Instructions to candidates
 Write your name and index number in the spaces provided above.
 Sign and write the date of examination 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 calculator and KNEC mathematical tables may be used, except where stated otherwise.
 Candidates should check the question paper to ascertain that all the pages are printed as indicated and that no questions are missing.
 Candidates should answer the questions in English.
For examiner’s use only
Section 1
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 (50 marks)
Answer all the questions in section I
 Evaluate (3 marks)
 Simplify 8x^{2} + 6x – 9 (3 marks)
16x^{2} – 9  Two similar solid cones made of the same material have masses of 8000g and 1000g respectively. If the base area of the smaller cone is 77cm^{2}, calculate;
 The base area of the larger cone (3 marks)
 The radius of the larger cone (2 marks)
 Given that cos(2x)°  sin(4x+30)° = 0. Calculate the value of x (3 marks)
 A line L passes through point (3,1) and is perpendicular to the line 2y=4x+5. Determine the equation of the line L. (3 marks)
 A Kenyan tourist left Germany for Kenya through Switzerland. While in Switzerland, he bought a watch worth 52 Deutsche marks. Using the exchange rates below.
1 Swiss Franc = 1.28 Deutsche Marks
1 Swiss Franc = 45.21 Kenya shillings
Find the value of the watch to the nearest (2 marks) Swiss Franc
 Kenya shillings (2 marks)
 State all integral values of x which satisfy the following pair of inequalities. (3 marks)
 A man is now three times as old as his daughter. In twelve years time he will be twice as old as his daughter. Find their present ages. (3 marks)
 The point A(3, 2) is mapped onto A^{1}(7, 1) under a translation T. Find the coordinates of the image of B(4, 6) under the same translation. (3 marks)
 Calculate the area of the trapezium below. (3 marks)
 Two machines A and B working together can do some work in 6 days. After 2 days machine A breaks down and it takes machine B 10 days to finish the remaining work. How long will it take machine A alone to finish the whole work if it does not break down. (3 marks)
 Solve for K in the equation. (3 marks)
 A square brass plate is 2mm thick and has a mass of 1.05kg. The density of the brass plate is 8.4g/cm^{3}. Calculate the length of the plate in cm. (3 marks)
 The sum of interior angles of two regular polygons of side n1 and n are in the ratio 4:5. Calculate;
 the value of interior angle of the polygon with side (n1) (2 marks)
 exterior angle (1 mark)
 Four athletes Onyango, Korir, Njuguna and Mutua can complete a 2km lap in a field in 12 minutes, 15 minutes, 18 minutes and 20 minutes respectively. If they start the race together, find the number of times the slowest athlete will be overlapped by the fastest athlete by the time they next cross the finish line simultaneously. (3 marks)
 The figure below shows a triangular prism. Draw its net. (3 marks)
 Four towns P, Q, R and S are such that Q is 160km from town P on a bearing of 065°. R is 280km on a bearing of 152° from Q. S is due west of R on a bearing of 155° from P. Using a scale of 1cm to represent 40km.
 Show the relative positions of P, Q, R and S. (6 marks)
 Find the bearing of;
 S from Q (1 mark)
 P from R (1 mark)
 Find the distance
 PS (1 mark)
 RS (1 mark)

 Complete the table below for y = 4x^{2}  x^{3} (2 marks)
x
1
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
y=4x^{2}x^{3}
5
1.1
0
6.1
 On the grid provided, draw the graph of y = 4x^{2} – x^{3} for 1≤x≤4 (3 marks)

 use the graph to find the roots of x3 – 4x2 + 4 = 0 (2 marks)
 On the same axis, draw the graph of 2y = x + 6 and state the values of x for which the two graphs intersect. (3 marks)
 Complete the table below for y = 4x^{2}  x^{3} (2 marks)
 A particle moves from rest and attains a velocity of 10m/s after two seconds it then moves with 10m/s velocity for 4 seconds. It finally decelerates uniformly and comes to rest after 6 seconds.
 Draw a velocity time graph for the motion of this particle (3 marks)
 From the graph find;
 the acceleration during the first two seconds. (2 marks)
 the uniform deceleration during the last six seconds. (2 marks)
 the total distance covered by the particle (3 marks)

 Find the gradient of a line L1 perpendicular to the line whose equation is y=4x+4 (2 marks)
 Calculate the angle in which line L1 is making with
 xaxis (2 marks)
 yaxis (1 mark)
 Line L_{2} is passing through the xaxis at 2 and point T(2, k) and it is parallel to line L_{1}. Calculate the value of K. (2 marks)
 Another line L_{3} is perpendicular to line L_{2} and passes through point T. Calculate the equation of line L_{3} leaving your answer in the form ax + by + c = 0 (3 marks)
 In the figure below P, Q, R and S are points on the circle centre O. PRT and USTV are straight lines. Line UV is a tangent to the circle at S. Angle RST is 50° and angle RTV is 150°.
 Calculate the size of:
 angle ORS (2 marks)
 angle USP (1 mark)
 angle PQR (2 marks)
 Given that RT=7cm and ST=9cm, calculate to three significant figures:
 the length of line PR (2 marks)
 the radius of the circle. (3 marks)
 Calculate the size of:
 In the triangle below OA = a and OB = b. M is the midpoint of AB and N is a point on OB such that ON = ^{1}/_{3} OB. AN and OM intersect at P.
 Express the following vectors in terms of a and b
 AB (1 mark)
 OM (1 mark)
 AN (2 marks)
 If OP = tOM and AP = sAN, express OP in two different ways hence find the value of t and s. (5 marks)
 State the ratio AN:NP (1 mark)
 Express the following vectors in terms of a and b
 Two circles with centres O_{1} and O_{2} have radii 10cm and 8cm respectively and intersect at points A and B. Angle AO_{1}B = 90° and angle AO_{2}B = 124.23°. Calculate to two decimal places;
 The length AB (2 marks)
 The length O_{1}O_{2} (2 marks)
 Area of minor segment centre O_{1} (3 marks)
 Area of quadrilateral O_{1}AO_{2}B (3 marks)
 A quadrilateral ABCD with vertices A(2, 6), B(4, 8), C(5, 6) and D(3, 4) is mapped onto quadrilateral A^{I}B^{I}C^{I}D^{I} by a reflection in the line y = x+5.
 On the grid provided draw the quadrilateral ABCD and its image A^{I}B^{I}C^{I}D^{I} under reflection in the line y= x+5 (5 marks)
 Quadrilateral A^{II}B^{II}C^{II}D^{II} is the image of quadrilateral A^{I}B^{I}C^{I}D^{I }under a negative quarter turn about (1, 1). On the same grid, draw quadrilateral A^{II}B^{II}C^{II}D^{II} and state the coordinates of the image (3 marks)
 Quadrilateral A^{III}B^{III}C^{III}D^{III}is the image of quadrilateral A^{II}B^{II}C^{II}D^{II} under an enlargement with scale factor 1 about (1,1). On the same grid, draw A^{III}B^{III}C^{III}D^{III }and state the coordinates of the image. (2 marks)
 On the grid provided draw the quadrilateral ABCD and its image A^{I}B^{I}C^{I}D^{I} under reflection in the line y= x+5 (5 marks)
MARKING SCHEME
SECTION I (50 marks)
Answer all the questions in section I
 Evaluate (3 marks)
Num  ^{3}/_{4} + ^{12}/_{7} ÷ ^{4}/_{7 }× ^{7}/_{3}
^{3}/_{4} + ^{9}/_{7} =^{ 57}/_{28}
Den (^{10}/_{7} ^{ 5}/_{8}) × ^{2}/_{3} = ^{15}/_{28}
^{57}/_{28} × ^{28}/_{15} = ^{34}/_{5} A_{1}  Simplify 8x^{2} + 6x – 9 (3 marks)
16x^{2} – 9
Num  8x^{2} + 12x  6x  9
4x(2x + 3)  3(2x + 3)
(4x  3)(2x + 3)
Den  16x2  9
(4x + 3)(4x  3)
(4x  3)(2x + 3)
(4x + 3)(4x  3)
2x + 3
4x + 3  Two similar solid cones made of the same material have masses of 8000g and 1000g respectively. If the base area of the smaller cone is 77cm^{2}, calculate;
 The base area of the larger cone (3 marks)
v.s.f = 8000 = 8
1000
a.s.f = 4
1
l.s.f =
Base area = 4 × 77 = 308cm^{2} A_{1}  The radius of the larger cone (2 marks)
πr^{2} = 308
^{22}/_{7} r^{2}= 308
r^{2 }= (2156 ×7) ÷ 22
r = √98
r = 9.9cm^{2} A_{1}
 The base area of the larger cone (3 marks)
 Given that cos(2x)°  sin(4x+30)° = 0. Calculate the value of x (3 marks)
 A line L passes through point (3,1) and is perpendicular to the line 2y=4x+5. Determine the equation of the line L. (3 marks)
 A Kenyan tourist left Germany for Kenya through Switzerland. While in Switzerland, he bought a watch worth 52 Deutsche marks. Using the exchange rates below.
1 Swiss Franc = 1.28 Deutsche Marks
1 Swiss Franc = 45.21 Kenya shillings
Find the value of the watch to the nearest (2 marks) Swiss Franc
 Kenya shillings (2 marks)
 State all integral values of x which satisfy the following pair of inequalities. (3 marks)
3  x ≤ 1  ^{1}/_{2}x
6  2x ≤ 2  x
x ≤ 4
x ≤ 4
^{1}/_{2}(x  5) ≥ 7 + x
x  5 ≥  14 + 2x
x ≥ 9
x ≤ 9
4 ≤ x ≤ 9
4,5,6,7,8,9  A man is now three times as old as his daughter. In twelve years time he will be twice as old as his daughter. Find their present ages. (3 marks)
Daughters age = x years
mans age = 3xyears
In 12 years time D = x + 12
M = 3x + 12
3x + 12 = 2(x + 12)
3x + 12 = 2x + 24
x = 12
Daughter 12 years
Man 36 years  The point A(3, 2) is mapped onto A^{1}(7, 1) under a translation T. Find the coordinates of the image of B(4, 6) under the same translation. (3 marks)
 Calculate the area of the trapezium below. (3 marks)
sin 60 = h
4√5
√3 × 4√5 = h
h = 6
A = ^{1}/_{2}[11 + 7]6
=18 × 3
= 54cm^{2}  Two machines A and B working together can do some work in 6 days. After 2 days machine A breaks down and it takes machine B 10 days to finish the remaining work. How long will it take machine A alone to finish the whole work if it does not break down. (3 marks)
A + B = 6
1 + 1 = 1
A B 6
Work done in 2 days = 2  1
6 3
Remaining work = 1  ^{1}/_{3}
= ^{2}/_{3}
^{2}/_{3 }= 10 days
1 day
^{2}/_{3 } × 1 × ^{1}/_{10} = ^{1}/_{15}
Therefore B does ^{1}/_{15 }work in 1 day
1 + 1 = 1
A 15 6
1 = 1
A 10
A = 10 days  Solve for K in the equation. (3 marks)
let log_{3}k = x
x^{2} = ^{1}/_{2}x + ^{3}/_{2}
2x^{2}  x  3 = 0
2x^{2}  3x + 2x  3 = 0
(x +1)(2x  3) = 0
x = 1, x = 1^{1}/_{2}
log_{3}k = 1
k = ^{1}/_{3}
log_{3}k = ^{3}/_{2} , k =3^{3}/_{2}
k = √27  A square brass plate is 2mm thick and has a mass of 1.05kg. The density of the brass plate is 8.4g/cm^{3}. Calculate the length of the plate in cm. (3 marks)
vol plate = 1.05 × 1000g
8.4g/cm^{3}
= 125cm^{3}
L × L × L = 125
L^{2} ×0.2 = 125
L^{2} = 125
0.2
L = √1250
2
L = √625
L = 25cm  The sum of interior angles of two regular polygons of side n1 and n are in the ratio 4:5. Calculate;
 the value of interior angle of the polygon with side (n1) (2 marks)
(n  1  2)180 = 4
(n  2)180 5
4n  8 = 5n  15
n = 7
(6  2)180 = 120
6
= 120º  exterior angle (1 mark)
180º  120º = 60º
 the value of interior angle of the polygon with side (n1) (2 marks)
 Four athletes Onyango, Korir, Njuguna and Mutua can complete a 2km lap in a field in 12 minutes, 15 minutes, 18 minutes and 20 minutes respectively. If they start the race together, find the number of times the slowest athlete will be overlapped by the fastest athlete by the time they next cross the finish line simultaneously. (3 marks)
LCM 2 12 15 18 20 2 6 15 9 10 3 3 15 9 5 3 1 5 3 5 5 1 15 1 5 1 1 1 1
180 = 9 times
20
180 = 15 times
12
15  9 = 6 times  The figure below shows a triangular prism. Draw its net. (3 marks)
 Four towns P, Q, R and S are such that Q is 160km from town P on a bearing of 065°. R is 280km on a bearing of 152° from Q. S is due west of R on a bearing of 155° from P. Using a scale of 1cm to represent 40km.
 Show the relative positions of P, Q, R and S. (6 marks)
 Find the bearing of;
 S from Q (1 mark) = 198º
 P from R (1 mark) = 304º
 Find the distance
 PS (1 mark) = 204km
 RS (1 mark) = 184km
 Show the relative positions of P, Q, R and S. (6 marks)

 Complete the table below for y = 4x^{2}  x^{3} (2 marks)
x
1
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
y=4x^{2}x^{3}
5
1.1
0
0.9 3 5.7 8 9.4 9 6.1
0  On the grid provided, draw the graph of y = 4x^{2} – x^{3} for 1≤x≤4 (3 marks)

 use the graph to find the roots of x3 – 4x2 + 4 = 0 (2 marks)
y = x^{3} + 4x^{2}
0 = x^{3}  4x^{2} + 4
y = 4
x = 0.9, 1.2, 3.8  On the same axis, draw the graph of 2y = x + 6 and state the values of x for which the two graphs intersect. (3 marks)
y = ^{1}/_{2 }x 3
x 0 2 y 3 4
 use the graph to find the roots of x3 – 4x2 + 4 = 0 (2 marks)
 Complete the table below for y = 4x^{2}  x^{3} (2 marks)
 A particle moves from rest and attains a velocity of 10m/s after two seconds it then moves with 10m/s velocity for 4 seconds. It finally decelerates uniformly and comes to rest after 6 seconds.
 Draw a velocity time graph for the motion of this particle (3 marks)
 From the graph find;
 the acceleration during the first two seconds. (2 marks)
a = 10  0
2
= 10
2
= 5m/s^{2}  the uniform deceleration during the last six seconds. (2 marks)
0  10 = 10
6 6
= 1.667m/s^{2}  the total distance covered by the particle (3 marks)
A = [^{1}/_{2}(4 + 6)10] + ^{1}/_{2} x 6 x 10
= 50 + 30
= 80m
 the acceleration during the first two seconds. (2 marks)
 Draw a velocity time graph for the motion of this particle (3 marks)

 Find the gradient of a line L1 perpendicular to the line whose equation is y=4x+4 (2 marks)
4 x m_{2} = 1
Grad = ^{1}/_{4}  Calculate the angle in which line L1 is making with
 xaxis (2 marks)
tanØ = ^{1}/_{4}
Ø = tan^{1}(^{1}/_{4})
Ø = 14.04º  yaxis (1 mark)
90  14.04
= 75.96º
 xaxis (2 marks)
 Line L_{2} is passing through the xaxis at 2 and point T(2, k) and it is parallel to line L_{1}. Calculate the value of K. (2 marks)
(2,0)(2,k)
k  0 =  1
2  2 4
4k = 4
k = 1  Another line L_{3} is perpendicular to line L_{2} and passes through point T. Calculate the equation of line L_{3} leaving your answer in the form ax + by + c = 0 (3 marks)
^{1}/_{4}m_{2} = 1
GL_{3} = 4
(2,1)(x,y)
y  1 = 4
x + 2 1
y  1 = 4x + 8
y = 4x + 9
4x  y + 9 = 0
 Find the gradient of a line L1 perpendicular to the line whose equation is y=4x+4 (2 marks)
 In the figure below P, Q, R and S are points on the circle centre O. PRT and USTV are straight lines. Line UV is a tangent to the circle at S. Angle RST is 50° and angle RTV is 150°.
 Calculate the size of:
 angle ORS (2 marks)
= 40º  angle USP (1 mark)
= 80º  angle PQR (2 marks)
= 130º
 angle ORS (2 marks)
 Given that RT=7cm and ST=9cm, calculate to three significant figures:
 the length of line PR (2 marks)
= 4.57  the radius of the circle. (3 marks)
= 2.98
 the length of line PR (2 marks)
 Calculate the size of:
 In the triangle below OA = a and OB = b. M is the midpoint of AB and N is a point on OB such that ON = ^{1}/_{3} OB. AN and OM intersect at P.
 Express the following vectors in terms of a and b
 AB (1 mark)
= b  a  OM (1 mark)
= ^{1}/_{2}a + ^{1}/_{2}b  AN (2 marks)
= ^{1}/_{3}b  a
 AB (1 mark)
 If OP = tOM and AP = sAN, express OP in two different ways hence find the value of t and s. (5 marks)
OP = tom
t(^{1}/_{2}a + ^{1}/_{2}b)
= ^{1}/_{2}ta + ^{1}/_{2}tb
^{1}/_{2}ta = (1  s)a
^{1}/_{2}t = 1  s .......(i) t = ^{2}/_{3}s
^{1}/_{2}t = ^{1}/_{3}s .........(ii) ^{1}/_{2}(^{2}/_{3}s) = 1  s
OP = a + s (^{1}/_{3}b  a)
=a + ^{1}/_{3}sb  sa
=a  sa + ^{1}/_{3}sb
=(1  s)a + ^{1}/_{3}sb
s = ^{6}/_{8} = ^{3}/_{4}t = ^{2}/_{3} × ^{3}/_{4} = ^{1}/_{2}  State the ratio AN:NP (1 mark)
AP = ^{3}/_{4}AN
AP:PN = 3:1
 Express the following vectors in terms of a and b
 Two circles with centres O_{1} and O_{2} have radii 10cm and 8cm respectively and intersect at points A and B. Angle AO_{1}B = 90° and angle AO_{2}B = 124.23°. Calculate to two decimal places;
 The length AB (2 marks)
AM = sin45º
10
AM = 10sin45º
AB = 2(10sin45º)
AB = 14.14 cm  The length O_{1}O_{2} (2 marks)
O_{1}M = cos45º
10
O_{1}M = 10cos45º
= 7.072cm
O_{2}M = cos62.115º
8
O_{2}M = 8cos62.115º
= 3.742cm
O_{1}O_{2} = 7.072 + 3.742
=10.81 cm  Area of minor segment centre O_{1} (3 marks)
90πr^{2}  ^{1}/_{2} × 10 × 10
360
^{1}/_{4} × 3.142 × 100  50
78.55  50
= 28.55cm  Area of quadrilateral O_{1}AO_{2}B (3 marks)
^{1}/_{2 }× 10 × 10 sin 90º + ^{1}/_{2} × 8 × 8 sin 124.23º
50 + 32 sin 124.23º
= 76.457
= 76.46cm
 The length AB (2 marks)
 A quadrilateral ABCD with vertices A(2, 6), B(4, 8), C(5, 6) and D(3, 4) is mapped onto quadrilateral A^{I}B^{I}C^{I}D^{I} by a reflection in the line y = x+5.
 On the grid provided draw the quadrilateral ABCD and its image A^{I}B^{I}C^{I}D^{I} under reflection in the line y= x+5 (5 marks)
y = x + 5x 0 2 1 y 5 3 4  Quadrilateral A^{II}B^{II}C^{II}D^{II} is the image of quadrilateral A^{I}B^{I}C^{I}D^{I }under a negative quarter turn about (1, 1). On the same grid, draw quadrilateral A^{II}B^{II}C^{II}D^{II} and state the coordinates of the image (3 marks)
A^{II}(5,1)B^{II}(3,3)C^{II}(2,1)D^{II}(4,1)  Quadrilateral A^{III}B^{III}C^{III}D^{III}is the image of quadrilateral A^{II}B^{II}C^{II}D^{II} under an enlargement with scale factor 1 about (1,1). On the same grid, draw A^{III}B^{III}C^{III}D^{III }and state the coordinates of the image. (2 marks)
A^{III}(3,3)B^{III}(1,5)C^{III}(0,3)D^{III}(2,1)
 On the grid provided draw the quadrilateral ABCD and its image A^{I}B^{I}C^{I}D^{I} under reflection in the line y= x+5 (5 marks)
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