## Mathematics P2 Questions and Answers - Nambale Mock Exams 2021/2022

Instructions to candidates:

1. Write your name, Index number, in the spaces provided above.
2. Sign and write the date of examination in the spaces provided above.
3. The paper contains two sections: Section I and Section II.
4. Answer All the questions in Section I and only five questions from Section II
5. All answers and working must be written on the question paper in the spaces provided below each question.
6. Show all the steps in your calculations, giving your answers at each stage in the spaces below each question.
7. Non – programmable silent electronic calculators and KNEC Mathematical tables may be used, except where stated otherwise.
8. Candidates should answer the questions in English.

For Examiner’s Use Only
Section I

 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 (50MARKS)
Answer ALL the questions in this section in the spaces provided.

1. Use logarithm tables to evaluate (3 marks)
2. Without using a calculator or mathematical table evaluate  leaving your answer in simplified form. (3 marks)
3. Expand up to the term in x in ascending powers of x3 .Hence find the value of (1.005)10 correct to four decimal places. (3 marks)
4. Solve for x in the equation
2log10x +log105 = 1 + 2log104 (3marks)
5. In the figure below OS is the radius of a circle centre O. Chords SQ and TU are extended to meet at P and OR is perpendicular to QS at R. OS = 61 cm, PU = 50 cm, UT = 40 and PQ = 30cm.

Calculate the length of
1. QS (2 marks)
2. OR to 2 decimal places (2 mark)
6. Simplify as far as possible leaving your answer in surd form (3marks)
7. In the figure below angle A=68º, B= 39º, BC= 8.4cm and CN is the bisector of angle ACB. Calculate the length CN to 1decimal place. (3 marks)
8. Given that the matrix  is a singular matrix, find the values of x. (3marks)
9. Make x the subject of the equation (3 marks)
10. The equation of the circle is given by x2 + y2 + 8x -2y -1 = 0 . Determine the radius and the centre of the circle. (4marks)
11. A coffee blender mixes 6 parts of type A with 4 parts of type B. if type A cost him sh. 24 per kg and type B cost him sh. 22 per kg, at what price per kg should he sell the mixture in order to make 5% profit. Give your answer to 2 decimal places (3marks)
12. Musau invested a sum of money which earned him 10% compound interest in the first year. In the second year, the investment earned him 20% compound interest and in the third year, it earned him 25% compound interest. At the end of the three years, the investment was worth sh. 11,550,000. What sum did he invest. (3marks)
13. Line AB is 8cm long. On the same side of line AB draw the locus of point P such that the area of triangle APB is 12cm2 and angle APB=90º (3marks)
14. In a class of 20 students, there are 12 boys and 8 girls. If two students from the class are chosen at random to go to trip, what is the probability that both of them are boys (3marks)
15. After transformation T represented by the matrix (2 1), the triangle ABC was mapped onto triangle A1B1C1 where A1,B1,C1had coordinates (2,0), (4,0) and (4,6) respectively. Determine the coordinates A, B, and C (3marks)
16. The length and breadth of a rectangular floor were measured and found to be 4.1m and 2.2m respectively. If a possible error of 0.01m was made in each of the measurements; find the:
1. Maximum and minimum possible area of the floor (2marks)
2. Maximum wastage in the carpet ordered to cover the whole floor. (1mark)

SECTION II (50 MARKS)
INSTRUCTIONS: Answer ANY FIVE questions only in this section

1.
1. complete the table below, giving the values correct to 2 decimal places (2mks)
 X0 0 150 300 450 600 750 900 1050 1200 1350 1500 1650 1800 Cos 2X0 1 0.87 0 -0.5 -1 -0.5 0.00 0.5 0.87 1 Sin (X0+300) 0.5 0.71 0.87 0.97 1 0.87 0.71 0.5 0 -0.5
2. Using the grid provided draw on the same axes the graph of y=cos 2Xº and y=sin(Xº+30º) for
0º≤X≤180º. (4mks)
3. Find the period of the curve y=cos 2xº (1mk)
4. Using the graph, estimate the solutions to the equations;
1. sin(Xº+30º)=cos 2Xº (1mk)
2. Cos 2Xº=0.5 (1mk)
2. A Quantity P varies partly as the square of m and partly as n. When p= 3.8, m = 2 and n = -3, When p = - 0.2, m = 3 and n= 2.
1. Find
1. The equation that connects p, m and n (4marks)
2. The value of p when m = 10 and n = 4 (1mark)
2. Express m in terms of p and n (2marks)
3. If P and n are each increased by 10%, find the percentage increase in m correct to 2 decimal place. (3marks)
3.
1. The 5th term of an AP is 16 and the 12th term is 37.
Find;
1. The first term and the common difference ( 3 marks)
2. The sum of the first 21 terms (2 marks)
2. The second, fourth and the seventh term of an AP are the first 3 consecutive terms of a GP. If the common difference of the AP is 2.
Find:
3. The common ratio of the GP ( 3 marks)
4. The sum of the first 8 terms of the GP (2 marks)
4. The table below shows the rates of taxation in a certain year.

In that period, Juma was earning a basic salary of sh. 21,000 per month. In addition, he was entitled to a house allowance of sh. 9000 p.m. and a personal relief of ksh.1056 p.m He also has an insurance scheme for which he pays a monthly premium of sh. 2000. He is entitled to a relief on premium at 15% of the premium paid.
1. Calculate how much income tax Juma paid per month. (7mks)
2. Juma’s other deductions per month were cooperative society contributions of sh. 2000 and a loan repayment of sh. 2500. Calculate his net salary per month. (3mks)
5. A cupboard has 7 white cups and 5 brown ones all identical in size and shape. There was a blackout in the town and Mrs. Kamau had to select three cups, one after the other without replacing the previous one.
1. Draw a tree diagram for the information. (2mks)
2. Calculate the probability that she chooses.
1. Two white cups and one brown cup. (2mks)
2. Two brown cups and one white cup. (2mks)
3. At least one white cup. (2mks)
4. Three cups of the same colour. (2mks)
6. The For a sample of 100 bulbs, the time taken for each bulb to burn was recorded. The table below shows the result of the measurements.
 Time(in hours) 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 Number of bulbs 6 10 9 5 7 11 15 13 8 7 5 4
1. Using an assumed mean of 42, calculate
1. the actual mean of distribution (4mks)
2. the standard deviation of the distribution (3mks)
2. Calculate the quartile deviation (3mks)
7. The position of town A and B on the earth’s surface are (36ºN, 49ºE) and (36ºN, 131ºW) respectively.
1. Find the difference in longitude between town A and town B (2marks)
2. Given that the radius of the earth is 6370km, calculate the distance between town A and B along;
1. Parallel of longitude (2marks)
2. A great circle (3marks)
3. Another town C is 840km east of town B and on the same latitude as town A and B. find the longitude of town C (3marks)
8. A trader is required to supply two types of shirts, type A and type B. the total number of shirts must not be more than 400. He has to supply more of type A than type B shirts. However the number of type A shirts must not be more than 300 and the number of type B shirts must not be less than 80. Let x be the number of type A shirts and y be the number of type B shirts.
1. Write down in terms of x and y all the linear inequalities representing the information above (4marks)
2. On the grid provided, draw the inequalities and shade the unwanted regions (4marks)
3. The profits were as follows;
Type A: sh. 600 per shirt
Type B: sh. 400 per shirt
1. Use the graph to determine the number shirts of each type that he should make to maximize the profit (1mark)
2. Calculate the maximum possible profit (1mark)

## MARKING SCHEME

SECTION I (50MARKS)
Answer ALL the questions in this section in the spaces provided.

1. Use logarithm tables to evaluate (3 marks)
 No standard form log 1.19111     3=0.6370 0.4239 0.4239 × 10-1 ¯1.6272 149.6 1.496 × 102 2.1750 1.9022 log 6 = 0.7782 7.782 × 10-1 ¯1.8911 1.9111 4.335 4.335 × 100 0.6370
2. Without using a calculator or mathematical table evaluate  leaving your answer in simplified form. (3 marks)
3. Expand up to the term in x in ascending powers of x3 .Hence find the value of (1.005)10 correct to four decimal places. (3 marks)
4. Solve for x in the equation
5log10x +log105 = 1 + 2log104 (3marks)
5log10x +log105 = 1 + 2log104
log105x5 = log10160
5x5 = 160
x5 = 32
x5 = 25 =
x = 2
5. In the figure below OS is the radius of a circle centre O. Chords SQ and TU are extended to meet at P and OR is perpendicular to QS at R. OS = 61 cm, PU = 50 cm, UT = 40 and PQ = 30cm.

Calculate the length of
1. QS (2 marks)
PT.PU = PS.PQ
90 × 50 = (30 × QS) × 30
4500 = 900 + 30QS
30QS = 3600
QS = 120
2. OR to 2 decimal places (2 mark)

OR =  √612 - 602
= √121
= 11.00 cm
6. Simplify as far as possible leaving your answer in surd form (3marks)

7. In the figure below angle A=68º, B= 39º, BC= 8.4cm and CN is the bisector of angle ACB. Calculate the length CN to 1decimal place. (3 marks)

∠ACN = 180 - (68 + 39)
2
=365º
8.4   =     x
sin68º    sin39º
x = 5.701 cm
8. Given that the matrix  is a singular matrix, find the values of x. (3marks)
x (x - 1) - 0  = 0
x = 0
x = 1
9. Make x the subject of the equation (3 marks)

(t)2 =      b2
(s)       x - 4
t2(x - 4) = s2b2
t2x - 4t = s2b2
t2x = s2b2 + 4t
x = s2b2 + 4t
t2
10. The equation of the circle is given by x2 + y2 + 8x -2y -1 = 0 . Determine the radius and the centre of the circle. (4marks)
x2 + 8x + 16 + y2 - 2y + 1 = 1 + 16 + 1
(x + 4)2 + (y - 1)2 = 18
center x(-4,1)
radius = √18 = 4.243 units
or
r  = 3√2 units
11. A coffee blender mixes 6 parts of type A with 4 parts of type B. if type A cost him sh. 24 per kg and type B cost him sh. 22 per kg, at what price per kg should he sell the mixture in order to make 5% profit. Give your answer to 2 decimal places (3marks)
avearage cost = total cost
total
=(24 × 6) + (22 × 4)
6 + 4
144 + 88 = 222 = 23.2
10          10
105 × 23.2 = 24.35
100
12. Musau invested a sum of money which earned him 10% compound interest in the first year. In the second year, the investment earned him 20% compound interest and in the third year, it earned him 25% compound interest. At the end of the three years, the investment was worth sh. 11,550,000. What sum did he invest. (3marks)
1st year
A = P(1 +   r   )n
100
A = P(1 +   10   )1
100
A = 1.1P
2nd year
A = 1.1P(1 +   20   )1
100
= 1.1P(1 + 0.2)
A = 1.1P × 1.2
A = 1.32P
3rd year
A = 1.32P(1 +   25   )1
100
= 1.65P = 1150000
P = sh 7 000 000
13. Line AB is 8cm long. On the same side of line AB draw the locus of point P such that the area of triangle APB is 12cm2 and angle APB=90º (3marks)

p is a point substended by chord AB to the circumference of a semi-circle or circle
14. In a class of 20 students, there are 12 boys and 8 girls. If two students from the class are chosen at random to go to trip, what is the probability that both of them are boys (3marks)
( 12 × 11) = 132 = 33
20   19     380    95
15. After transformation T represented by the matrix (2 1), the triangle ABC was mapped onto triangle A1B1C1 where A1,B1,C1had coordinates (2,0), (4,0) and (4,6) respectively. Determine the coordinates A, B, and C (3marks)
A   B  C              A   B   C
(2 1) ( x1   x2   x3)  =      (2  4  4)
(0  1) (y1  y2  y3)            (0  0  6)
2x1 + y1 = 2
0 + y1 = 0
y1 = 0
x1 = 1

2x2 + y2 = 4
0 + y2 = 0
y2 = 0
x2 = 2

2x3 + y3 = 4
0 + y3 = 6
y3 = 6
2x3 = -2
x3 = -1

A(1,0)
B(2,0)
C(-1,6)
16. The length and breadth of a rectangular floor were measured and found to be 4.1m and 2.2m respectively. If a possible error of 0.01m was made in each of the measurements; find the:
1. Maximum and minimum possible area of the floor (2marks)
maximum area = 4.11 m × 2.21m = 9.0831m2
minimum area = 4.09m × 2.19m = 8.9571m2
2. Maximum wastage in the carpet ordered to cover the whole floor. (1mark)
actual area = 4.1 ×2.2
=9.022
wastage = (9.02 - 8.9571) + (9.0831 - 9.02)
2
= 0.063

SECTION II (50 MARKS)
INSTRUCTIONS: Answer ANY FIVE questions only in this section

1.
1. complete the table below, giving the values correct to 2 decimal places (2mks)
 X0 0 150 300 450 600 750 900 1050 1200 1350 1500 1650 1800 Cos 2X0 1 0.87 0.5 0 -0.5 -0.87 -1 -0.87 -0.5 0 0.5 0.87 1 Sin (X0+300) 0.5 0.71 0.87 0.97 1 0.97 0.87 0.71 0.5 0.26 0 -0.26 -0.5
2. Using the grid provided draw on the same axes the graph of y=cos 2Xº and y=sin(Xº+30º) for
0º≤X≤180º. (4mks)
3. Find the period of the curve y=cos 2xº (1mk)
360
b
= 360 = 180º
2
4. Using the graph, estimate the solutions to the equations;
1. sin(Xº+30º)=cos 2Xº (1mk)
x = 18.5 ± 2º
x = 139º ± 2º
2. Cos 2Xº=0.5 (1mk)
x= 30º
2. A Quantity P varies partly as the square of m and partly as n. When p= 3.8, m = 2 and n = -3, When p = - 0.2, m = 3 and n= 2.
1. Find
1. The equation that connects p, m and n (4marks)
p = xm2 + yn
3.8 = 4x - 3y
-0.2 = 9x + 2y
7.6 = 8x - 6y
- 0.6 = 27x + 6y +
7       35x
7 = x
35
x = 1/5 = 0.2
3.8 = 0.8 - 3y
-1 = y
p = 0.2m2 - n
2. The value of p when m = 10 and n = 4 (1mark)
p = 0.2m2 - n
p = 20 - 4
p = 16
2. Express m in terms of p and n (2marks)
p = 0.2m2 - n
0.2m2 = p + n
m2 = p + n
0.2
m = √p + n
0.2
m = ± √p + n
0.2
3. If P and n are each increased by 10%, find the percentage increase in m correct to 2 decimal place. (3marks)
m0 = √p + n
0.2
m1 = √1.1(p + n)
0.2
m1 = √1.1p + 1.1n = √5.5(p + n)
0.2
= 2.3452√(p + n)
m0 = √p + n = √5(p + n)
0.2
= 2.2361√(p + n)
% change in m = (m1 - m0) × 100%
m0
2.3452√(p + n) - 2.2361√(p + n)
2.2361√(p + n)
2.3452- 2.2361 × 100%
2.2361
= 0.1001 × 100%
2.2361
= 0.04879 × 100%
= 4.88% 2dp
3.
1. The 5th term of an AP is 16 and the 12th term is 37.
Find;
1. The first term and the common difference ( 3 marks)
Tn = a + (n - 1)d
T5 = a + 4d = 16
T2 = a + 11d = 37
-7d = -21
d = 3
a + 4(3) = 16
a + 12 = 16
a = 4
2. The sum of the first 21 terms (2 marks)
Sn = n/2(2a + (n - 1)d)
=21/2((2 × 4) + (20 × 3))
= 714
2. The second, fourth and the seventh term of an AP are the first 3 consecutive terms of a GP. If the common difference of the AP is 2.
Find:
3. The common ratio of the GP ( 3 marks)
a + d, a + 3d, a + 6d
a + 2, a + 6, a + 12
a + 6 = a + 12
a + 2   a + 6
(a + 6)2 = (a + 2) (a + 12)
a2 + 12a + 36 = a2 + 14a + 24
2a = 12
a = 6
G.p
8, 12, 18
r = 12
8
= 1½
4. The sum of the first 8 terms of the GP (2 marks)
S8 = 8 ((3/2)8 - 1)
1½ - 1
= 394.0625
4. The table below shows the rates of taxation in a certain year.

In that period, Juma was earning a basic salary of sh. 21,000 per month. In addition, he was entitled to a house allowance of sh. 9000 p.m. and a personal relief of ksh.1056 p.m He also has an insurance scheme for which he pays a monthly premium of sh. 2000. He is entitled to a relief on premium at 15% of the premium paid.
1. Calculate how much income tax Juma paid per month. (7mks)
taxable income  = 21000 + 9000
= sh.30,000
p.a. = 30,000 × 12 = k.f 18,000 p.a.
20
2 × 3900 = sh. 7800
3 × 3900 =sh.11700
4 × 3900 = sh.15600
5 × 3900 = sh.19500
7 × 2400 = sh.16800
sh.71400
tax paid = 71400 - 16272
= sh 55128
P.A.Y.E = 55128
12
= sh. 4594
15 × 2000 = 300
100
total relief p.a. = (300 + 1056) × 12 = sh. 16272
2. Juma’s other deductions per month were cooperative society contributions of sh. 2000 and a loan repayment of sh. 2500. Calculate his net salary per month. (3mks)
total deductions = 4594 + 2000 + 2000 + 2500
sh 11094 per month
net salary = 30000 - 11094
=sh 18,906.00
5. A cupboard has 7 white cups and 5 brown ones all identical in size and shape. There was a blackout in the town and Mrs. Kamau had to select three cups, one after the other without replacing the previous one.
1. Draw a tree diagram for the information. (2mks)
2. Calculate the probability that she chooses.
1. Two white cups and one brown cup. (2mks)
(7 × 6 × 5) + (× 5 × 6) + (5 × 7 × 6)
12   11  10     12  11  10    12   11  10
= 21
44
2. Two brown cups and one white cup. (2mks)
(7 × 5 × 4) + (5× 7 × 4) + (5 × 4 × 7)
12   11  10     12  11  10    12   11  10
= 7
22
3. At least one white cup. (2mks)
(5 × 4 × 5) + (5× 7 × 4) + (7 × 5 × 4) + (5 × 7 × 6) + (× 5 × 6) + (7 × 6 × 5) + (7 × 6 × 5
12   11  10    12  11  10     12   11  10    12   11  10    12   11  10   12   11  10    12   11  10
= 427
440
4. Three cups of the same colour. (2mks)
(7 × 6 × 5) + (5 × 4 × 3)
12   11  10    12  11  10
=
44
6. The For a sample of 100 bulbs, the time taken for each bulb to burn was recorded. The table below shows the result of the measurements.
 Time(in hours) 15-19 20-24 25-29 30-34 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 Number of bulbs 6 10 9 5 7 11 15 13 8 7 5 4
1. Using an assumed mean of 42, calculate
1. the actual mean of distribution (4mks)
2. the standard deviation of the distribution (3mks)
2. Calculate the quartile deviation (3mks)
7. The position of town A and B on the earth’s surface are (36ºN, 49ºE) and (36ºN, 131ºW) respectively.
1. Find the difference in longitude between town A and town B (2marks)
A(36ºN, 49ºE)  B(36ºN, 131ºW)
longitudinal difference
49º + 131º = 180º
2. Given that the radius of the earth is 6370km, calculate the distance between town A and B along;
1. Parallel of longitude (2marks)
Distance = 2πRcosθ
360 ×  22 ×  6370 cos36
dist = 16196.52023km
2. A great circle (3marks)
dist = σ2πR
360
= 108 × 2 × 22 × 6370
360           7
dist = 12,012km
3. Another town C is 840km east of town B and on the same latitude as town A and B. find the longitude of town C (3marks)

B(36º, 131ºw)
D = σ2πRcosσ
360
840 =  σ  × 2 × 22 × 6370cos 36º
360        7
840 = 89.9807σ
9.34º = σ
131º - 9.34º = 121.66º
longitude of town c = 121.66ºw
8. A trader is required to supply two types of shirts, type A and type B. the total number of shirts must not be more than 400. He has to supply more of type A than type B shirts. However the number of type A shirts must not be more than 300 and the number of type B shirts must not be less than 80. Let x be the number of type A shirts and y be the number of type B shirts.
1. Write down in terms of x and y all the linear inequalities representing the information above (4marks)
x + y ≤ 400
x ≤ y
x > 0

x ≤ 300
y ≥ 80
y > 0
2. On the grid provided, draw the inequalities and shade the unwanted regions (4marks)
3. The profits were as follows;
Type A: sh. 600 per shirt
Type B: sh. 400 per shirt
1. Use the graph to determine the number shirts of each type that he should make to maximize the profit (1mark)
sample
(150, 200)
(200, 200)
(180,220)
(150,250)
(100,300)
200 type A
200 type B
2. Calculate the maximum possible profit (1mark)
600A + 4000p = max profits
(600 × 200) + (400 × 200)
120000 + 80000
=sh. 20000

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