Instructions to candidates
- Write your name and class 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 any 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.
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
Attempt ALL questions in this section.
- Use logarithms to evaluate the value of
Give your answer correct to 4 significant figures. (3marks) - Make x the subject of the formula (3marks)
- Without using a calculator or mathematical tables, express in surd form and simplify. (3marks)
- Find the centre and radius of a circle whose equation is 3x2 + 3y2 + 18x - 12y +39 = 12 (3marks)
- Under a transformation whose matrix is , an object of area 12cm2 mapped onto an image whose area is 60cm2. Find the possible values of x. (3marks)
-
- Expand the binomial (2marks)
- Using the first 4 terms of the binomial above solve for 1.755 (2 marks)
- Solve for x in the equation -3 sin2x + 8cosx = 0 for 0° ≤ x ≤ 360° (3marks)
- Given 4.50 ÷ 5.0; Find the percentage error in the quotient. Give your answer to 4 s.f. (3marks)
- Find the equation of tangent to the curve y = (2x2+1) (x-3) at x = 1. (3 marks)
- Given that (3 marks)
- Use the trapezoidal rule to approximate the area bounded by the curve y = -x2-3x+10 and the x-axis by using 4 trapezoids of equal width from x = -4 to x = 0 (3 marks)
- Maina bought a new laptop on hire purchase. The cash value of the laptop was Ksh. 56,000. He paid a deposit of Ksh. 14,000 followed by 24 equal monthly installments of Ksh.3500 each. Calculate the monthly rate at which the compound interest was charged. (3marks)
- Water flowing at a rate of 2m/sec through two pipes of diameter 3cm and 5cm respectively deliver water to a 6 cm diameter pipe. Calculate the speed of flow in the 6 cm pipe if a1. are kept full. Give your answer in m/sec (3 marks)
- A quantity A is partly constant and partly varies inversely as a quantity B. Given that A = -10 when B= 2.5 and A = 10 when B = 1.25, find the value of A when B = 1.5. (4marks)
- A coffee dealer mixes two brands of coffee, x and y to obtain 40kg of the mixture worth Ksh. 2,600. If brand x is valued at Ksh. 70 per kg and brand y is valued at Ksh. 55 per kg. Calculate the ratio in its simplest form in which brands x and y are mixed. (3marks)
- The figure below shows a circle centre O. AB and PQ are chords intersecting externally at a point C. AB = 9cm, PQ= 5cm and QC = 4cm. Find the length of BC. (3marks)
SECTION II
Answer five questions only in this section.
- The marks of 50 students in a mathematics test were taken from a form 4 class and recorded in the table below.
Marks 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100 Frequency 2 5 7 9 11 8 5 3 - On the grid provided, draw a cumulative frequency curve of the data, using 1cm to represent students and icm to represent 10 marks.
(3marks) - From your curve in (a) above;
- Estimate the median mark (1mark)
- Determine the interquartile deviation (2marks)
- Determine the 10th to goth percentile range. (2marks)
- It is given that students who score over 45 marks pass the test. Use your graph in (a) above to estimate the percentage of students that pass.
(2marks)
- On the grid provided, draw a cumulative frequency curve of the data, using 1cm to represent students and icm to represent 10 marks.
- The table below shows the rate at which income tax was charged for all income earned in the year 2015
Taxable income per month in k£ Rate of tax per k£ 1 - 236 10% 237-472 15% 473 - 708 20% 709 - 944 25% 945 and above 30% - A tax of Ksh 1200 was deducted from Mr. Rono's monthly salary. He was entitled to a personal relief of Ksh 1064 per month. Calculate his monthly
- Gross tax in k£. (1mark)
- Taxable income in Ksh (5marks)
- He was entitled to a house allowance of Ksh 3000 and medical allowances of Ksh 2000 calculate his monthly basic salary in Ksh. (2mark)
- Every month the following deductions were made from his salary electricity bill of sh. 680, water bill of sh 460, co-operative shares of sh 1250 and loan repayment of sh 2000 calculate his net salary in Ksh. (2marks)
- A tax of Ksh 1200 was deducted from Mr. Rono's monthly salary. He was entitled to a personal relief of Ksh 1064 per month. Calculate his monthly
-
- An arithmetic progression is such that the first term is -5, the last term is 135 and the sum of progression is 975.
- The number of terms in the series. (3marks)
- The common difference of the progression. (3marks)
- The sum of the first three terms of a geometric progression is 27 and first term is 36. Determine the common ratio and the value of the fourth term. (4marks)
- An arithmetic progression is such that the first term is -5, the last term is 135 and the sum of progression is 975.
- The diagram below shows a straight line 2x + y = 8 intersecting the curve y = 2x2 - 4x + 4 at the points P and Q.
- Find the coordinates of P and Q. (3marks)
- Calculate the area of the shaded region. (4 marks)
- Find the coordinates of the stationary points on the curve y = 2x2 - 4x + 4 (3 marks)
- The diagram below shows the frustum of a rectangular based pyramid. The base ABCD is a rectangle of side 24cm by 12cm. The top EFGH is a rectangle of side 14cm by 7cm.Each of the slanting edges of the frustum is 13cm.
Determine the:- Altitude of the frustum (4marks)
- Angle between the line AG and the base ABCD (3marks)
- Volume of the frustum (3marks)
- The probability that three candidates; Anthony, Beatrice and Caleb will pass an examination are 3/4, 2/3 and 4/5 and respectfully. Find the probability that:
- All the three candidates will pass (2marks)
- All the three candidates will not pass. (2marks)
- Only one of them will pass (2marks)
- Only two of them will pass. (2marks)
- At most two of them will pass. (2marks)
- In the figure below PQR is a tangent to the circle at Q. TS is a diameter and TSR and UV are straight lines. QS is parallel to TV. Angles SQR = 40° and TQV = 55°.
- Find the following angles giving reasons each case.
- ∠QTS (2marks)
- ∠QRS (2marks)
- ∠QVT (2marks)
- ∠QUT (2marks)
- Given that QR-8cm, and SR = 4cm. Find the radius of the circle. (2marks)
- Find the following angles giving reasons each case.
- A firm has a fleet of vans and trucks. Each van can carry 9 crates and 3 cartons. Each truck can carry 4 crates and 10 cartons. The firm has to deliver not more than 36 crates and at least 30 cartons.
- If x vans and y trucks are available to make the delivery. Write down inequalities to represent the above information. (4 marks)
- Use the grid provided, to represent the inequalities in (a) above (4 marks)
- Given that the cost of using a truck is four times that of using a van, determine the number of vehicles that may give minimum cost (2 Marks)
MARKING SCHEME
- Use logarithms to evaluate value of. Give your answer correct to 4 significant figures
- Make x the subject of the formula
(2p)2 = x+2w
4x + 3R
4P2(4x + 3R) = x+ 2w
16P2x + 12P2R=x+2w
16P2x - x = 2w -12P2R
x2(16P2 -1)=2w-12P2R
X = 12w - 12P2R
16P2 - 1 - Without using calculator or mathematical tables, express in surd form and simplify
- Find centre and radius of a circle
x2 +y2 +6x-4y= -9
(x + 3)2 + (y-2)2 = -9 + 9 + 4
Centre (-3,2)
Radius = 2 units - Find the possible values of x
(x - 1)(x + 1)-3 = 60
12
x2 -1-3=5
x2 = 9
x = ±3 -
- Expand binomial
coef: 1, 5, 10, 10, 5,1
1.25 (-1)° +5.24(-1)1 + 10.23(-1)2 + 10.22(-1)3 +5.21(-1)4 + 1.2°(-1)5
4x 4x 4x 4x 4x 4x
32 - 20 + 5 - 5 + 5 - 1
x x2 8x3 128x4 1024x5 - Using first 4 terms of the binomial above solve or 1.755
(2-0.25)5 = (2- 1/4x) -0.25 = -1/4x
x = -1
4 x 0.25
X = 1
= 32-20+5 - 5
= 16.375
- Expand binomial
- solve for x
- find percentage error in the quotient in 4 s.f
- Find equation of tangent to the curve
y = (2x2 + 1)(x - 3)
y=2x3- 6x2 + x - 3
dy = 6x2 -12x+1
dx
X = 1
dy = 6 -12 +1 = -5
dx
@x=1
y = 6
Equation
y + 6= -5
x - 1
y + 6 = -5+5
y=-5x - 1 - Find /AB/
AB = 2i-j-2k
i +21 - 3h-
i-3j+k - Approximate area bounded by the curve
x -4 -3 -2 -1 0 y 6 10 12 12 10
A = 1/2(16 + 2(34))
A = 1/2 × 84
A = 42 square units - Calculate monthly rate at which compound interest was charged
Total interest amount = 34,000
Principal = 56,000
(C.P. Dep14.000)
42,000
84,000 - 42,000 (1 + R)24
100
84.000 = (1 +B)24
42,000 100
24√2 = (1+R)
100
1.0293-1 = R
100
R = 2.93% P.m - Calculate speed of flow in the 6cm pipe if all are kept frull
Vol. delivered by 2 pipes in 1 sec.- 22/7 x 1.52 x 200 = 14142/7cm
- 22/7 x 2.52 x 200 = 27500cm3
= 39284/7cm3
Total Vol. 53426/7
22/7 x 32 x h=53426/7
h =53426/7 x 7
22 x 9
h= 1883/9
speed = 1.889m/sec
- Find value o A when B-1.5
- Calculate the ratio in its simplest form in which brands x and y are mixed Cost of 1kg mixture
2600 = 65
40
2 : 1 - Find length of BC
x(x +9)= 9 x 4
x2 + 9x -36 = 0
Section II
-
- On graph provided draw a cumulative frequency curve of the data, using 1 cm to represents students and lcm to represent 10 marks
- from curve
- Estimate median mark
1/2 x 60 = 30th = 65 - Determine interquartile deviation
Q3 = 3/4 x 60 = 45th 71.5
Q1 = 1/4 x 60 = 15th 51.5
20.0
Interquartile=1/2 x 20 = 10 - Determine 10th to 90th percentile range
10th = 10 x 60 = 6th = 39
100
90th = 90 x 60 - 54th = 84
100
- Estimate median mark
- Use graph to estimate percentage of students that pass
50 students passed
= 50 x 100
60
= 831/3% passed
- On graph provided draw a cumulative frequency curve of the data, using 1 cm to represents students and lcm to represent 10 marks
-
- Calculate his monthly
- Gross tax in K£
= 1200 + 1064 = Ksh. 2264
2264
20
= K£ 113.2 - Taxable income in Ksh.
1st brachet = 236 x 10/100 = K£ 23.6
2ndbrachet = 236 x 0.15 = K£ 35.4
3rdbrachet = 236 x 0.2 = K£ 47.2
4thbrachet = a x 0.25 K£ 7.0
a = 7 - 28
0.25
Taxable income
2.6 x 3+28 736
= K£ 736 x 20
=Kshs. 14,720
- Gross tax in K£
- Calculate his monthly basic salary in Ksh.
Basic salary - 14,720-(3,000+ 2,000)
= Ksh. 9.720 - Calculate net salary in Ksh.
Net salary
14,720 - (1,200 +680 + 460 + 1250 + 2000)
=Kshs. 9,130
- Calculate his monthly
-
-
- The number of terms in the series
Sn = n(a+1)
2
975 = n(-5+135)
2
n = 975 x 2
130
n = 15 - The common difference of the progression
S15 = n (2a + (n - 1)d
975 = 15 (2x-5 + (15-1)d)
2
975 = -10 + 14d
7.5
130 + 10 = 14d
d=13
- The number of terms in the series
- Determine common ratio and the value of the fourth term
a+ar+ar2 = 27
36 + 36r + 36r2 = 27
36r2+ 36r+9 = 0
4r2 + 4r + 1 = 0
-
-
- Find the coordinates of P and Q
2x2 - 4x + 4 = -2x + 8
2x2 - 4x + 2x + 4-8 = 0
2x2 - 2x - 4 = 0
x2 - x -2 = 0
x=1 ± 3
2
x = 2 or-1
P(-1, y=8-(2x-1)
P(-1, 10)
X = 2, Y = 8- (2 x 2)
= 4
Q = (2,4) - Calculate the area of the shaded region
Area of trapezium
A= 1/2 (10 + 4) × 3
= 7x 3 = 21
Area under curve
*(2x2 - 4x + 4)dx
12x3 - 2x2 + 4x + c)2-t
3
= (16 - 8 + c) - (-2 - 2 - 4 + c)
3
= 16 + c + 20 - c
3 3
Area = 12 sq units
Shaded area
= 21-12 = 9 square units - Find coordinates of stationery points on the curve
@ Stationary point
oy=0
ox
oy = 4x - 4 = 0
ox
4π = x
X = 1
When x=1
y = 2 - 4 +4
y = 2
Coordinates A stationary point is (1,2)
- Find the coordinates of P and Q
- Determine
- Altitude of the frustum
- Angle between line AG and the base ABCD
= √720 - 5.590
= 21.24
TanD = 11.74
21.24
= 28.93° - Volume of the frustum
12 = x + 11.74
7 x
12x = 7x + 82.18 X = 15.436
V=1/3 x 24 x 12 x 28.176 - 1/3 x 14 x 7 x 16.436
= 2704.896-536.909
= 2167.987cm2
- Altitude of the frustum
- Find probability that:
- All the three candidates will pass
P(ABC)= 3/4 x 2/3 x 4/5 = 2/3 - All the three candidates will not pass
P(A'B'C')= 1/4 x 1/3 x 1/5 = 1/60 - Only one of them will pass
=P (A'B'C') + P(A'B'C) + P(ABC)
=(3/4 x 1/3 x 1/5) + (1/4 x 2/3 x 1/5) + (1/4 x1/3 x 4/5) - Only two of them will pass
= P (ABC') + P (AB'C) + P (A'BC)
= (3/4 × 2/3 × 1/5) + (3/4 × 1/3 × 4/5) + (1/4 × 2/3 × 4/5)
= 1/10 + 1/5 + 2/15 = 13/30 - Almost two of them will pass
P (one or 2 passes)
=3/20 + 12/20
=7/12
- All the three candidates will pass
-
- Fins the following angles giving reasons each case
- <QTS
<QTS = <SOR = 40°
(Alternate seg. Theorem) - <ORS
<TSQ = 90 - 40 - 50°
<OSR = 180 - 50o = 130°
(Angles on a straight line adds up to 180°)
<QRS-180-(130+40) -10°
(Angles in a triangle adds up to 180) - <OVT
<SQR-90-55-35°
<SOR = <QVT = 35°
(Alternate angles)
(check for a complete reason) - <OUT
<QUT = <OST 50°
(angles subtended by the same chord in the same segment are equal)
- <QTS
- Given that QR-8cm, and SR = 4 cm, Find the radius of the circle
Ts=diameter = x
82 = (x + 4) 4
64 = 4x + 16
4x = 48
x= 12
Diameter = 12cm
Radius - 6cm
- Fins the following angles giving reasons each case
-
- if x vans and y trucks are available to make delivery. Write down inequalities to represent the above information
- 9x + 4y ≤ 36
- 3x + 10y ≥ 30
- y > 0
- x>0
- Use grid provided, to represent the inequalities in (a) above
- Given that cost of using a tuck is four times that of using a van, Determine number of vehicles that may give minimum cost
y=4 cost of trucks=y
x = then cost of van = 1/4y
1 van, 3 trucks
- if x vans and y trucks are available to make delivery. Write down inequalities to represent the above information
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