Mathematics Paper 2 Questions and Answers - Kigumo Mocks 2021 Exams

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INSTRUCTIONS TO CANDIDATES

  • This paper consists of TWO sections: Section I and Section II.
  • Answer ALL the questions in Section I and any five questions from Section II.
  • Answers and working must be written on the question paper in the spaces provided below each question.
    Marks may be given for correct working even if the answer is wrong.
  • Non-programmable silent electronic calculators and KNEC Mathematical tables may be used, except where stated otherwise

FOR EXAMINER’S USE ONLYSECTION I

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

TOTAL

                                 

SECTION II GRAND TOTAL

17

18

19

20

21

22

23

24

TOTAL

                 

QUESTIONS

SECTION I (50 marks)
Answer all the questions in this section in the spaces provided.

  1. Evaluate without using tables or calculators. (3 marks)
    1 auyygduyda
  2. Make w the subject of the formulae. 2x = 2 aidhuad(3marks)
  3. Two pipes, P and Q can fill an empty tank in 3 hours and 4 hours respectively. It takes 5 hours to fill the tank when an outlet pipe R is opened the same time with the inlet pipes. Calculate the time pipe R takes to empty the tank. (3 marks)
  4. Given that OM=i – 3j + 4k, ON= 6i + 3j – 5k and OQ = 2OM + 5ON, find the magnitude of OQ to 3 significant figures. (3 marks)
  5. A triangle ABC is such that a =14.30 cm, b =16.50 cm and B =56º. Find the radius of a circle that circumscribes the triangle. (3 marks)
  6. The third and sixth terms of a geometric progression (G.P.) are -64 and 8 respectively. Find;
    1. The common ratio, (2marks)
    2. The first term of the G.P. (1mark)
  7. Calculate the standard deviation of the set of numbers
    29. 31, 28, 29, 31. 46, 39, 31, (3 marks)
  8. Grace deposited Ksh 16 000 in a bank that paid simple interest at the rate of 14% per annum. Joyce deposited the same amount of money as Grace in another bank that paid compound interest semi- annually. After 4 years, they had equal amounts of money in the banks.
    Determine the compound interest rate per annum, to I decimal place, for Joyce’s deposit.(3 marks)
  9. Simplify 9 uhgauygda, leaving the answer in the form a + b√c , where a, b and c are rational numbers (3 marks)
  10. The table below shows income tax rates in a certain year.

    Mo      Monthly income in Ksh

    Tax r  Rate in each Ksh

    1 – 9680

    10%

    9681 – 18800

    15%

    18801 – 27920

    20%

    27921 – 37040

    25%

    Over 37040

    30%

    In that year, a monthly personal tax relief of Ksh. 1056 was allowed. If the monthly income tax paid by an employee was Ksh 4249 calculate his monthly taxable income. (3 marks)
  11. Grade I coffee cost sh 500 per kilogram while grade II coffee costs sh 400 per kilogram. The grades are mixed to obtain a mixture that costs sh 420 per kilogram. In what ratio should the two grades be mixed? (3 marks)
  12. The base length and height of parallelogram were measured as 8.4 cm and 4.5 cm respectively. Calculate the percentage error in the area of the parallelogram. (3 marks)
  13. The equation of a circle is x2 +y2 + 6x — 14y + 58 =r2 If the circle passes through the point (2, 7). Determine its radius and the coordinates of it centre. (4 marks)
  14. Find the gradient of the curve y = x214 a ajygdudagd at point 14 b auygdyuad(3marks)
  15. In the figure below, the tangent HXY meets chord PQ produced at Y. Chord XZ passes through the centre, O, of the circle and intersects PQ at T. Line XY = 16 cm and QY = 10 cm.
    15 agduyagda
    1. Calculate the length PQ. (2 marks)
    2. If ZT = 4 cm and PT: TQ = 3:5, find XT. (2 marks)
  16. Quantity P varies partly as Q and partly varies inversely as square of Q. When Q = 1, P = 1 and when Q = ½, P = -3. Find the equation of the relationship connecting P and Q. (3 marks) 

SECTION II (50 Marks)
Answer any five questions from this section.

  1. A certain Sub-county advertised for a tender to construct its headquarters. Two contractors A and B assessed the work. Contractor A indicated would do the same work in 12 months while contractor B indicated would do the same work in 18 months. The two contractors were awarded the tender. Contractor B did the work for three months then it was joined by contractor A.
    1. Determine;
      1. The fraction of the work done by contractor B in 3 months, (2 marks)
      2. How long the two contractors took to complete the remaining work. (4 marks)
    2.  Given that contractors A and B would incur expenditure amounting to sh 120 000 per month and sh 90 000 per month respectively, calculate the total expenditure of each contractor. (4 marks)
  2. An examination involves a written test and a practical test. The probability that a candidate passes the written test is 6/11 if the candidate passes the written test, then the probability of passing the practical test is 3/5, otherwise it would be 2/7
    1. Illustrate this information on a tree diagram. (2marks)
    2. Determine the probability that a candidate is awarded
      1. Credit for passing both tests. (2marks)
      2. Pass for passing the written test. (2marks)
      3. Retake for passing one test. (2marks)
      4. Fail for not passing the written test. (2marks)
  3.                    
    1. Construct triangle PQR with PQ = 7.2cm, QR = 6.5cm and angle PQR = 480 (3marks)
    2. The locus L1, of points equidistant from P and Q, and locus , L2 of points equidistant from P and R, meet at M. Locate M and measure QM (4marks)
    3. A point x moves within triangle PQR such that QX ≥ QM. Shade and label the locus of X. (3marks)
  4. The figure below shows a square ABCD point V is vertically above middle of the base ABCD. AB = 10cm and VC = 13cm.
    16 jhggaygdasd
    Find;
    1. The length of diagonal AC (2marks)
    2. The height of the pyramid (2marks)
    3. The acute angle between VB and base ABCD. (2marks)
    4. The acute angle between BVA and ABCD. (2marks)
    5. The angle between AVB and DVC. (2marks)
  5. The table below shows the distribution of ages in years of 50 adults who attended a clinic:-

    Age

    21-30

    31-40

    41-50

    51-60

    61-70

    71-80

    Frequency

    15

    11

    17

    4

    2

    1

    1. State the median class (1 mark)
    2. Using a working mean of 45.5, calculate:-
      1. The mean age (3 marks)
      2. The standard deviation (3 marks)
      3. Calculate the 6th decile. (3 marks)
  6. An aircraft leaves A(60ºN,13ºW) at 1300 hours and arrives at B(60ºN, 47ºE) at 1700 hrs
    1. Calculate the average speed of the aircraft in knots. (3marks)
    2. Town C (60ºN, 133ºW)has a helipad. Two helicopters S and T leaves B at the same time. S moves due West to C while T moves due North to C. If the two helicopters are moving at 600 knots.
      1. The time taken by S to reach C (2marks)
      2. The time taken by T to reach C (2marks)
    3. The local time at a town D (23ºN, 5ºW) is 1000 hours. What is the local time at B.? (3marks)
  7. A certain uniform supplier is required to supply two types of shirts: one for girls labelled G and the other for boys labelled B. The total number of shirts must not be more than 400.He has to supply more of type G than of type B. However the number of type G shirts must not be more than 300 and the number of type B shirts must not be less than 80. By taking x to be the number of type G shirts and y the number of type B shirts,
    1. Write down in terms of x and y all the inequalities representing the information above. (3 marks)
    2. On the grid provided draw the inequalities and shade the unwanted regions. (4marks)
      graph paper aidghuahda
    3. Given that type G costs Shs. 500 per shirt and type B costs Shs. 300 per shirt.
      1. Use the graph in (b) above to determine the number of shirts of each type that should be made to maximize profit. (1mark)
      2. Calculate the maximum possible profit. (2marks)
  8. A transformation represented by the matrix 24 auydguagdamaps the points A(0, 0), B(2, 0), C(2, 3) and D(0, 3) of the quad ABCD onto A¹B¹C¹D¹ respectively.
    1. Draw the quadrilateral ABCD and its image A¹B¹C¹D¹. (3marks)
      graph paper aidghuahda
    2. Hence or otherwise determine the area of A¹B¹C¹D¹. (2 marks)
    3. Another transformation 24 ba yujgduyadmaps A¹B¹C¹D¹ onto A¹¹B¹¹C¹¹D¹¹.
      Draw the image A¹¹B¹¹C¹¹D¹¹. (2mks)
    4. Determine the single matrix which maps A¹¹B¹¹C¹¹D¹¹ back to ABCD. (3mks)

MARKING SCHEME

SECTION I (50 marks)
Answer all the questions in this section in the spaces provided.

  1. Evaluate without using tables or calculators. (3 marks)
    1 auyygduyda
    log (1/2 x 64)   
    log (1/32 x 8/1)
    log 32
    log 1/4
    log 25
    log 2-2
     5 = -21/2
    -2
  2. Make w the subject of the formulae. 2x = 2 aidhuad(3marks)
    (2x)2 = 2w + 8
                 3w -5
    4x2(3w - 5) = 2w + 8
    12x2w - 20x2 = 2w + 8
    12x2w - 2w = 8 + 20x2
    w(12x2 - 2) = 8 + 20x2
    w = 8 + 20x2
           12x2 - 2
    or
    w = -8 - 20x2
           2 - 12x2
  3. Two pipes, P and Q can fill an empty tank in 3 hours and 4 hours respectively. It takes 5 hours to fill the tank when an outlet pipe R is opened the same time with the inlet pipes. Calculate the time pipe R takes to empty the tank. (3 marks)
    p = 3 hrs
    q = 4 hrs
    r = x hrs
    1 w hr part of the tank filled
    1 hr = (1/3 + 1/4) - 1/x = 7/12 - 1/x
    7x - 12 = 1
          12x      5
    5(7x - 12) = 12x
    35x - 60 = 12x
    35x - 12x = 60
    23x = 60
    x = 60/23 hrs     or   2 14/23 hrs
  4. Given that OM=i – 3j + 4k, ON= 6i + 3j – 5k and OQ = 2OM + 5ON, find the magnitude of OQ to 3 significant figures. (3 marks)
    4 sauhdiuahda
  5. A triangle ABC is such that a =14.30 cm, b =16.50 cm and B =56º. Find the radius of a circle that circumscribes the triangle. (3 marks)
    5 ajgdhad
    b = 2R
        sin B
    16.5 = 2R
    sin 56
    19.90:2R
        2     2
    9.95cm = R
  6. The third and sixth terms of a geometric progression (G.P.) are -64 and 8 respectively. Find;
    1. The common ratio, (2marks)
      ar2 = -64
      ar5 = 8
      ar2(r3) = 8
      -64(r3) =  8  
      -64        -64
      r = -1/2
    2. The first term of the G.P. (1mark)
      a = -256
  7. Calculate the standard deviation of the set of numbers
    29. 31, 28, 29, 31. 46, 39, 31, (3 marks)
      d = x - 33  d2 
     28   -5  25 
     29  -4  16
     29  -4  16
     31  -2  8
     31  -2  8
     31  -2  8
     39  6  36
     46 13 169
    sd = √Σd2
               n
    = √286
           8
    = 5.979
    28 + 29 + 29 + 31 + 31 + 31 + 39 + 46
                               8
    x = 264 = 33
            8
  8. Grace deposited Ksh 16 000 in a bank that paid simple interest at the rate of 14% per annum. Joyce deposited the same amount of money as Grace in another bank that paid compound interest semi- annually. After 4 years, they had equal amounts of money in the banks.
    Determine the compound interest rate per annum, to I decimal place, for Joyce’s deposit.(3 marks)
    total amount for Grace in 4 years
    16000 x 14/100 x 4 = 8960
    16000 + 8960 = 24960

    Joyce
    A = P(1 + r/100)n
    24960 = 1600 (1 + (r/2)/100)8
    1.56 = (1 + r/200)8
    1.0572 = 1 + r/200
    0.0572 = r/200
    11.4% = r
  9. Simplify 9 uhgauygda, leaving the answer in the form a + b√c , where a, b and c are rational numbers (3 marks)
    ans = 2 + √10
  10. The table below shows income tax rates in a certain year.

    Mo      Monthly income in Ksh

    Tax r  Rate in each Ksh

    1 – 9680

    10%

    9681 – 18800

    15%

    18801 – 27920

    20%

    27921 – 37040

    25%

    Over 37040

    30%

    In that year, a monthly personal tax relief of Ksh. 1056 was allowed. If the monthly income tax paid by an employee was Ksh 4249 calculate his monthly taxable income. (3 marks)
    gross tax = 4249
                    + 1056  
                       5304
    9680 x 10/100 = 968
    9120 x x15/100 = 1368
    9120 x 20/100 = 1824
    x × 25/100 = 1144
    x =  1144 x 100 = 4576
                25
    taxable income = 32,496 /=
  11. Grade I coffee cost sh 500 per kilogram while grade II coffee costs sh 400 per kilogram. The grades are mixed to obtain a mixture that costs sh 420 per kilogram. In what ratio should the two grades be mixed? (3 marks)
    500x + 400y = 420
       x + y
    500x + 400y = 420x + 420y
    500x - 420x = 420y - 400y
    80x = 20y
    20y    20y
    x =
    y    4
    x:y = 1:4
  12. The base length and height of parallelogram were measured as 8.4 cm and 4.5 cm respectively. Calculate the percentage error in the area of the parallelogram. (3 marks)
    A = b x h
    max area = 8.45 x 4.55
    =38.4475
    min area = 8.35 x 4.45
    =37.1575
    A.E = 38.4475 - 37.1575
                           2
    =0.645
    P.E = 0.645 x 100
              37.8
    = 1.706%
  13. The equation of a circle is x2 +y2 + 6x — 14y + 58 =r2 If the circle passes through the point (2, 7). Determine its radius and the coordinates of it centre. (4 marks)
    22 + 72 + 6[2] - 14(7) + 58 = r2
    4 + 49 + 12 - 98 + 58 = r2
    25 = r2
    5 = r
    x2 + y2 + 6x - 14y + 58 = 25
    x2 + 6x + 9 + y2 - 14y + 49 = 33 + 9 + 49
    (x + 3)2 + (y - 7)2 = 25
    centre (-3, 7)
    r = 5
  14. Find the gradient of the curve y = x214 a ajygdudagd at point 14 b auygdyuad(3marks)
    y = x3 + 1/2x2 - x
    dy/dx = 3x2 + x - 1
    = 3(12) + (1) - 1
    =3 + 1 - 1
    = 3
  15. In the figure below, the tangent HXY meets chord PQ produced at Y. Chord XZ passes through the centre, O, of the circle and intersects PQ at T. Line XY = 16 cm and QY = 10 cm.
    15 agduyagda
    1. Calculate the length PQ. (2 marks)
      let PQ = y
      (10 + y)10 = 16
      100 + 10y = 256
      10y = 156
      y = 15.6 cm
      PQ = 15.6cm
    2. If ZT = 4 cm and PT: TQ = 3:5, find XT. (2 marks)
      PQ = 15.6
      PT = 3/8 x 15.6
      =5.85
      TQ= 5/8 x 15.6
      =9.75cm
      PT x TQ = ZT.TX
      5.85 x 9.75 = 4 x x
      x =  5.85 x 9.75
                   4
      = 14.26cm
  16. Quantity P varies partly as Q and partly varies inversely as square of Q. When Q = 1, P = 1 and when Q = ½, P = -3. Find the equation of the relationship connecting P and Q. (3 marks) 
    ans = P = Q - 1/Q2

SECTION II (50 Marks)
Answer any five questions from this section.

  1. A certain Sub-county advertised for a tender to construct its headquarters. Two contractors A and B assessed the work. Contractor A indicated would do the same work in 12 months while contractor B indicated would do the same work in 18 months. The two contractors were awarded the tender. Contractor B did the work for three months then it was joined by contractor A.
    1. Determine;
      1. The fraction of the work done by contractor B in 3 months, (2 marks)
        A = 12m
        B = 18m
        1 month = 1/18
        3 months - ?
        1/18 x 3
        = 1/6th of the week
      2. How long the two contractors took to complete the remaining work. (4 marks)
        remaining week
        1 - 1/6 = 5/6
        both  working
        1 month = 1/18 + 1/12
        2 + 3 = 5
          36     36
        5/36 → 1 month
        5/6→?
        = 6 months
    2.  Given that contractors A and B would incur expenditure amounting to sh 120 000 per month and sh 90 000 per month respectively, calculate the total expenditure of each contractor. (4 marks)
      total months for B
      3 + 6 = 9 months
      9 x 90000 = 810 000/=
      total months for A 
      6 months
      6 x 120 000 = 720 000
  2. An examination involves a written test and a practical test. The probability that a candidate passes the written test is 6/11 if the candidate passes the written test, then the probability of passing the practical test is 3/5, otherwise it would be 2/7
    1. Illustrate this information on a tree diagram. (2marks)
      a adada
    2. Determine the probability that a candidate is awarded
      1. Credit for passing both tests. (2marks)
        p(w and p)
        p = 6/11 x 3/5 = 18/55
      2. Pass for passing the written test. (2marks)
        p(w and p) or p(w and p1)
        (6/11 x 3/5) + (6/11 x 2/5)
        18/55 + 12/55 = 30/55
      3. Retake for passing one test. (2marks)
        p(w and p1) or p(w1 and p1)
        (6/11 x 2/5) + (5/11 x 2/7)
        12/55 + 10/77 = 134/385
      4. Fail for not passing the written test. (2marks)
        p(w1 and p1)
        5/11 x 2/7 = 10/77
  3.                    
    1. Construct triangle PQR with PQ = 7.2cm, QR = 6.5cm and angle PQR = 480 (3marks)
      angles ahgda
    2. The locus L1, of points equidistant from P and Q, and locus , L2 of points equidistant from P and R, meet at M. Locate M and measure QM (4marks)
      QM = 3.8 ±0.1 cm
    3. A point x moves within triangle PQR such that QX ≥ QM. Shade and label the locus of X. (3marks)
  4. The figure below shows a square ABCD point V is vertically above middle of the base ABCD. AB = 10cm and VC = 13cm.
    20 ans jkhadh
    Find;
    1. The length of diagonal AC (2marks)
      AC = √102 + 102
      Ac = 14.14cm
    2. The height of the pyramid (2marks)
      angles 2iaghuda
      √132 - 7.072
      =10.91cm
    3. The acute angle between VB and base ABCD. (2marks)
      angles 3 auhduad
      cos Φ = A
                   H 
      =7.07
         13
      cos Φ = 0.5438º
      Φ = 57.05º
    4. The acute angle between BVA and ABCD. (2marks)
      d auhduiad
      Tan θ = 0/A
      = 10.91
            5
      =65.39º
    5. The angle between AVB and DVC. (2marks)
      e ajhdgyuad
      ∠ mvn
      Tan θ = o/A
      = 5/10.91
      θ = 24.62 x 2
      = 49.24º
  5. The table below shows the distribution of ages in years of 50 adults who attended a clinic:-

    Age

    21-30

    31-40

    41-50

    51-60

    61-70

    71-80

    Frequency

    15

    11

    17

    4

    2

    1

    1. State the median class (1 mark)
      41-50
    2. Using a working mean of 45.5, calculate:-
      class  mid-point 
      x - 45.5  xf   d2f  c.f
       21-30  25.5  -20 15   -300  6000  15
       31-40  35.5  -10  11  -110  1100 26 
       41-50  45.5  0 17   00  0000  43
       51-60  55.5  10  4  40  400  47
       61-70  65.5  20  2  40  800 49 
       71-80  75.5  30  1  30 900   50
            Σf = 50 Σxf = -300 Σd2f = 9200  
      1. The mean age (3 marks)
        mean(x) = 45.5 + -300
                                       50
        = 45.5 - 6
        = 39.5
      2. The standard deviation (3 marks)
        ans = 12.17
      3. Calculate the 6th decile. (3 marks)
        6th decile = 6/10 x 50 = 30
        = 40.5 + (30 - 26)10
                           17
        =40.5 + 2.353
        =42.853
  6. An aircraft leaves A(60ºN,13ºW) at 1300 hours and arrives at B(60ºN, 47ºE) at 1700 hrs
    1. Calculate the average speed of the aircraft in knots. (3marks)
      arcraft uhgauygda
      angle at the centre  = (13 + 47) = 50
      1 = 60 cos
      distance between A & B  = 50 x 60 cos 60 = 1500n.m
      time taken 1700hrs - 1300hrs = 4 hrs
      speed = 1500 = 375knots
                       4
    2. Town C (60ºN, 133ºW)has a helipad. Two helicopters S and T leaves B at the same time. S moves due West to C while T moves due North to C. If the two helicopters are moving at 600 knots.
      1. The time taken by S to reach C (2marks)
        distance from B to C for S = 180 x 60 x cos 60 = 5400 n.m
        distance for B to C for T = 60 x 60 = 3600n.m
        time taken by s = 5400 = 9 hrs
                                    600
      2. The time taken by T to reach C (2marks)
        time taken by T = 3600 = 6hrs
                                     600
    3. The local time at a town D (23ºN, 5ºW) is 1000 hours. What is the local time at B.? (3marks)
      1 = 4 min
      diff in longitude (5 + 47) = 52
      52 x 4 = 208m
      =3hrs 28min
      local time at B = 1000hrs + 3 hrs 28 min
      =1328 hrs
  7. A certain uniform supplier is required to supply two types of shirts: one for girls labelled G and the other for boys labelled B. The total number of shirts must not be more than 400.He has to supply more of type G than of type B. However the number of type G shirts must not be more than 300 and the number of type B shirts must not be less than 80. By taking x to be the number of type G shirts and y the number of type B shirts,
    1. Write down in terms of x and y all the inequalities representing the information above. (3 marks)
      girls = G = x
      Boys = B= y
      1. G + B ≤ 400
      2. G>B
      3. G≤300
      4. B>80
      5. x + y ≤ 400
      6. x > y
      7. x ≤ 300
      8. y > 80
      9. x ≥ 0
      10. y ≥ 0
    2. On the grid provided draw the inequalities and shade the unwanted regions. (4marks)
      graph filled
    3. Given that type G costs Shs. 500 per shirt and type B costs Shs. 300 per shirt.
      1. Use the graph in (b) above to determine the number of shirts of each type that should be made to maximize profit. (1mark)
        (300,100)
        300 - G
        100 - B
      2. Calculate the maximum possible profit. (2marks)
        500(300) + 300(100)
        150 000 + 30 000
        = 180 000
  8. A transformation represented by the matrix 24 auydguagdamaps the points A(0, 0), B(2, 0), C(2, 3) and D(0, 3) of the quad ABCD onto A¹B¹C¹D¹ respectively.
    1. Draw the quadrilateral ABCD and its image A¹B¹C¹D¹. (3marks)
      filled 2
    2. Hence or otherwise determine the area of A¹B¹C¹D¹. (2 marks)
    3. Another transformation 24 ba yujgduyadmaps A¹B¹C¹D¹ onto A¹¹B¹¹C¹¹D¹¹.
      Draw the image A¹¹B¹¹C¹¹D¹¹. (2mks)
    4. Determine the single matrix which maps A¹¹B¹¹C¹¹D¹¹ back to ABCD. (3mks)

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