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 I, II.
 Answer all the questions in section 1 and any five questions from section II.
 All working and answers must be written on the question paper in the spaces provided below each question.
 Show all steps in your calculations, giving 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.
QUESTIONS
SECTION 1 (50MARKS)
 Use logarithms to evaluate (4mks)
 A rectangular card measures 5.3cm by 2.5cm. find
 The absolute error in the area of the card. (2mks)
 The relative error in the area of the card (2mks)
 Solve the equation (4mks)
Sin (2x +10)º=0.5 for 0º ≤ x≤ 360º  In a transformation, an object with an area of 52cm2 is mapped onto an image whose area is 30cm^{2}. Given that the matrix of the transformation is find the value of x (3mks)
 Simplify leaving your answer in the form of a√b + c where a, b and c are integers. (3mks)
 A customer deposited sh 14000 in a saving account. Find the accumulated amount after one year if interest was paid at 12% p.a compounded quarterly (3mks)
 Expand (1+x)^{5}, hence use the expansion to estimate (1.04)^{5} correct to 4 decimal place (3mks)
 Find the centre and the radius of circle whose equation is (3mks)
x^{2}+4x+y^{2}5=0  Make d the subject of the formula (3mks)
P=1/2mn^{2}gd^{2} n  In what proportion should grades of sugar costing sh 45 and sh 50 per kg be mixed in order to produce a blend worth sh 48 per kg (3mks)
 Simplify the expression (3mks)
 Find the equation of the tangent to the curve (3mks)
y=2x^{2} at (2, 3)  Use matrix method to solve the given simultaneous equation (3mks)
3x+y=7
5x+2y=12  The sum of n terms of the sequence 3, 9, 15, 21 ... is 7500. Determine the value of n (3mks)
 The figure below (not drawn to scale) shows a triangle ABC in which AB=6cm, BC=9cm, AC=10cm. calculate the radius of the circle touching the three vertices of the triangle. (3mks)
 The point p (40ºS, 45ºE) and point Q (40ºS, 60ºW) are on the surface of the earth. Calculate the shortest distance along a circle of latitude between the two points. (3mks)
 The table below shows monthly income tax rates.
Monthly taxable pay K£
Rate of tax sh per K£
1342
343684
6851026
10271368
13691710
Over 1710
2
3
4
5
6
7
 calculate the civil servant taxable pay in K£ (4mks)
 Calculate the total tax (4mks)
 If the employee is entitled to a tax relief of sh 600 per month. What is the net tax paid? 2mks)
 In an agricultural research centre, the length of a sample of 50 maize cobs were measured and recorded as shown in the frequency distribution table below.
Length in cm
Number of cobs
810
1113
1416
1719
2022
2325
4
7
11
15
8
5
 The mean (2mks)

 the variance (5mks)
 The standard deviation (3mks)
 In the diagram shown below O is the centre of the circle, angle RTV=1500,and angle RST=500,
 Calculate the size of
 <ORS (2mks)
 <USP (1mk)
 <PQR (2mks)
 Given that RT =7cm and ST=9cm, calculate to 3.s.f
 The length of line PR (2mks)
 The radius of the circle (3mks)
 Calculate the size of
 The position of two towns A and B on the earth surface are (36ºN, 49ºE) and (36ºN, 131ºW) respectively
 Find the difference in longitude between town A and town B (2mks)
 Given that the radius of the earth is 6370km calculate the distance between town A and B (4mks)
 Another town C is 840km east of town B and on the same latitude as towns A and B. find the longitude of town C (4mks)
 The distance sm from a fixed point O, covered by a particle after ts is given by the equation
S=t^{3}6t^{2}+9t+5 Calculate the gradient to the curve at t=0.5s (3mks)
 Determine the values of s at the maximum and minimum turning points of the curve. (4mks)
 On the space provided sketch the curve of s=t^{3}6t^{2}+9t+5 (3mks)
 In the figure below OQ =a and OB=b. m is the midpoint of OA and AN:NB =2:1
 Express in terms of a and b
 BA (1mk)
 BN (1mk)
 ON (2mks)
 Given that BX=hBM and OX=KON determine the values of h and k (6mks)
 Express in terms of a and b

 Complete the table below, giving the values correct to 1 d.p (2mks)
X^{0}
0
40
80
120
160
200
240
2sin (x+20)^{0}
0.7
2.0
0.0
2.0
√3 cos x
1.7
1.3
0.9
1.6
 On the grid provided, using the same scale and axes, draw the graphs of y=2sin (x+20)º and y=√3 cos x for 0≤x≤ 240º (5mks)
 Use the graphs drawn in (b) above to determine:
The values of x for which 2 sin (x+20)º =√3 cos x (2mks)
The difference in the amplitudes of y=2sin(x+20) and y=√3 cox x (1mk)
 Complete the table below, giving the values correct to 1 d.p (2mks)
 The probabilities that a husband and wife will be a live 25 years from now are 0.7 and 0.9 respectively. Find the probability that in 25 years time;
 Both will be a live (2mks)
 Neither will be a live (3mks)
 One will be a live (2mks)
 At least one will be a live (3mks)
MARKING SCHEME
SECTION 1 (50MARKS)
 Use logarithms to evaluate (4mks)
No std log 0.8932 8.932 x 10^{1} 1.9509 x 2 582.3 5.823 x 10^{2} 2.7651
309.36 0.935 x 10^{1} 1.8410  A rectangular card measures 5.3cm by 2.5cm. find
 The absolute error in the area of the card. (2mks)
max V 5.35 min 5.25 max 2.55 min 2.45
max A  min A = M1
2
13.3875  12.8625
2
= 0.2626%  The relative error in the area of the card (2mks)
 The absolute error in the area of the card. (2mks)
 Solve the equation (4mks)
Sin (2x +10)º=0.5 for 0º ≤ x≤ 360º
2x + 10 = sin10.5
= 30º
2x + 10 = 210º, 330º, 570º, 690º
x = 100º, 160º, 280º, 335º  In a transformation, an object with an area of 52cm2 is mapped onto an image whose area is 30cm^{2}. Given that the matrix of the transformation is find the value of x (3mks)
ASF = 30/52
= 15/26 = 4x  2x + 2
15/26 = 2x + 2
x = 37/26  Simplify leaving your answer in the form of a√b + c where a, b and c are integers. (3mks)
 A customer deposited sh 14000 in a saving account. Find the accumulated amount after one year if interest was paid at 12% p.a compounded quarterly (3mks)
A = 14000 (1 + 12/100 x 1/4)^{4}
= 14000(1.03)^{4}
= 15757  Expand (1+x)^{5}, hence use the expansion to estimate (1.04)^{5} correct to 4 decimal place (3mks) r = 3
 Find the centre and the radius of circle whose equation is (3mks)
x^{2}+4x+y^{2}5=0  Make d the subject of the formula (3mks)
P=1/2mn^{2}gd^{2} n
2pn  mn^{3} = 2gd^{2}
d^{2} = mn^{3}  2pn
2q  In what proportion should grades of sugar costing sh 45 and sh 50 per kg be mixed in order to produce a blend worth sh 48 per kg (3mks)
let a:b = ratio
45a + 50b = 48
a + b
45a  48a = 48b  50b
a:b = 2:3  Simplify the expression (3mks)
N:(4m  3n)(4m + 3n)
D: 4m^{3}  4mn + 3mn  3n^{2}
4m(m  n) + 3n(m  n)
(4m + 3n)(m  n)
= 4m  3n
m  n  Find the equation of the tangent to the curve (3mks)
y=2x^{2} at (2, 3)
dy/dx = 4x
at x = 2
m = 2(4)
= 8
(x,y)(2,3)
m = 8
y  3 = 8
x  2 1
y  3 = 8 x 16
y = 8x + 15  Use matrix method to solve the given simultaneous equation (3mks)
3x+y=7
5x+2y=12  The sum of n terms of the sequence 3, 9, 15, 21 ... is 7500. Determine the value of n (3mks)
n/2(2a(n  1)d) = 7500
n/2(6 + (n  1)6) = 7500
6n + 6n^{2} = 15000
6n^{2} = 15000
n^{2} = 2500
n = 50  The figure below (not drawn to scale) shows a triangle ABC in which AB=6cm, BC=9cm, AC=10cm. calculate the radius of the circle touching the three vertices of the triangle. (3mks)
 The point p (40ºS, 45ºE) and point Q (40ºS, 60ºW) are on the surface of the earth. Calculate the shortest distance along a circle of latitude between the two points. (3mks)
Ò = 105º
= 105 x 2 x 22 x 6370 cos 40
360 7
= 8946 km  The table below shows monthly income tax rates.
Monthly taxable pay K£
Rate of tax sh per K£
1342
343684
6851026
10271368
13691710
Over 1710
2
3
4
5
6
7
 calculate the civil servant taxable pay in K£ (4mks)
20000 + 15/100 x 20000  700
22300/20 = 1115t  Calculate the total tax (4mks)
ans = 3523  If the employee is entitled to a tax relief of sh 600 per month. What is the net tax paid? 2mks)
3523
 600
29231
 calculate the civil servant taxable pay in K£ (4mks)
 In an agricultural research centre, the length of a sample of 50 maize cobs were measured and recorded as shown in the frequency distribution table below.
Length in cm
Number of cobs
x x^{2} fx fx^{2} 810
1113
1416
1719
2022
2325
4
7
11
15
8
5
9
12
15
18
21
24
81
144
225
324
441
576
36
84
165
270
168
120
324
1008
2475
4860
3528
2880
∑x = 843 ∑fx = 15075  The mean (2mks)
= 16.86 
 the variance (5mks)
= 17.25  The standard deviation (3mks)
= 4.153
 the variance (5mks)
 The mean (2mks)
 In the diagram shown below O is the centre of the circle, angle RTV=1500,and angle RST=500,
 Calculate the size of
 <ORS (2mks)
= 40º  base < s isosceles angles are equal  <USP (1mk)
= 80º  alternate segments theorem  <PQR (2mks)
= 130º  opposites <s a cycle quadrilateral add up to 180º
 <ORS (2mks)
 Given that RT =7cm and ST=9cm, calculate to 3.s.f
 The length of line PR (2mks)
(x + 7)7 = 9^{2} 7x + 49 = 81 7x = 22
x = 3.142  The radius of the circle (3mks)
PR = 7 + 3.142
= 10.142
3.142 = 2R
sin 50
R = 2.05cm
 The length of line PR (2mks)
 Calculate the size of
 The position of two towns A and B on the earth surface are (36ºN, 49ºE) and (36ºN, 131ºW) respectively
 Find the difference in longitude between town A and town B (2m
= 49 + 131
180= )  Given that the radius of the earth is 6370km calculate the distance between town A and B (4mks)
= 16196 km  Another town C is 840km east of town B and on the same latitude as towns A and B. find the longitude of town C (4mks)
= 121.66ºW
 Find the difference in longitude between town A and town B (2m
 The distance sm from a fixed point O, covered by a particle after ts is given by the equation
S=t^{3}6t^{2}+9t+5 Calculate the gradient to the curve at t=0.5s (3mks)
ds/dt = 3t^{2}  12t + 9
= 3(1/4)  (12)1/2 + 9
= 3/4  6 + 9
= 15/4  Determine the values of s at the maximum and minimum turning points of the curve. (4mks)
 On the space provided sketch the curve of s=t^{3}6t^{2}+9t+5 (3mks)
intercepts at t = 0 at s = s
=> (0,5) min (3,0) max(1,9)
 Calculate the gradient to the curve at t=0.5s (3mks)
 In the figure below OQ =a and OB=b. m is the midpoint of OA and AN:NB =2:1
 Express in terms of a and b
 BA (1mk)
b + a  BN (1mk)
1/3b + 1/3a  ON (2mks)
a + 2/3AB = 1/3a + 2/3b
 BA (1mk)
 Given that BX=hBM and OX=KON determine the values of h and k (6mks)
k = 3/2h
 Express in terms of a and b

 Complete the table below, giving the values correct to 1 d.p (2mks)
X^{0}
0
40
80
120
160
200
240
2sin (x+20)^{0}
0.7
1 2.0
0.0
2.0
√3 cos x
1.7
1.3
0.9
1.6
 On the grid provided, using the same scale and axes, draw the graphs of y=2sin (x+20)º and y=√3 cos x for 0≤x≤ 240º (5mks)
 Use the graphs drawn in (b) above to determine:
The values of x for which 2 sin (x+20)º =√3 cos x (2mks)
The difference in the amplitudes of y=2sin(x+20) and y=√3 cox x (1mk)
 Complete the table below, giving the values correct to 1 d.p (2mks)
 The probabilities that a husband and wife will be a live 25 years from now are 0.7 and 0.9 respectively. Find the probability that in 25 years time;
 Both will be a live (2mks)
0.7 x 0.9 = 0.63  Neither will be a live (3mks)
0.3 x 0.37 = 0.111  One will be a live (2mks)
(0.7 x 0.37) + (0.9 x 0.3) = 0.529  At least one will be a live (3mks)
0.529 x 0.63 = 0.33327
 Both will be a live (2mks)
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