Mathematics Paper 2 Questions and Answers - Mangu High School Mock Exams 2023

INSTRUCTIONS TO  CANDIDATES

• Answer All questions in Section I  and only Five questions  from section II
• All answers and working must be written on the question paper in the spaces provided below each
• Show all the steps in your calculations giving answers at each stage in the spaces provided below each
• Marks may be given for correct working even if the answer is wrong
• Candidates should answer questions in English.

SECTION 1 (50 MARKS)
1.  Factorise x2 – y2, hence evaluate 32822 − 32722               (3mks)
2. Find cos x − Sin x, if tan x = ¾  and 90°≤ x ≤360º                (3mks)
3. Expand  [1-2x]6   up to the fourth term. Hence use your expansion to evaluate (1.02) 6  to four decimal places. (4mks)
4. The average of the first and fourth terms of a GP is 140. Given that the first term is 64. Find the common ratio.                (3mks)
5. Make b the subject of the formula.                (3mks)

6. Two variables P and Q are such that P varies partly as Q and partly as the square root of Q. Determine the equation connecting P and Q. When Q=16, P=500 and when Q = 25, P = 800 (4mks)
7. Calculate the interest on sh 10,000 invested for 1 ½ years at 12 % p.a. Compounded semi-annually.                (3 mks)
8. Given that x=2i+j-2k, y= -3i+4j-k and z =5i + 3j+2k and that P= 3x-y+2z, find the magnitude of vector p to 3 significant figure         (4mks)
9. Eighteen labourers dig a ditch 80m long in 5 days. How long will it take 24 labourers to dig a ditch 64 m long? (3mks).
10. The expression 1+ x/2 is taken as an approximation for √(1+x) Find the percentage error in doing so if x = 0.44                (3mks)
11. The matrices are  and  such that AB = A + B Find a, b, and c.             (3mks)
12. Simplify (3mks)

2x2 – x−1
x2 – 1

13. On map of scale 1:25000 a forest has an area of 20cm2. What is the actual area in Km                (3mks)
14. In the figure below, DC = 6cm, AB = 5cm. Determine BC if DC is a  tangent. (3mks).

15. Evaluate without using logarithm tables.           (3mks)
3 log 10 2 + log 10 750 – log 10 6
16. A bag contains 10 balls of which 3 are  red, 5 are white and 2 green. Another bag contains 12 balls of which 4 are red, 3 are white and 5 are green. A bag is chosen at random and a ball picked at random from the bag. Find the probability that the ball so chosen is red.                   (4mks).
SECTION II (50 MARKS)
17. Income tax is charged on annual income at the rates shown below
 Taxable Income K£ Rate (shs per K£) 1 – 1500  1501 – 3000  3001 – 4500  4501 – 6000  6001 – 7500  7501 – 9000  9001 – 12000  Over 12000 2 3 5 7 9 101213

A certain headmaster earns a monthly salary of Ksh. 8570.. He is entitled to  tax relief of Kshs. 150 per month.

1. How much tax does he pay in a year.                (6 mks)

2. From the headmaster’s salary the following deductions are also made every month;

• W.C.P.S 2% of gross salary

• N.H.I.F Kshs. 1200

• House rent, water and furniture charges Kshs. 246 per month.
Calculate the headmaster’s net salary.                    (4 mks)
18.
1.
1. Taking the radius of the earth, R = 6370 km and p = 22/7 calculate the shorter distance between the two cities P (60°N , 29°W) and  Q (60°N, 31°E) along the parallel of latitude.           (3mks)
2. If it is 1200Hrs at P, what is the local time at Q.         (3mks)
2. An aeroplane flew due South from a point A (60°N, 45°E) to a point B. The distance covered by the aeroplane was 800km. Determine the position of B.          (4mks).
19. Triangle  PQR whose vertices are p(2,2), Q(5,3) and R(4,1) is mapped onto triangle P'Q'R' by a transformation whose matrix is
1. On the grid draw PQR and P1Q1R1.                  (4mks)
2. The triangle P1Q1R1 is mapped onto triangle P11Q11R11 whose vertices are P11(-2,-2), Q11(-5,-3) and R11 (-4,-1)           (2mks)
1. Find the matrix of transformation which maps triangle P1Q1R1 onto P11Q11R11              (2mks)
2. Draw the image P11Q11R11  on the same grid and describe the transformation that maps PQR onto P11Q11R11   (2mks)
3. Find a single matrix of transformation which will map P1Q1R1 on to P11Q11R11 .(2mks)

20.
1. Complete the table for y = Sin x + 2 Cos x.                                 (2mks)
 X 0 30 60 90 120 150 180 210 240 270 300 SinX 0 1 0.5 -0.5 −0.87 2CosY 2 0 −1.73 -1.73 1.0 Y 2 1 −1.23 -2.23 0.13

2. Draw the graph of y = Sin x + 2 cos x.                                               (3mks)
3. Solve sinx + 2 cos x = 0 using the graph.                                           (2mks)
4. Find the range of values of x for which y ≤ −0.5                                 (3mks).
21. A bag contains 3 red, 5 white and 4 blue balls. Two balls are picked without replacement. Determine the probability of picking.
1. 2 red balls                    (2mks)

2. Only one red ball                     (2mks)
3. At least a white ball                    (2mks)
4. Balls of same colour.                     (2mks)
5. Two white balls                     (2mks)
22.
1. Draw the graph of the function                      (2mks)
y = 10+3x – x2 for –2<x <5
2. Use of the trapezoidal rule with 5 stripes, find the area under the curve from x = −1 to x = 4                     (4mks)

3. Find the actual area under the curve from x = – 1 to x = 4.                                                                         (2mks)

4. Find the percentage error introduced by the approximation.                                                                       (2mks)
23. The figure below is a cuboid ABCDEFGH such that AB = 8cm, BC  = 6cm and CF 5cm.

Determine
1. the length
1. AC                    (2mks)
2. AF                   (2mks)
2. The angle AF makes with the plane ABCD. (3mks)
3. The angle AEFB makes with the base ABCD. (3mks)
24. A manager wishes to hire two types of machine. He considers the following facts.
Machine A              Machine B
Floor space                                                 2m2                         3m2
Number of men required to operate             4                              3
He has a maximum of 24m2 of floor space and a maximum of 36 men available. In addition he is not allowed to hire more machines of type B than of type A.
1. If he hires x machines of type A and y machines of type B, write down all the inequalities that satisfy the above conditions. (3mks)
2. Represent the inequalities on the grid and shade the unwanted region.                            (3mks)
3. If the profit from machine A is sh. 4 per hour and that from using B is kshs8 per hour. What number of machines of each type should the manager choose to give the maximum profit?                            (4mks

MARKING SCHEME

1.

(x- y) ( x+y)
( 3282 – 3272) ( 3282 + 3272)
65540

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2.
Tan x = is positive 3rd quadrant
Then sin x = -3/

h =  √( 42 + 32) =  √ 25 = 5
Sin  x = -3
5
Cos x – sin x = - 4   − -   =  -3
5         5       5
=  -1
5

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Identification of the hypotenuse

Cao
Accept  (−0.2)

3. 16 + 6(- ½  x ) + 15(- ½  x )2 + 20(- ½  x )3

= 1 – 3x + 15/4x2  - 5 x3
X = -0.04
1-3 (0.04) +  15/4(0.04)2  − 5/4( 0.04)3
= 1 + 0.12 + 0.006 + 0.000616
= 1.12616
= 1.1262

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For ✓ simplification

For ✓ substitution of x

4. a + ar3 = 140

2

64 + 64 r3 = 140
2
64 + 64 r3 = 280 64 → r3 = 280 – 64
64 r3 = 216 → r = √ 216
64
r =  3
2

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5.

a2  b2d2
b2 – d

a2b2 - a2d = b2d2
a2b2 - b2d2 = a2d
b2(a2 - d2) = a2d

b2    a2d
a2 - d2

=  √( a2)
(a2d2)

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✓ sq on both sides

6. P = aQ + √Q

P = 16a + 4b

( 500 = 16a + 4b)
(800 = 25a + 5b)

2500 = 80a + 20b
3200 = 100a + 20b
-700 = -20a
35 = a

Then b = -15
Equation connecting P and Q
p = 35Q – 15√Q

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For ✓ equation

For ✓ formation of simultaneous equations

For ✓ values of both a and b

7. 1000 (1+ 6/100)3

1000 x 1.063
Ksh 11910.16
Ksh 11910

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8. 4.562 x 0.38 = 1.73356
4 √1.73356      = 1.14745÷ 0.82
= 1.3993
= 1.4
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9. 18 x 64 x 5
24 x 80
6  x 64
8 x 16
3 days
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For ✓ simplification

10 True value = √ (1 + n )= 1.44 = 1.2
Approx. value
1 + = 1 +  44  = 1.22
2             2
= 1.22 – 1.2
= 0.02
1.2 x 100 =
1.2
= 1.67 %

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11.

3a + 0 = 3  + a
3b  + 0  = b
3a = 3 + a   →   a =
2
3b + 0 = b
2b = 0
B = 0
0 + 4c = 4 + c
3c = 4
C=  4
3

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For matrix equation

For forming of simultaneous equation

For values of a, b and c ( correct)

12.  2x2 – 2x  + x -1
( x + 1 ) ( x – 1)
2x ( x – 1 ) + 1 ( x- 1)
( x + 1 ) (n- 1 )
( 2x + 1)
( x + 1 )
= 2x + 1
x + 1
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13. 1    cm = 25000cm
2   1cm = 250m
3   1cm = 0.25
1cm2 = 0.0625
20cm2  = 20 x 0.0625
= 1.25/ cm2
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3
14. AB . BC = DC -2
5: BC = 36
BC =  36
5
= 7.2 cm
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3
15. Log108 + Log10750 - Log106

Log10 Log10 ( 8 x 750)
6
= Log101000

= 3

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3
16.

=   1    +   3
6        20
=   20 + 18
120
= 38      =   19
120          60

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4
17.

Taxable income 115 x 8570 = 9855.50
100
9855.50 x 12 p.a
20
= 5913.30

Tax
1 – 1500                        →     150
1501 – 3000                  →     225
3001 – 4500                  →     375
4501 – 5913.30             →    494.30
1244.30     -
90.00

K £ 1154.30 pa . or
Ksh 1923.83 per month

Total Decuctions
2      x  9855.5
100

= 197.11 ( wcps)
+
20.00
246.00
+
Tax per month 1923.83
2386.94

Net salary
9855.50 – 2386.94
Ksh 7468.65

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10
18 i)   θ   ( 2∏R cos θ)
360
60  x 2 x 22 x 6370 cos 60
360          7
= 1/3 x 22 x 910 x 0.5
= 3336.7
ii) Time (4 x 60) hrs
60
4 hrs.
Local time 1200 + 4
= 1600hrs
b)  θ  x 2πR = 800
360
=   θ     x 2 x πx 6370 = 800
360
θ =  800 x 360
2 x π x 6370
= 7.196°
∠ (60 – 7.196) = 52.80°
( 52.8°N 45°E)

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19.
20.  (a)
 X 30 60 120 180 240 270 Sin X 0.5 0.87 0.87 0 -0.87 -1 2 cos X 1.73 1 -1 -2 -1 0 Y 2.23 1.87 -0.13 -2 -1.87 -1

(b)

c) x = 114 ± 3° and
x = 294 ± 3°

d) line thro y = − 1.5

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B2

For all 6 values of y
B1 for at least 4

Appropriate scale use

✓ plotting

✓ curve

Points identified and stated B1 only stated

✓line

Points identified and stated

B1 only stated

21.

1. P ( RR) = 3/12 x 2/11
=1/22
2. P(IR) = RW or RB or WR or BR
15/132 + 12/132 + 15/132 + 12/132
= 9/22
3. p( At least white Ball ) =
P(RW) + P(WR) + P(WW) + P ( WB) + P(B)
15/132 + 9/132 + 20/132 + 20/132 + 20/132
= 84/132  or  7/11
4. P(RR or WW or BB)
= 6/ 132 + 20/132 + 12/132
= 19/66
B2

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✓ prob, tree

Or equivalent 0.04545

Or equivalent 0.4091

Or equivalent 0.6364

Or equivalent 0.2424

22
 X -2 -1 0 1 2 3 4 5 Y 0 6 10 12 12 10 6 0

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8 values

B1 at least 6

Appropriate scale use

✓ Plotting

✓ Curve

10
23.
1.
1. AC2 = 82 + 62 = 100
AC  = 10cm
2. AF2  = 102 + 52  = 125
AF = 11.18cm
2. Tan x = 5/11 = 0.5
x = 26.52°
3. Tan x =  5/6  = 0.8333
x = 39.7°
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Sketch

Sketch

10
24.

(c) P profit function object we function
P = 4x + 3y
Max profit at point (6,4)
P = 4(6) + 4,88)(4)
= 56
Hence he should here 6 medium of type A and 4 machine of type B

Inequalities
x≥O
y≥O
4x + 3y ≤ 36
4x + 3y = 24
y = n
x≥o and y ≥ o
2x + 3y ≤ 36
y ≤ n

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B1 - shading and line drawn

B1 - for shading and line drawn

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10.

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