INSTRUCTIONS TO CANDIDATES
 The paper contains two sections: Section I and II
 Answer ALL questions in section I and ONLY FIVE questions from section II.
 All working and answers must be written on the question paper in the spaces provided below each question.
 Marks may be awarded for correct working even if the answer is wrong.
 Negligence and slovenly work will be penalized.
 Nonprogrammable silent electronic calculators and mathematical tables are allowed for use.
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
 Without using a calculator, evaluate (3mks)
 A number n is such that when it is divided by 3,7,11 or 13, the remainder is always one. Find the number n (2mks)
 Solve the inequality and show the solution on the number line (3mks)
 In the figure below AB is parallel to DE and area of triangle DEC is 8cm^{2}.
Find The value of x (2mks)
 The area of quadrilateral ABED (2mks)
 A bank in Kenya buys and sells foreign currencies as follows;
Currency
Buying(Ksh)
Selling (Ksh)
1 sterling Pound
1 US dollar
132.40
70.40
132.75
70.84
 Solve for x given that; (2mks)
Cos(2x – 40)° = sin (20 +3x)°  The size of an interior angle of a regular polygon is 3x° while its exterior angle is (x20)°. Find the number of sides of the polygon. (3mks)
 Log 5 2 +log (2x+10) = log (x4) (3 mks)
 Find the equation of the perpendicular bisector of a straight line passing through the points P(2,7) and Q(4,3) giving your answer in the form ax +by +c =0 (4mks)
 The figure below shows a sector of a circle with centre O, radius 10cm. The chord PQ=16cm Calculate the area of the sector PXQ (4mks)
 Simplify the expression (3mks)
 Use the reciprocal tables to evaluate (3mks)
 The volumes of two similar solid cylinders are 4752cm^{3} and 1408cm^{3}. If the area of the curved surface of the smaller cylinder is 352cm^{2}, find the area of the curved surface of the larger cylinder (4mks)
 A bus takes 195 minutes to travel a distance of (2x +30) km at an average speed of (x20)km/h. Calculate the actual distance travelled. Give your answer in kilometres (3mks)
 Given that OA =3i 2j and OB=4i+j. Find the distance between points A and B correct to 1 decimal place (2mks)
 Murimi and Naliaka had each 288 tree seedlings.Murimi planted equal number of seedlings per row in x rows while Naliaka planted equal number of seedlings in (x+1) rows. The number of tree seedlings planted by Murimi in each row were 4 more than those planted by Naliaka in each row. Calculate the number of seedling Murimi planted in each row. (4 mks)
SECTION II.(ANSWER FIVE QUESTIONS ONLY
 Two business partners Mary and John contributed Ksh.60000 and Ksh90000 respectively to start up a business. After 6 months, Lucy joined the business partnership and contributed Ksh.100000. At the end of the year, a gross profit of Ksh.720000 was realized. They then agreed to reinvest 30% of the profit accrued back into the business and use 20% of the profit for running the business operations. The remainder was to be shared among the business partners in the ratio of their contribution.
 Calculate the amount;
 Reinvested into the business (2mks)
 Used for business operations (1mk)
 Calculate the amount of profit each partner got (4mks)
 If the amount put back into the business was added to individual’s shares proportional to their initial contribution, find the amount of each partner’s shares (3mks)
 Calculate the amount;
 A bus left Nairobi at 7.00a.m and travelled towards Eldoret at an average speed of 80km/h. At 7.45a.m a car left Eldoret towards Nairobi at an average speed of 120km/h. Given that the distance between Nairobi and Eldoret is 300km. Calculate;
 The time the bus arrived at Eldoret (2mks)
 The time of the day, the two met (3mks)
 The distance from Nairobi where the two met (2mks)
 The distance of the bus from Eldoret when the car arrived in Nairobi (3 mks)
 The figure below shows a solid which is a rectangular based pyramid of height 12cm.
 Calculate the slanting length VC (2mks)
 Calculate the surface area of the pyramid (4mks)
 Calculate the volume of the pyramid (2mks)
 Determine the density of the metal which make this solid if its mass is 1.632kg (2mks)
 In the figure below OJKL is a parallelogram in which OJ = 3p ̃ and OL = 2 r
̃ If A is a point on LK such that LA = ½ AK and point B divide the line JK externally in the ratio 3:1, express OB and AJ in terms of p and r. (2 marks)
 Line OB interests AJ at X such that OX = mOB and AX = nAJ.
 Express OX in terms of p, r and m. (1 mark)
 Express OX in terms of p, r and n (1 mark)
 Determine the value of m and n and hence the ratio in which point x divides line AJ. (6 marks)
 The figure below shows two triangles, ABC and BCD with a common base BC=3.4cm, AC=7.2cm, CD=7.5cm and <ABC=90°. The area of triangle ABC=Area of triangle BCD.
Calculate correct to one decimal place; The area of triangle ABC (3mks)
 The size of <BCD (3mks)
 The length of BD (2mks)
 The size of <BDC (2mks)
 The equation of a curve is y=x^{3}+4x^{2} 3.
 Fill in the table below for the curve y=x^{3}+4x^{2} – 3 (2mks)
x
4
3
2
1
0
1
y
 On the grid provided, draw the curve y=x^{3}+4x^{2}  3 for the range 4≤x≤1 scale: Horizontal axis 2cm represent 1 unit vertical axis 1cm rep 1 unit (3mks)
 Use your graph to solve
 x^{3} +4x^{2} – 3 = 0 (2mks)
 4x^{3} +16x^{2} –x – 16 = 0 (3mks)
 Fill in the table below for the curve y=x^{3}+4x^{2} – 3 (2mks)

 A straight line L1 whose equation is y2x=4 meets the xaxis at M. Determine coordinates of M. (1mk)
 A second line L_{2} is perpendicular to L_{1} at M. Find the equation of Line L_{2} in the form ax+by+c=0 where a,b and c are integers. (3mks)
 A third line L_{3} passes through (2,3) and is parallel to L_{1}. Find;
 The equation of L_{3} in the form y=mx +c (3mks)
 Point N, the intersection of L_{2} and L_{3} (3mks)
 A curve is represented by the function y=x^{3} +x^{2} – x
 Find ^{dy}/_{dx} (1mk)
 Determine the values of y at the turning points of the curve y =x^{3} +x^{2} –x (4mks)
 Determine the nature of the turning points (2mks)
 In the space provided below, sketch the curve of y=x^{3}+x^{2} –x (3mks)
MARKING SCHEME
 Without using a calculator, evaluate (3mks)
9/4 + 3/10 = 45 + 6
20
= 51 x 10
20 17
= 3/2
1^{1}/_{2}  A number n is such that when it is divided by 3,7,11 or 13, the remainder is always one. Find the number n (2mks)
3, 7 , 11, 13
lcm = 3 x 7 x 1 x 13
= 3003
number = 3003 + 1
= 3004  Solve the inequality and show the solution on the number line (3mks)
3  2x ≤ x
3x ≤ 3
x ≥ 1
x ≤ 2x + 5
3
3x ≤ 2x + 5
x ≤ 5  In the figure below AB is parallel to DE and area of triangle DEC is 8cm^{2}.
Find The value of x (2mks)
4/10 = 6/6 + x
24 + 4x = 60
4x = 36
x = 9  The area of quadrilateral ABED (2mks)
8 = 4 (A.SF)
Area of Δ ABC 25
Area of Δ ABC = 8 x 25 = 50cm^{2}
4
Area of quadrilateral = 50  8
= 42 cm^{2}
 The value of x (2mks)
 A bank in Kenya buys and sells foreign currencies as follows;
Currency
Buying(Ksh)
Selling (Ksh)
1 sterling Pound
1 US dollar
132.40
70.40
132.75
70.84
3500 x 70.40 = 246, 400 ksh
246 400  115 000 = ksh 131 400
= 131 400
132.75
= 989.83 pounds  Solve for x given that; (2mks)
Cos(2x – 40)° = sin (20 +3x)°
(2x  40) + (20 + 3x) = 90
2x  40  20 + 3x = 90
5x = 150
x = 30º  The size of an interior angle of a regular polygon is 3x° while its exterior angle is (x20)°. Find the number of sides of the polygon. (3mks)
3x + (x  20) = 180
4x = 20º
x = 50
exterior angle = 50  20 = 30º
number of sides = 360
30
= 12 sides  Log 5 2 +log (2x+10) = log (x4) (3 mks)
ans
= x = 5  Find the equation of the perpendicular bisector of a straight line passing through the points P(2,7) and Q(4,3) giving your answer in the form ax +by +c =0 (4mks)
g_{1} = 7  3 = 4 = 2
2  4 2
g_{2} = 1/2
midpoint =( 2 + 4 7 + 3)
2 2
= (3, 5)
y  7 = 1
x  2 2
x  2 = 2y  14
x  2y + 12 = 0  The figure below shows a sector of a circle with centre O, radius 10cm. The chord PQ=16cm Calculate the area of the sector PXQ (4mks)
sin 1/2x = 8/10
sin 1/2 = 0.8
1/2x = 53.13
x = 106.26
sector area = (360  106.26) x 22 x 102
360
= 2253.74 x 22 x 100
360 7
= 221.52  Simplify the expression (3mks)
num = 3x^{2}  4xy + y^{2}
3x^{2}  3xy  xy + y^{2}
3x (x  y)  y(x  y)
(3x  y)(x  y)
den = 9x^{2}  y^{2}
= (3x  y)(3x  y)
num = (3x  y) (x  y)
den (3x  y) (3x  y)
= x  y
3x  y  Use the reciprocal tables to evaluate (3mks)
(3.21 x 10^{3})^{1} = 0.3115 x 10^{3}
= 311.5
(5.86 x 10^{1})^{1} = 0.1706 x 10
= 1.706
=(5 x 311.5) + 23(1.906)
=1557.5 + 39.238
= 1596.738  The volumes of two similar solid cylinders are 4752cm^{3} and 1408cm^{3}. If the area of the curved surface of the smaller cylinder is 352cm^{2}, find the area of the curved surface of the larger cylinder (4mks)
v.s.f = 4752/1408 = 3.375
l.s.f = (3.375)^{1/3} = 1.5
a.s.f = 1.62 = 2.25
area of larger cylinder = 2.25 x 352
= 792 cm^{2}  A bus takes 195 minutes to travel a distance of (2x +30) km at an average speed of (x20)km/h. Calculate the actual distance travelled. Give your answer in kilometres (3mks)
195 min = 13/4 hr
13/4 = 2x + 30
x  20
13x  260 = 8x + 120
5x = 380
x = 76 km
distsance = 2x + 30
=2(76) + 30
= 182  Given that OA =3i 2j and OB=4i+j. Find the distance between points A and B correct to 1 decimal place (2mks)
 Murimi and Naliaka had each 288 tree seedlings.Murimi planted equal number of seedlings per row in x rows while Naliaka planted equal number of seedlings in (x+1) rows. The number of tree seedlings planted by Murimi in each row were 4 more than those planted by Naliaka in each row. Calculate the number of seedling Murimi planted in each row. (4 mks)
288  288 = 4
x x + 1
288 (x  1)  288x = 4
x(x + 1)
288x + 288  288x = 4x(x + 1)
288 = 4x^{2} + 4x
x^{2} + x  72 = 0
x2 + 9x  8x  72 = 0
x(x + 9)  8(x + 9) = 0
(x  8)(x + 9) = 0
x = 8 or 9 hence
x = 9
SECTION II.(ANSWER FIVE QUESTIONS ONLY
 Two business partners Mary and John contributed Ksh.60000 and Ksh90000 respectively to start up a business. After 6 months, Lucy joined the business partnership and contributed Ksh.100000. At the end of the year, a gross profit of Ksh.720000 was realized. They then agreed to reinvest 30% of the profit accrued back into the business and use 20% of the profit for running the business operations. The remainder was to be shared among the business partners in the ratio of their contribution.
 Calculate the amount;
 Reinvested into the business (2mks)
0.3 x 720, 000 = 216 000  Used for business operations (1mk)
0.2 x 720,000 = 144,000
 Reinvested into the business (2mks)
 Calculate the amount of profit each partner got (4mks)
60,000 x 1 : 90,000 x 1 : 100,000 x 0.5
60 : 90 : 50
6 : 9 : 5
720, 000  (216 000 + 144,000) = 360,000
6/20 x 360000 = 108,000
9/20 x 360000 = 160000
5/20 x 360000 = 90000  If the amount put back into the business was added to individual’s shares proportional to their initial contribution, find the amount of each partner’s shares (3mks)
60000 + 6/20 x 216000 = 124,800
90,000 + 9/20 x 216,000 = 187,200
100,000 + 5/20 x 216,000 = 154,000
 Calculate the amount;
 A bus left Nairobi at 7.00a.m and travelled towards Eldoret at an average speed of 80km/h. At 7.45a.m a car left Eldoret towards Nairobi at an average speed of 120km/h. Given that the distance between Nairobi and Eldoret is 300km. Calculate;
 The time the bus arrived at Eldoret (2mks)
time taken = 300
80
= 3 hr 45 min
7.00 + 3.45 = 10.45 am  The time of the day, the two met (3mks)
distance covered by the bus at 7.45 am = 80 x 3/4 = 60 km
remaining distance = 300  60 = 240km
time taken to meet = 240/200
= 1 hr 12 min
time = 7.45 + 1.12
= 8.57 am  The distance from Nairobi where the two met (2mks)
60 km + (80 x 1.2)
60 + 96 = 156 km  The distance of the bus from Eldoret when the car arrived in Nairobi (3 mks)
time taken by the car to reach
Nairobi = 300 = 2^{1}/_{2}
120
Distance of bus from nairobi =
60 + (80 x 2.5) = 260 km
distance from Eldoret = 300  260
= 40 km
 The time the bus arrived at Eldoret (2mks)
 The figure below shows a solid which is a rectangular based pyramid of height 12cm.
 Calculate the slanting length VC (2mks)
 Calculate the surface area of the pyramid (4mks)
 Calculate the volume of the pyramid (2mks)
volume = 1/3 BA x H
1/3 x 48 x 13
= 192 cm^{3}  Determine the density of the metal which make this solid if its mass is 1.632kg (2mks)
density = m/v
= 1.632 x 1000
= 1632 g
D = 1632/192
= 8.5 g/cm^{3} or 8500 kg/m^{3}
 Calculate the slanting length VC (2mks)
 In the figure below OJKL is a parallelogram in which OJ = 3p ̃ and OL = 2 r
̃ If A is a point on LK such that LA = ½ AK and point B divide the line JK externally in the ratio 3:1, express OB and AJ in terms of p and r. (2 marks)
OB = 3P +3r
AJ = 2P  2r  Line OB interests AJ at X such that OX = mOB and AX = nAJ.
 Express OX in terms of p, r and m. (1 mark)
OX = m(OB) = m(3p + 3r)
=3mp + 3mr  Express OX in terms of p, r and n (1 mark)
OX = 2r + p + n(2p  2r)
= (1 + 2n)p + (2  2n)r  Determine the value of m and n and hence the ratio in which point x divides line AJ. (6 marks)
3mp + 3mr = (1 + 2n)p + (2  2n)r
3m = 1 + 2n .... (i)
3m = 2  2n .....(ii)
1 + 2n = 2  2n
4n = 1
n = 1/4
substituting n = 1/4 in eqn (i)
3m = 1 + 2(1/4)
3m = 1 + 1/2
3m = 3/2
m = 1/2
the ratio in which x divides AJ
AX = nAJ = 1/4 AJ
Ratio 1:3
 Express OX in terms of p, r and m. (1 mark)
 If A is a point on LK such that LA = ½ AK and point B divide the line JK externally in the ratio 3:1, express OB and AJ in terms of p and r. (2 marks)
 The figure below shows two triangles, ABC and BCD with a common base BC=3.4cm, AC=7.2cm, CD=7.5cm and <ABC=90°. The area of triangle ABC=Area of triangle BCD.
Calculate correct to one decimal place; The area of triangle ABC (3mks)
AB = √7.2^{2}  3.4^{2}
= 6.3 x 67
A = 1/2 x 3.4 x 6.3467
= 10.789
=10.8 1 d.p  The size of <BCD (3mks)
area of traingle ABC = Area of triangle BCD
10.8 = 1/2 x 3.4 x 7.5 sin θ
10.8 = 12.75 sin θ
sin θ = 10.8/12.75 = 0.84705
θ = 57.9 acute angle
obtuse angle = 180  57.9
= 122.1 1 d.p  The length of BD (2mks)
a^{2} = b^{2} + c^{2}  2bc cos A
a^{2} = 3.42 + 7.52  2 x 3.4 x 7.5 cos 122.1
a^{2} = 94.91
a = 9.742 = 9.7 )1 d.p)  The size of <BDC (2mks)
3.4 = 9.7
sinθ sin 122.1
sin θ = 3.4 x sin 122.1 = 0.2969
9.7
θ = 17.273
= 17.3 1 d.p
 The area of triangle ABC (3mks)
 The equation of a curve is y=x^{3}+4x^{2} 3.
 Fill in the table below for the curve y=x^{3}+4x^{2} – 3 (2mks)
x
4
3
2
1
0
1
y
3 6 5 0 3 2  On the grid provided, draw the curve y=x^{3}+4x^{2}  3 for the range 4≤x≤1 scale: Horizontal axis 2cm represent 1 unit vertical axis 1cm rep 1 unit (3mks)
 Use your graph to solve
 x_{3 }+4x^{2} – 3 = 0 (2mks)
ans = x_{3} = 0.8 ± 0.05  4x^{3} +16x^{2} –x – 16 = 0 (3mks)
x_{3} = 0.95 ± 0.05
 x_{3 }+4x^{2} – 3 = 0 (2mks)
 Fill in the table below for the curve y=x^{3}+4x^{2} – 3 (2mks)

 A straight line L1 whose equation is y2x=4 meets the xaxis at M. Determine coordinates of M. (1mk)
y  2x = 4
0  2x = 4
2x = 4
x = 2
(2, 0)  A second line L_{2} is perpendicular to L_{1} at M. Find the equation of Line L_{2} in the form ax+by+c=0 where a,b and c are integers. (3mks)
m_{1} x m_{2} = 1
m_{1} = 2
m_{2} = 1/2
y  0 = 1/2
x + 2
2y = x + 2
x  2y  2 = 0
x + 2y + 2 = 0  A third line L_{3} passes through (2,3) and is parallel to L_{1}. Find;
 The equation of L_{3} in the form y=mx +c (3mks)
ans
y = 2x  1  Point N, the intersection of L_{2} and L_{3} (3mks)
l_{2} = x + 2y = 2
l_{3} = 2x  y = 1
2x + 4y = 4
2x  y = 1
5y = 5
y = 1
x + 2(1) = 2
x  2 = 2
x = 0
p (0, 1)
 The equation of L_{3} in the form y=mx +c (3mks)
 A straight line L1 whose equation is y2x=4 meets the xaxis at M. Determine coordinates of M. (1mk)
 A curve is represented by the function y=x^{3} +x^{2} – x
 Find ^{dy}/_{dx} (1mk)
y = x^{3} + x^{2}  x
dy/dx = 3x^{2} + 2x  1  Determine the values of y at the turning points of the curve y =x^{3} +x^{2} –x (4mks)
 Determine the nature of the turning points (2mks)
 In the space provided below, sketch the curve of y=x^{3}+x^{2} –x (3mks)
 Find ^{dy}/_{dx} (1mk)
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