## Mathematics Paper 2 Questions and Answers - Sukellemo Joint Mock Examinations July 2020

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.

1. Use logarithms to evaluate the value of

2. Make x the subject of the formula (3marks)
3. Without using a calculator or mathematical tables, express  in surd form and simplify. (3marks)
4. Find the centre and radius of a circle whose equation is 3x2 + 3y2 + 18x - 12y +39 = 12  (3marks)
5. 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)
6.
1. Expand the binomial  (2marks)
2. Using the first 4 terms of the binomial above solve for 1.755  (2 marks)
7. Solve for x in the equation -3 sin2x + 8cosx = 0 for 0° ≤ x ≤ 360° (3marks)
8. Given 4.50 ÷ 5.0; Find the percentage error in the quotient. Give your answer to 4 s.f. (3marks)
9. Find the equation of tangent to the curve y = (2x2+1) (x-3) at x = 1. (3 marks)
10. Given that (3 marks)
11. 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)
12. 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)
13. 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)
14. 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)
15. 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)
16. 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.

1. 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
1. On the grid provided, draw a cumulative frequency curve of the data, using 1cm to represent students and icm to represent 10 marks.
(3marks)
2. From your curve in (a) above;
1. Estimate the median mark (1mark)
2. Determine the interquartile deviation (2marks)
3. Determine the 10th to goth percentile range. (2marks)
3. 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)
2. 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%
1. 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
1. Gross tax in k£. (1mark)
2. Taxable income in Ksh (5marks)
2. He was entitled to a house allowance of Ksh 3000 and medical allowances of Ksh 2000 calculate his monthly basic salary in Ksh. (2mark)
3. 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)
3.
1. An arithmetic progression is such that the first term is -5, the last term is 135 and the sum of progression is 975.
1. The number of terms in the series. (3marks)
2. The common difference of the progression. (3marks)
2. 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)
4. The diagram below shows a straight line 2x + y = 8 intersecting the curve y = 2x2 - 4x + 4 at the points P and Q.

1. Find the coordinates of P and Q. (3marks)
2. Calculate the area of the shaded region. (4 marks)
3. Find the coordinates of the stationary points on the curve y = 2x2 - 4x + 4  (3 marks)
5. 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:
1. Altitude of the frustum (4marks)
2. Angle between the line AG and the base ABCD (3marks)
3. Volume of the frustum (3marks)
6. 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:
1. All the three candidates will pass (2marks)
2. All the three candidates will not pass. (2marks)
3. Only one of them will pass (2marks)
4. Only two of them will pass. (2marks)
5. At most two of them will pass. (2marks)
7. 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°.
1. Find the following angles giving reasons each case.
1. ∠QTS (2marks)
2. ∠QRS (2marks)
3. ∠QVT (2marks)
4. ∠QUT (2marks)
2. Given that QR-8cm, and SR = 4cm. Find the radius of the circle. (2marks)
8. 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.
1. If x vans and y trucks are available to make the delivery. Write down inequalities to represent the above information. (4 marks)
2. Use the grid provided, to represent the inequalities in (a) above (4 marks)
3. 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

1. Use logarithms to evaluate value of. Give your answer correct to 4 significant figures
2. 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
3. Without using calculator or mathematical tables, express in surd form and simplify
4. Find centre and radius of a circle
x2 +y2 +6x-4y= -9
(x + 3)2 + (y-2)2 = -9 + 9 + 4
Centre (-3,2)
5. Find the possible values of x
(x - 1)(x + 1)-3  = 60
12
x2 -1-3=5
x2 = 9
x = ±3
6.
1. 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
2. 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
7. solve for x
8. find percentage error in the quotient in 4 s.f
9. 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
10. Find /AB/
AB = 2i-j-2k
i +21 - 3h-
i-3j+k
11. Approximate area bounded by the curve
 x -4 -3 -2 -1 0 y 6 10 12 12 10
A = 1/2 × 1[(6 + 10) + (10 + 12 + 12)]
A = 1/2(16 + 2(34))
A = 1/2 × 84
A = 42 square units
12. 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
13. Calculate speed of flow in the 6cm pipe if all are kept frull
Vol. delivered by 2 pipes in 1 sec.
1. 22/7 x 1.52 x 200 = 14142/7cm
2. 22/7 x 2.52 x 200 =  27500cm3
= 39284/7cm3
Total Vol. 53426/7
22/x 32 x h=53426/7
h =53426/x 7
22 x 9
h= 1883/9
speed = 1.889m/sec
14. Find value o A when B-1.5
15. Calculate the ratio in its simplest form in which brands x and y are mixed Cost of 1kg mixture
2600 = 65
40

2 : 1
16. Find length of BC
x(x +9)= 9 x 4
x2 + 9x -36 = 0

Section II

1.
1. On graph provided draw a cumulative frequency curve of the data, using 1 cm to represents students and lcm to represent 10 marks
2.  from curve
1. Estimate median mark
1/2 x 60 = 30th = 65
2. 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
3. Determine 10th to 90th percentile range
10th =  10 x 60 = 6th = 39
100
90th = 90 x 60 - 54th = 84
100
3. Use graph to estimate percentage of students that pass
50 students passed
= 50 x 100
60
= 831/3% passed
2.
1. Calculate his monthly
1. Gross tax in K£
= 1200 + 1064 = Ksh. 2264
2264
20
= K£ 113.2
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
2. Calculate his monthly basic salary in Ksh.
Basic salary - 14,720-(3,000+ 2,000)
= Ksh. 9.720
3. Calculate net salary in Ksh.
Net salary
14,720 - (1,200 +680 + 460 + 1250 + 2000)
=Kshs. 9,130
3.
1.
1. The number of terms in the series
Sn  = n(a+1)
2
975 = n(-5+135)
2
n = 975 x 2
130
n = 15
2. 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
2. 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
4.
1. 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)
2. 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
= 21-12 = 9 square units
3. 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)
5. Determine
1. Altitude of the frustum
2.  Angle between line AG and the base ABCD
= √720 - 5.590
= 21.24
TanD = 11.74
21.24
= 28.93°
3. Volume of the frustum
12 =  x + 11.74
7                x
12x = 7x + 82.18        X = 15.436
V=1/x 24 x 12 x 28.176 - 1/x 14 x 7 x 16.436
= 2704.896-536.909
= 2167.987cm2
6. Find probability that:
1. All the three candidates will pass
P(ABC)= 3/4 x 2/3 x 4/5 = 2/3
2. All the three candidates will not pass
P(A'B'C')= 1/4 x 1/3 x 1/5 = 1/60
3. 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)
4. 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
5. Almost two of them will pass
P (one or 2 passes)
=3/20 + 12/20
=7/12
7.
1. Fins the following angles giving reasons each case
1. <QTS
<QTS = <SOR = 40°
(Alternate seg. Theorem)
2. <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)
3. <OVT
<SQR-90-55-35°
<SOR = <QVT = 35°
(Alternate angles)
(check for a complete reason)
4. <OUT
<QUT = <OST 50°
(angles subtended by the same chord in the same segment are equal)
2. 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
8.
1. if x vans and y trucks are available to make delivery. Write down inequalities to represent the above information
1. 9x + 4y ≤ 36
2. 3x + 10y ≥ 30
3. y > 0
4. x>0
2. Use grid provided, to represent the inequalities in (a) above
3. 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

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