INSTRUCTIONS TO CANDIDATES
 Write your name, admission number, class and date in the spaces provided above.
 The paper consists of two sections: section I and section II.
 Section I has sixteen questions and section II has eight questions.
 Answer all the questions in section I and any five in 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
 KNEC Mathematical table and silent nonprogrammable calculators may be used.
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 (50 marks)
Answer ALL the questions in this section
 Evaluate 3⁄4 + 1 5⁄7 ÷ 4⁄7 of 2 1⁄3 (3 marks)
1 3⁄7  5⁄8 × 2⁄15  Simplify 4x^{2}  9 (3 marks)
8x^{2} + 6x  9  Two similar solid cones made of the same material have masses of 800g and 100g respectively. If the base area of the smaller cone is 38.5cm^{2}, calculate;
 The base area of the larger cone (2 marks)
 The radius of the larger cone (2 marks)
 Given that cos(2x)°sin(2x30)° = 0. Calculate the value of sin x (3 marks)
 A line L passes through point (5, 3) and is parallel to the line y+ 1⁄2 x5=0. Determine the equation of the line L in the y=mx+c. (3 marks)
 A Kenyan bank buys and sells foreign currency as shown in the table below.
Buying (Kshs.)
Selling (Kshs.)
1 US dollar
95.34
95.87
1 UK pound
124.65
125.13
 Find all the integral values of x which satisfy the inequalities (3 marks)
20x > 5+2x ≥ x+5  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 P(5, 4) is mapped onto P1(9, 3) under a translation T. Find the coordinates of the image of Q(6, 8) under the same translation. (3 marks)
 The figure below shows a solid wedge PQRSTU. Complete the solid showing all the hidden edges with dotted lines. (3marks)
 Two machines X and Y working together can do some work in 6 days. After 2 days machine X breaks down and it takes machine Y 10 days to finish the remaining work. How long will it take machine X alone to finish the whole work if it does not break down. (3 marks)
 Solve for X in the equation. (3 marks)
(log_{4}X )^{2}= ^{1}/_{2} log_{4}X+ ^{3}/_{2}  The area of a rhombus is 60cm^{2} given that one of its diagonals is 15cm long, calculate the perimeter of the rhombus. (4 marks)
 The sum of interior angles of a regular polygon is 24 times the size of the exterior angle. Find the number of sides of the polygon and hence name it. (3 marks)
 Using tables, find the reciprocal of 0.432 and hence evaluate ^{(√0.1225)}/_{0.432} (3 marks)

 Matrices P and Q are given by P = (3 1 2)and Q = Find the product PQ. (1 mark)
 Given A = and B find AB (2 marks)
SECTION II (50 marks)
Answer ANY five questions in this section
 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 the function y = 2x^{2} + 3x – 5. (2 marks)
x 4 3 2 1 0 1 2 2x^{2} 18 0 3x 12 3 6 5 y  On the grid provided draw the graph of y = 2x^{2} + 3x – 5 for 4 ≤ x ≤ 2 (4 marks)
 Use your graph to state the roots of
 2x^{2} + 3x – 5 = 0 (1mark)
 2x^{2} + 6x – 2 = 0 (3marks)
 Complete the table below for the function y = 2x^{2} + 3x – 5. (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=x+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 50º and angle RTV is 150º.
 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:
 The figure below is triangle OAB in which OA = a and OB = b. M and N are points on OA and OB respectively such that OM:MA = 1:3 and ON:NB = 2:1.
 Express the following vectors in terms of a and b
 AN (1 mark)
 BM (1 mark)
 AB (1 mark)
 Lines AN and BM intersect at X such that AX=hAN and BX=kBM. Express OX in two different ways and find the value of h and k. (6 marks)
 OX produced meets AB at Y such that AY:YB =3:2. Find AY in terms of a and b. (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)
 PQR is a triangle with coordinates; P(3, 3), Q(5, 1) and R (2, 1). P’Q’R’ is the image of PQR under an enlargement such that the coordinates are P'(3, 0), Q'(7, 4) and R'(1, 4). Using a scale of 1:1 on both axes;

 Plot PQR and P’Q’R’ hence locate the centre of enlargement by construction. (4 marks)
 State the scale factor of the enlargement. (2 marks)
 P’’Q’’R’’ is the image of PQR under a translation T . Plot P''Q''R''. (2 marks)
 P’’’Q’’’R’’’ is the image of PQR under a reflection whose mirror line is y=2. Plot P’’’Q’’’R’’ (2 marks)

MARKING SCHEME
SECTION I (50 marks)
Answer ALL the questions in this section
 Evaluate 3⁄4 + 1 5⁄7 ÷ 4⁄7 of 2 1⁄3 (3 marks)
1 3⁄7  5⁄8 × 2⁄15
Num: 3⁄4 + 1 5⁄7 ÷ 4⁄7 of 2 1⁄3
3⁄4 + ^{12}/_{7} ÷ 4⁄7 × ^{7}/_{3}3⁄4 + ^{12}/_{7} x ^{3}/_{4}
=3⁄4 + ^{9}/_{7} = 21 + 36
28
=^{57}/_{28 }Den:
= ^{10}/_{7}  ^{5}/_{8} x ^{2}/_{15}
= 80  35 x ^{2}/_{15}
56
= ^{45}/_{56} x ^{2}/_{15} = ^{3}/_{28}
∴ ^{57}/_{28 }÷ ^{3}/_{28} = ^{57}/_{28} x ^{26}/_{3} = 19  Simplify 4x^{2}  9 (3 marks)
8x^{2} + 6x  9
num: (2x + 3)(2x  3)
Den: 8x^{2} + 6x  9
p = 72
s = 6
12, 6
8x^{2} + 12x  6x  9
4x(2x + 3)  3(2x + 3)
(4x  3)(2x + 3)
= (2x + 3)(2x  3)
(4x  3)(2x + 3)
= 2x  3
4x  3  Two similar solid cones made of the same material have masses of 800g and 100g respectively. If the base area of the smaller cone is 38.5cm^{2}, calculate;
 The base area of the larger cone (2 marks)
v.s.f = 800:100
=8:1
l.s.f = 3√8:1
=2:1
a.s.f = (2:1)^{2}
=4:1
x = 38.5 x 4
=154 cm^{2}  The radius of the larger cone (2 marks)
πr^{2} = 154
r^{2} = 154 x ^{7}/_{22}
r^{2} = 49
r = √49
r = 7 cm
 The base area of the larger cone (2 marks)
 Given that cos(2x)°sin(2x30)° = 0. Calculate the value of sin x (3 marks)
cos 2x = sin (2x  30)º
2x + 2x  30º = 90º
4x = 120º
x= 30º
sin 30 = 0.5 or ^{1}/_{2}  A line L passes through point (5, 3) and is parallel to the line y+ 1⁄2 x5=0. Determine the equation of the line L in the y=mx+c. (3 marks)
m_{1} = m_{2} = ^{1}/_{2}
y3 = ^{1}/_{2}
x+5
2y  6 = x  5
2y = x + 1
y = ^{1}/_{2}x + ^{1}/_{2}  A Kenyan bank buys and sells foreign currency as shown in the table below.
Buying (Kshs.)
Selling (Kshs.)
1 US dollar
95.34
95.87
1 UK pound
124.65
125.13
= 15000 x 124.65 x 92/100
=1,720,170 /=
spent = 1/2 x 1,720,170
=860, 085 /=
1 → 95.87
? → 860 085
=1 x 860 085
95.87
=8971 USD  Find all the integral values of x which satisfy the inequalities (3 marks)
20x > 5+2x ≥ x+5
20  x > 5 + 2x
15 > 3x
5 > x or x < 5
5 + 2x ≥ x + 5
0 ≥ x
0 ≤ x
0 ≤ x < 5
values are
0, 1 , 2 , 3/4  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)
Now 12 yrs time
DX x  12
3x (x + 12)2
(3x + 12) = (x + 12)2
3x + 12 = 2x + 24
x = 12 yrs (daughter)
man = 3 x 12
= 36 yrs  The point P(5, 4) is mapped onto P1(9, 3) under a translation T. Find the coordinates of the image of Q(6, 8) under the same translation. (3 marks)
 The figure below shows a solid wedge PQRSTU. Complete the solid showing all the hidden edges with dotted lines. (3marks)
 Two machines X and Y working together can do some work in 6 days. After 2 days machine X breaks down and it takes machine Y 10 days to finish the remaining work. How long will it take machine X alone to finish the whole work if it does not break down. (3 marks)
total work by both = 6 days
work done in one day = ^{1}/_{6}
2 days by both = ^{2}/_{6} = ^{1}/_{3}
remainder = ^{2}/_{3}
y ⇒ 10 = ^{2}/_{3}
alone x = 1
^{2}/_{3}x = 10
x = ^{30}/_{2} = 15 days
∴^{1}/_{15} + ^{1}/_{A} = ^{1}/_{6}
^{1}/_{A} = ^{1}/_{6}  ^{1}/_{15} = 5 + 2
30
^{1}/_{A} = ^{3}/_{30} = ^{1}/_{10}
A alone = 10 days  Solve for X in the equation. (3 marks)
(log_{4}X )^{2}= ^{1}/_{2} log_{4}X+ ^{3}/_{2}let log_{4}X = y
y^{2} = ^{1}/_{2}y + ^{3}/_{2}
y^{2} = ^{1}/_{2}y + ^{3}/_{2} = 0
2y^{2}  y  3 = 0
p = 6 s = 1
3 & 2
2y^{2} + 2y  3y  3 = 0
2y (y + 1)  3(y + 1) = 0
y = 1 or 1.5(^{3}/_{2})  The area of a rhombus is 60cm^{2} given that one of its diagonals is 15cm long, calculate the perimeter of the rhombus. (4 marks)
A = 1/2 d1d2
60 = 1/2 x 15 x d2
d2 = 8 cm
l2 = 42 + 7.52
l = √72.25
= 8.5 cm
perimeter = 4 x 8.5
= 34 cm  The sum of interior angles of a regular polygon is 24 times the size of the exterior angle. Find the number of sides of the polygon and hence name it. (3 marks)
(2n  4) 90º = (360) 24
n
180ºn  360º = 8640
n
180n^{2}  360n  8640 = 0
n^{2}  2n  48 = 0
p = 48
s = 2
8, 6
n^{2} + 6n  8n  48 = 0
n(n + 6)  8(n  6) = 0
(n  8) (n + 6) = 0
n = 8 or 6
8 sided (octagon)  Using tables, find the reciprocal of 0.432 and hence evaluate ^{(√0.1225)}/_{0.432} (3 marks)
= 1
4.32 x 10^{1}
= 2.315 x 10^{1}
= 2.315
√0.1225 = √12.25 x 10^{2}
= 3.4999 x 10^{1}
=0.34999
=0.34999 x 2.315
=0.81022685
= 0.8102 (4 s.f) 
 Matrices P and Q are given by P = (3 1 2)and Q = Find the product PQ. (1 mark)
PQ = 3  Given A = and B find AB (2 marks)
= 30 + 7 + 0 40 + 42 + 15
27 + 11+ 0 36 + 66 + 36
AB = [37 97]
[38 138]
 Matrices P and Q are given by P = (3 1 2)and Q = Find the product PQ. (1 mark)
SECTION II (50 marks)
Answer ANY five questions in this section
 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)
104º + 90º = 194 ± 1  P from R (1 mark)
270º + 33º = 303º ± 1
 S from Q (1 mark)
 Find the distance
 PS (1 mark)
5 x 40 = 200 km ± 4  RS (1 mark)
4.8 x 40 = 192km ± 4
 PS (1 mark)
 Show the relative positions of P, Q, R and S. (6 marks)

 Complete the table below for the function y = 2x^{2} + 3x – 5. (2 marks)
x 4 3 2 1 0 1 2 2x^{2} 32 18 8 2 0 2 8 3x 12 9 6 3 0 3 6 5 5 5 5 5 5 5 5 y 15 4 3 6 5 0 9  On the grid provided draw the graph of y = 2x^{2} + 3x – 5 for 4 ≤ x ≤ 2 (4 marks)
 Use your graph to state the roots of
 2x^{2} + 3x – 5 = 0 (1mark)
y = 2x^{2} + 3x  5
0=2x^{2} + 3x  5
y = 0
x= 1 or 2 5  2x^{2} + 6x – 2 = 0 (3marks)
y = 2x2 + 3x  5
0 = 2x2 + 6x  2
y = 0  3x  3
x = 0.25 ± 0.1
3.3 ± 0.1
 2x^{2} + 3x – 5 = 0 (1mark)
 Complete the table below for the function y = 2x^{2} + 3x – 5. (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 = Δv
Δc
= 100
20
=5m/s^{2} or 5ms^{2}  the uniform deceleration during the last six seconds. (2 marks)
= 0  10
12  6
= 10
6
= 1^{2}/_{5} m/s
=1.40 m/s^{2}  the total distance covered by the particle (3 marks)
D = (1/2 x 2 x 10) + (10 x 4) + (1/2 x 6 x 10)
=10m + 40m + 30m
=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=x+4 (2 marks)
m_{1}m_{2} = 1
m_{1} = 1
∴m_{2} = 1  Calculate the angle in which line L1 is making with
 xaxis (2 marks)
θ = Tan^{1} m2 = Tan^{1}1
= 45º  yaxis (1 mark)
θ = 45º
Alternative angles are equal
 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)
through points (2,0) & (2, k)
k  0 = 1
2 2
k = 1 (4)
k = 4
∴T(2, 4)  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)
m = 1 T= (2, 4)
y  4 = 1
x + 2
y  4 = x + 2
y  x  6 = 0
 Find the gradient of a line L1 perpendicular to the line whose equation is y=x+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 50º and angle RTV is 150º.
*no need to wait for the reasons
*inspect the diagram Calculate the size of:
 angle ORS (2 marks)
40º  base angles of an isoceles Δ  angle USP (1 mark)
80º  angles in a straight line add up 180º  angle PQR (2 marks)
130º opposite angles of a cyclic quadrilateral add up to 180º
 angle ORS (2 marks)
 Given that RT=7cm and ST=9cm, calculate to three significant figures:
 the length of line PR (2 marks)
PT:RT = ST^{2}
7(X + 7) = 9^{2}
7X = 81  49
X = 32/7 = 4.57CM  the radius of the circle. (3 marks)
2R = 4.57
sin 50º
2R = 4.64
R = 4.64
2
R = 2.32 cm
 the length of line PR (2 marks)
 Calculate the size of:
 The figure below is triangle OAB in which OA = a and OB = b. M and N are points on OA and OB respectively such that OM:MA = 1:3 and ON:NB = 2:1.
 Express the following vectors in terms of a and b
 AN (1 mark)
 BM (1 mark)
 AB (1 mark)
 Lines AN and BM intersect at X such that AX=hAN and BX=kBM. Express OX in two different ways and find the value of h and k. (6 marks)
 OX produced meets AB at Y such that AY:YB =3:2. Find AY in terms of a and b. (1 mark)
AY = 3/5 (b  a) or 3/5 b  3/5a
 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)
AB = 2 x 10 x sin 45º
=14.14cm
or
2 x 8 x sin 62.115º
=14.14 cm  The length O_{1}O_{2} (2 marks)
O_{1}O_{2} =(10 x cos 45º) + (8 x cos 62.115º)
=7.071 + 3.741
=10.812
=10.81cm  Area of minor segment centre O_{1} (3 marks)
= 28.57cm^{2}  Area of quadrilateral O_{1}AO_{2}B (3 marks)
=76.46cm^{2}
 The length AB (2 marks)
 PQR is a triangle with coordinates; P(3, 3), Q(5, 1) and R (2, 1). P’Q’R’ is the image of PQR under an enlargement such that the coordinates are P'(3, 0), Q'(7, 4) and R'(1, 4). Using a scale of 1:1 on both axes;

 Plot PQR and P’Q’R’ hence locate the centre of enlargement by construction. (4 marks)
(1,2)  State the scale factor of the enlargement. (2 marks)
(2)
 Plot PQR and P’Q’R’ hence locate the centre of enlargement by construction. (4 marks)
 P’’Q’’R’’ is the image of PQR under a translation T . Plot P''Q''R''. (2 marks)
=( 4 3 6 )
(6 4 4)  P’’’Q’’’R’’’ is the image of PQR under a reflection whose mirror line is y=2. Plot P’’’Q’’’R’’ (2 marks)

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