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

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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 1 jygauydga(3 marks)
  2. Without using a calculator or mathematical table evaluate 2 jgatgda leaving your answer in simplified form. (3 marks)
  3. Expand 3 kiahuhdaup 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.
    5 jgaugda
    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)
    6 iuhyauygda
  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)
    7 kuhauygda
  8. Given that the matrix 8 kuhaiyuhda is a singular matrix, find the values of x. (3marks)
  9. Make x the subject of the equation (3 marks)
    9 kgajda
  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

      00

      150

      300

      450

      600

      750

      900

      1050

      1200

      1350

      1500

      1650

      1800

      Cos 2X0

      1.00

      0.87

       

      0.00

      -0.5

       

      -1.00

       

      -0.5

      0.00

      0.50

      0.87

      1.00

      Sin (X0+300)

      0.50

      0.71

      0.87

      0.97

      1.00

       

      0.87

      0.71

      0.50

       

      0.00

       

      -0.50

    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)
      graph paper jagytda
    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.
    20 auygyda
    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)
      graph paper jagytda
    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 1 jygauydga(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 2 jgatgda leaving your answer in simplified form. (3 marks)
    2 ands ahbvygd
  3. Expand 3 kiahuhdaup 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)
    3 redo kag duya
  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.
    5 jgaugda
    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)
      5b jhbaugda
      OR =  √612 - 602
      = √121
      = 11.00 cm
  6. Simplify as far as possible leaving your answer in surd form (3marks)
    6 iuhyauygda

    6 jvgyuagdaq
  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)
    7 kuhauygda
    ∠ACN = 180 - (68 + 39)
                           2
    =365º
      8.4   =     x     
    sin68º    sin39º
    x = 5.701 cm
  8. Given that the matrix 8 kuhaiyuhda 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)
    9 kgajda
    (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)
    circle ghaud
    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

      00

      150

      300

      450

      600

      750

      900

      1050

      1200

      1350

      1500

      1650

      1800

      Cos 2X0

      1.00

      0.87

      0.50

      0.00

      -0.5

      -0.87

      -1.00

      -0.87

      -0.5

      0.00

      0.50

      0.87

      1.00

      Sin (X0+300)

      0.50

      0.71

      0.87

      0.97

      1.00

      0.97

      0.87

      0.71

      0.50

      0.26

      0.00

      -0.26

      -0.50

    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)
      graph iygaugyda
    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.
    20 auygyda
    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)
      prob tree jgfautda
    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)
        22 kagdya
      2. the standard deviation of the distribution (3mks)
        22 ii khauda
    2. Calculate the quartile deviation (3mks)
      22b jhgayda
  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
        ii kugauygdya
    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)
      c jygauygda
      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
      c jygauygda
  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
      graph 2 khaguygda
    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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