × 4) + (20 × 3))
= 714
 The first term and the common difference ( 3 marks)
 The second, fourth and the seventh term of an AP are the first 3 consecutive terms of a GP. If the common difference of the AP is 2.
Find:  The common ratio of the GP ( 3 marks)
a + d, a + 3d, a + 6d
a + 2, a + 6, a + 12
a + 6 = a + 12
a + 2 a + 6
(a + 6)^{2} = (a + 2) (a + 12)
a^{2} + 12a + 36 = a2 + 14a + 24
2a = 12
a = 6
G.p
8, 12, 18
r = 12
8
= 1½  The sum of the first 8 terms of the GP (2 marks)
S8 = 8 ((3/2)^{8}  1)
1½  1
= 394.0625
 The 5th term of an AP is 16 and the 12^{th} term is 37.
 The table below shows the rates of taxation in a certain year.
In that period, Juma was earning a basic salary of sh. 21,000 per month. In addition, he was entitled to a house allowance of sh. 9000 p.m. and a personal relief of ksh.1056 p.m He also has an insurance scheme for which he pays a monthly premium of sh. 2000. He is entitled to a relief on premium at 15% of the premium paid. Calculate how much income tax Juma paid per month. (7mks)
taxable income = 21000 + 9000
= sh.30,000
p.a. = 30,000 × 12 = k.f 18,000 p.a.
20
2 × 3900 = sh. 7800
3 × 3900 =sh.11700
4 × 3900 = sh.15600
5 × 3900 = sh.19500
7 × 2400 = sh.16800
sh.71400
tax paid = 71400  16272
= sh 55128
P.A.Y.E = 55128
12
= sh. 4594
15 × 2000 = 300
100
total relief p.a. = (300 + 1056) × 12 = sh. 16272  Juma’s other deductions per month were cooperative society contributions of sh. 2000 and a loan repayment of sh. 2500. Calculate his net salary per month. (3mks)
total deductions = 4594 + 2000 + 2000 + 2500
sh 11094 per month
net salary = 30000  11094
=sh 18,906.00
 Calculate how much income tax Juma paid per month. (7mks)
 A cupboard has 7 white cups and 5 brown ones all identical in size and shape. There was a blackout in the town and Mrs. Kamau had to select three cups, one after the other without replacing the previous one.
 Draw a tree diagram for the information. (2mks)
 Calculate the probability that she chooses.
 Two white cups and one brown cup. (2mks)
(7 × 6 × 5) + (7 × 5 × 6) + (5 × 7 × 6)
12 11 10 12 11 10 12 11 10
= 21
44  Two brown cups and one white cup. (2mks)
(7 × 5 × 4) + (5× 7 × 4) + (5 × 4 × 7)
12 11 10 12 11 10 12 11 10
= 7
22  At least one white cup. (2mks)
(5 × 4 × 5) + (5× 7 × 4) + (7 × 5 × 4) + (5 × 7 × 6) + (7 × 5 × 6) + (7 × 6 × 5) + (7 × 6 × 5)
12 11 10 12 11 10 12 11 10 12 11 10 12 11 10 12 11 10 12 11 10
= 427
440  Three cups of the same colour. (2mks)
(7 × 6 × 5) + (5 × 4 × 3)
12 11 10 12 11 10
= 9
44
 Two white cups and one brown cup. (2mks)
 Draw a tree diagram for the information. (2mks)
 The For a sample of 100 bulbs, the time taken for each bulb to burn was recorded. The table below shows the result of the measurements.
Time(in hours)
1519
2024
2529
3034
3539
4044
4549
5054
5559
6064
6569
7074
Number of bulbs
6
10
9
5
7
11
15
13
8
7
5
4
 Using an assumed mean of 42, calculate
 the actual mean of distribution (4mks)
 the standard deviation of the distribution (3mks)
 the actual mean of distribution (4mks)
 Calculate the quartile deviation (3mks)
 Using an assumed mean of 42, calculate
 The position of town A and B on the earth’s surface are (36ºN, 49ºE) and (36ºN, 131ºW) respectively.
 Find the difference in longitude between town A and town B (2marks)
A(36ºN, 49ºE) B(36ºN, 131ºW)
longitudinal difference
49º + 131º = 180º  Given that the radius of the earth is 6370km, calculate the distance between town A and B along;
 Parallel of longitude (2marks)
Distance = 2πRcosθ
360 × 22 × 6370 cos36
dist = 16196.52023km  A great circle (3marks)
dist = σ2πR
360
= 108 × 2 × 22 × 6370
360 7
dist = 12,012km
 Parallel of longitude (2marks)
 Another town C is 840km east of town B and on the same latitude as town A and B. find the longitude of town C (3marks)
B(36º, 131ºw)
D = σ2πRcosσ
360
840 = σ × 2 × 22 × 6370cos 36º
360 7
840 = 89.9807σ
9.34º = σ
131º  9.34º = 121.66º
longitude of town c = 121.66ºw
 Find the difference in longitude between town A and town B (2marks)
 A trader is required to supply two types of shirts, type A and type B. the total number of shirts must not be more than 400. He has to supply more of type A than type B shirts. However the number of type A shirts must not be more than 300 and the number of type B shirts must not be less than 80. Let x be the number of type A shirts and y be the number of type B shirts.
 Write down in terms of x and y all the linear inequalities representing the information above (4marks)
x + y ≤ 400
x ≤ y
x > 0
x ≤ 300
y ≥ 80
y > 0  On the grid provided, draw the inequalities and shade the unwanted regions (4marks)
 The profits were as follows;
Type A: sh. 600 per shirt
Type B: sh. 400 per shirt Use the graph to determine the number shirts of each type that he should make to maximize the profit (1mark)
sample
(150, 200)
(200, 200)
(180,220)
(150,250)
(100,300)
200 type A
200 type B  Calculate the maximum possible profit (1mark)
600A + 4000p = max profits
(600 × 200) + (400 × 200)
120000 + 80000
=sh. 20000
 Use the graph to determine the number shirts of each type that he should make to maximize the profit (1mark)
 Write down in terms of x and y all the linear inequalities representing the information above (4marks)
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