Mathematics Paper 2 Questions and Answers - Kassu Jet Pre Mocks 2022

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INSTRUCTIONS TO CANDIDATES    
  • This paper consists of two sections: Section I and Section II.
  • Answer ALL questions in section 1 and ONLY FIVE questions from section II
  • All answers and workings must be written on the question paper in the spaces provided below each question.
  • Show all the steps in your calculation, giving your answer 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.

SECTION I. Answer all the questions  ( 50 marks)

  1. A student spends 3/8 of his time playing basketball, ¼ of the remaining in playing table tennis and ¾  of the remaining time playing volleyball. The rest is spent on reading novels. What fraction of the time is spent on reading novels?   
    (3 marks)
  2. Simplify;    √5 − 1         (3 marks)
                      √5 + 1
  3. Solve the equation    2 log 3 + log (x−2)  =  2 log x        (3 marks)
  4. The base and perpendicular height of a triangle measured to the nearest millimetre are 15.0 cm and 9.5 cm respectively. Find:
    1. The absolute error in calculating the area of the triangle          (1 mark)
    2. The percentage error in the area, giving the answer to 1 decimal place.  (3 marks)
  5. Find the value of θ, given that;  ½ sin = 0.35 for 0° ≤ θ ≤ 360° (3 marks)
  6. Make Q the subject of formula P = MathsPP2KJPreMQ6   (3 marks)
  7. The coordinates of the end points of a diameter of a circle are A2,4 and B(2,6). Find the equation of the circle in the form ax2 + by2 +cx + dy + e = 0       (4 marks)
  8. Kimani wants to buy a TV on hire purchase. It has a cash price of Ksh.30,000. He makes a down payment of Ksh.9,000 and 12 monthly instalments of ksh. 2,200 each. Calculate the rate of compound interest charged per month.(Give your answer to 1 dp).                   (3 marks)
  9. Expand (3 + 3x)6  in ascending powers of x. Hence use the expansion up to the 3rd term, to find the value of (3.03)6 correct to 2 decimal places.             (3 marks)
  10. The following are ages of students in a class 7,9,8,9,11,12,10 9,8,6,7,10,11,12,6,9,7, and 11.
    1. Complete the frequency distribution table below            (1 mark)
       Ages  6   7   8   9   10   11   12 
       No of students              
    2. Calculate the standard deviation of their ages in five years’ time.    (2 marks)
  11. Find the possible values of x given that MathsPP2KJPreMAns11 is a singular matrix.     (3 marks)
  12. Evaluate using the logarithm table;                                                  (4 marks)
    (        log 9.814         )−½
    [4.283 × (0.0094782)²]
  13. The figure below is that of a circumcircle of the triangle ∠ABC.  The  radius of the circle is 5cm. Given that  ∠ABC = 70° and ΔACB=40°. Calculate the area of ABC .                                                                                                          (3 marks)
    MathsPP2KJPreMQ13
  14. A quantity P varies partly as the cube of Q and partly varies inversely as the square of Q. when Q = 2, P = 108 and when Q = 3, P = 259. Find the value of P when Q = 6.                (3 marks)
  15. The table below shows the number of insects and corresponding number of days in breeding. 
     Number of insects  200  300  400  500  600  700  800  900
     Days  4.4  6.4  7.4  8.0  8.5  9.0  9.5  10
    1. On the grid provided, draw the graph of number of insects against the number of days.      (1 mark)
    2. Determine the rate of breeding between 5th  and 7th day.          (2 marks)
  16. Calculate the area of the minor segment of a circle of radius 5√2cm, cut off by a chord of length 10cm.        (3 marks)     

SECTION II: Answer any 5 questions from this section. ( 50 marks)

  1. Income rates for income earned were charged as shown in the table alongside:
    A civil servant earns a monthly salary of Ksh.27,000. He was also given a house allowance of Ksh. 12,000 , transport allowance  Ksh.1,800 and medical allowance Ksh.2,000. He is entitled to a family relief of Kshs. 1040 per month.
     Income in Ksh. pm  Rate in Ksh. Per Shs. 20
     1- 8400   2
     8401- 18,000   3
     18,001- 30,000   4
     30,000 - 36,000   5
      36,000-48,000   6
     48,001 and above   7
    Determine:
    1.  
      1. His taxable income per month in Ksh.     (2 marks)
      2. His net tax.        (6 marks)
    2. In addition, the following deductions were made
      NHIF Ksh.                        430
      Loan repayment               Ksh. 6500
      Bank  shares                     Ksh. 1000.
      Calculate his net pay per month.                                               (2 marks)
  2.  
    1. In the figure below, OY:YA= 1:3, AX:XB=1:2, OA= a and OB =b . n is the point of intersection of BY and OX. 
      MathsPP2KJPreMQ18
       Determine;                                               
      1. OX             (2 marks)           
      2. BY              (1 mark)
    2. Given that BN = mBY and ON = nOX, express ON in two ways in terms of  a, b, m and n            (3 marks)
    3. Find the values of m and n          (4 marks)
  3.  
    1. In a geometrical progression the sum of the second and third term is 12 and the sum of the third and fourth terms is −36.  Find the first term and the common ratio.                                                                (4 marks)
    2. In an arithmetic progression the 12th  term is 25 and the 7th  term is three times the second term, find;
      1. The first term and the common difference (4 marks)
      2. The sum of the first  10 terms of the arithmetic progression.(2 marks)
  4. The table below shows the frequency distribution of marks scored by students  in a  test.
     Marks   1-10  11-20   21-30   31-40   41-50 
     Frequency     2     4     8     4     2
    1. On the grid provided, draw a cumulative frequency curve for the data.    (3 marks)
    2. Use your graph to determine;
      1. The pass mark if only 6 students passed the exam.                        (2 marks)
      2. The upper quartile mark                                                                     (1 mark)
    3. Find the percentage change if the upper quartile in b(ii) above was found by calculation.          (3 marks)
  5. A gold urn contains 3 red balls and 4 white balls and a silver urn contains 5 red balls and 2 white balls. A die is rolled and if a 6 shows, balls will be selected at random from the gold urn. Otherwise balls are selected from the silver urn.
    1. Find the probability of selecting a red ball.          (3 marks)
    2. If two balls are selected at random without replacement,
      1. Draw a tree diagram to represent this information.                                (3 marks)
      2. Find the probability that two balls are white.                               (2 marks)
      3. Find the probability that there is at most one white ball from the silver urn.   (3 marks)
  6.  
    1. Using a ruler and a compass only construct triangle ABC where AB=7cm, Angle CBA=82.5° and BC=5cm   (4marks)
    2.  
      1. Locate a point T inside the triangle which is equidistant from points A and B and also equidistant from lines AB and AC                     (3 marks)
      2. Measure TB                      (1 mark)
    3. By shading the unwanted region show the area inside the triangle where P lies if it is nearer to point B than to point A and also nearer to the line AB than line AC.       (2 marks)
  7.  
    1. Complete the table for y = Sin x + 2 Cos x.          (2 marks)
       0  30  60  90  120  150  180  210  240  270  300
      Sin x  0.00      1.00    0.50    −0.50      −0.87
      2 cos x  2.00      0.00    −1.73    −1.73      1.00
      y  2.00      1.00    −1.23    −2.23      0.13
    2. Draw the graph of y = Sin x + 2 cos x.          (3 marks)
    3. Solve sin x + 2 cos x = 0 using the graph.                     (2 marks) 
    4. Find the range of values of x for which y < −0.5          (3 marks)
  8. A triangle ABC with vertices at A (1,-1) ,B (3,-1) and C (1, 3) is mapped onto triangle A1B1C1 by a transformation whose matrix is .Triangle A¹B¹C¹ is then mapped onto A¹¹B¹¹C¹¹ with vertices at A¹¹ (2, 2), B¹¹ (6, 2) and C¹¹ (2,−6) by a second transformation.
    1. Find the coordinates of A¹B¹C¹                 (3 marks)
    2. Find the matrix which maps A¹B¹C¹ onto A¹¹B¹¹C¹¹. (3 marks)
    3. Determine the ratio of the area of triangle A¹B¹C¹ to triangle A¹¹B¹¹C¹¹. (1 mark)
    4. Find the transformation matrix which maps A¹¹B¹¹C¹¹ onto ABC    (3 marks)

MARKING SCHEME

SECTION I. Answer all the questions  ( 50 marks)

  1. A student spends 3/8 of his time playing basketball, ¼ of the remaining in playing table tennis and ¾  of the remaining time playing volleyball. The rest is spent on reading novels. What fraction of the time is spent on reading novels?   
    (3 marks)
    3/8 → B.ball
    ¼ × 3/8 
    = 5/32
    3/8 + 5/32
    = 17/32
    Remaining 15/32
    ¾ × 15/32 = 45/128
    17/32 + 45/128 = 113/128
    1 − 113/128
    = 15/128
  2. Simplify;    √5 − 1         (3 marks)
                      √5 + 1
    √5 − 1   ×  √5 + 1
    √5 + 1       √5 + 1
    Numerator 
    6 − 2√5
    Denominator
    5 − 1 = 4
    6 − 2√5 = 3/2 − ½√5 or 1.5 − 0.5√5
       4
  3. Solve the equation    2 log 3 + log (x−2)  =  2 log x        (3 marks)
    log(32(x−2)) = log x2 
    9(x − 2) = x2 
    x2 − 9x + 18 = 0
    (x − 6) (x −3) = 0
    x = 6
    x = 3
  4. The base and perpendicular height of a triangle measured to the nearest millimetre are 15.0 cm and 9.5 cm respectively. Find:
    1. The absolute error in calculating the area of the triangle          (1 mark)
       Min product  Actual   Max product 
       14.95
         9.45
       15.0
        9.5
       15.05
         9.55
       141.2775  142.5  143.7275
      A.E = 143.7275 − 141.2775
                               2
             = 1.225
    2. The percentage error in the area, giving the answer to 1 decimal place.  (3 marks)
      1.225 × 100
      142.5
      = 0.859 = 0.9%
  5. Find the value of θ, given that;  ½ sin = 0.35 for 0° ≤ θ ≤ 360° (3 marks)
    Sin θ = 0.70
    θ = sin−1 0.70
    θ = 44.43°
    180 − 44.43 
    = 135.57
  6. Make Q the subject of formula P2MathsPP2KJPreMQ6   (3 marks)
    P2MathsPP2KJPreMQ6
    P2Q2 − P2 = Q2 
    P2Q2 − Q2 = P2 
    Q2 (P2 − 1) = P2 
    Q2 =    P²   
            P2 − 1

    MathsPP2KJPreMAns6
  7. The coordinates of the end points of a diameter of a circle are A2,4 and B(2,6). Find the equation of the circle in the form ax2 + by2 +cx + dy + e = 0       (4 marks)
    Centre
    (2+4, 4+6)
       2       2
    (0, 5)
    Radius = √((0−2)² + (5−4)²)
    r = √5
    (x−0)² + (y−5)² = (√5)²
    x² + y² − 10y + 25 = 5
    x² + y² − 10y + 20 = 0
  8. Kimani wants to buy a TV on hire purchase. It has a cash price of Ksh.30,000. He makes a down payment of Ksh.9,000 and 12 monthly instalments of ksh. 2,200 each. Calculate the rate of compound interest charged per month.(Give your answer to 1 dp).                   (3 marks)
    P = 30000 − 9000
       = 21000
    A = 12 × 2200
       = 26,400
    26,400 = 21,000(1+r/100)12 
    1.257 = (1+r/100)12 
    12√1.257 = 1.0193
    1.0193 = (1+r/100)
    r = 1.93 
    r = 1.9
  9. Expand (3 + 3x)6  in ascending powers of x. Hence use the expansion up to the 3rd term, to find the value of (3.03)6 correct to 2 decimal places.             (3 marks)
    (3 + 3x)6
    729 + 4356x + 10935x² + 14580x³ + 10935x4 + 4356x5 + 729x6 
    3 + 3x = 3.03
        x = 0.01
    729 + 4356x + 10935x² → 729 + 4356(0.01) + 10935(0.01)²
     = 773.6535
     ≅ 773.62 (2dp)
  10. The following are ages of students in a class 7,9,8,9,11,12,10 9,8,6,7,10,11,12,6,9,7, and 11.
    1. Complete the frequency distribution table below            (1 mark)
       Ages x  6   7   8   9   10   11   12 
       No of students  2  3  2  4  2  3  2
       (x − x̄)  −3  −2  −1  0  1  2  3
       (x − x̄)²   9    4    1   0   1   4   9
      x̄ = 162
             18
         = 9
    2. Calculate the standard deviation of their ages in five years’ time.    (2 marks)
      V = ∑d² 
              N
         =28/18 
         = 1.5
      s.d = 1.247
  11. Find the possible values of x given that MathsPP2KJPreMAns11 is a singular matrix.     (3 marks)
    x² + 8x − 48 = 0
    x² + 12x − 4x − 48 = 0
    x(x + 12) − 4 (x + 12) = 0
    (x + 12) ( x − 4) = 0
    x = −12
    x = 4
  12. Evaluate using the logarithm table;                                                  (4 marks)
    (        log 9.814         )−½ = (4.283 × (0.009478)²)½
    [4.283 × (0.0094782)²]                   0.9931
    MathsPP2KJPreMAns12
  13. The figure below is that of a circumcircle of the triangle ABC.  The  radius of the circle is 5cm. Given that  ∠ABC = 70° and ∠ACB=40°. Calculate the area of ΔABC .                                                                                                          (3 marks)
    MathsPP2KJPreMQ13
    BC = AC ≠ a
        a       = 2 × 5
    sin 70°
    a = 10 × sin 70
    A = ½ × (10sin70)² × sin 40°
    A = 28.38cm²
  14. A quantity P varies partly as the cube of Q and partly varies inversely as the square of Q. when Q = 2, P = 108 and when Q = 3, P = 259. Find the value of P when Q = 6.                (3 marks)
    P = Q³k +  n  
                     Q²
    108 = 8k + n/4
    259 = 27k + n/9
    432   = 32k + n
    2331 = 243k + n
    −1899 = −211k
    k = 9
    108 = 8(9) = n/4   
    n = 144
    P = 63(9) + 144
                        36   
    P = 1948     
  15. The table below shows the number of insects and corresponding number of days in breeding. 
     Number of insects  200  300  400  500  600  700  800  900
     Days  4.4  6.4  7.4  8.0  8.5  9.0  9.5  10
    1. On the grid provided, draw the graph of number of insects against the number of days.      (1 mark)
      MathsPP2KJPreMAns15a
    2. Determine the rate of breeding between 5th  and 7th day.          (2 marks)
      5th → 220
      7th → 350
      350 − 220
          7−5
      = 65 ± 1
  16. Calculate the area of the minor segment of a circle of radius 5√2cm, cut off by a chord of length 10cm.        (3 marks)   
    MathsPP2KJPreMAns15
    sin θ =   5   
                5√2
    Sin θ =   1   
                 √2
    θ = 45°
    90/360 × 3.142 × (5√2)
    = 39.275
    ½ (5√2) × sin 90° 
    = 25
    39.275 − 25
    = 14.275

SECTION II: Answer any 5 questions from this section. ( 50 marks)

  1. Income rates for income earned were charged as shown in the table alongside:
    A civil servant earns a monthly salary of Ksh.27,000. He was also given a house allowance of Ksh. 12,000 , transport allowance  Ksh.1,800 and medical allowance Ksh.2,000. He is entitled to a family relief of Kshs. 1040 per month.
     Income in Ksh. pm  Rate in Ksh. Per Shs. 20
     1- 8400   2
     8401- 18,000   3
     18,001- 30,000   4
     30,000 - 36,000   5
      36,000-48,000   6
     48,001 and above   7
    Determine:
    1.  
      1. His taxable income per month in Ksh.     (2 marks)
        27,000 + 12,000 + 1,800 + 2,000
        = 42,800
      2. His net tax.        (6 marks)
        MathsPP2KJPreMAns17a
    2. In addition, the following deductions were made
      NHIF Ksh.                        430
      Loan repayment               Ksh. 6500
      Bank  shares                     Ksh. 1000.
      Calculate his net pay per month.                                               (2 marks)
      Total deductions
      = 7930 + 7180
      = 15110
      42800 − 15110
      = 27,690
  2.  
    1. In the figure below, OY:YA= 1:3, AX:XB=1:2, OA= a and OB =b . n is the point of intersection of BY and OX. 
      MathsPP2KJPreMQ18
       Determine;                                               
      1. OX             (2 marks)           
        MathsPP2KJPreMAns18a
      2. BY              (1 mark)

        BY = ¼a − b
    2. Given that BN = mBY and ON = nOX, express ON in two ways in terms of  a, b, m and n            (3 marks)
      MathsPP2KJPreMAns18b
    3. Find the values of m and n          (4 marks)
      (1−m) b + ¼ma = 2/3na + 1/3nb
      ¼m = 2/3n
      m = 8/3n .....(i)
      1−m = 1/3n
      3 − 3m = n
      − 3(8/3n) = n
      3 = 9n
      n = 1/3
      m = 8/3(1/3)
      m = 8/9
  3.  
    1. In a geometrical progression the sum of the second and third term is 12 and the sum of the third and fourth terms is −36.  Find the first term and the common ratio.                                                                (4 marks)
      ar + ar² = 12
      ar² + ar³ = −36
      ar(1+r)=12
      ar²(1+r) = −36
      ar²  = −36
      ar        12
      r = −3
      From ar + ar² = 12
      −3a + 9a = 12
      a = 2
    2. In an arithmetic progression the 12th  term is 25 and the 7th  term is three times the second term, find;
      1. The first term and the common difference (4 marks)
        a + 11d = 25
        (a + d)3 = a + 6d
        a + 11d = 25
        2a = 3d
        a = 3/2d .......(i)
        3/2d + 11d = 25
        3d + 22d = 50
        25d = 50
        d = 2
        a = 3/2(2)
        a = 3
      2. The sum of the first  10 terms of the arithmetic progression.(2 marks)
        Sn = n/2(2a + (n−1)d)
             =10/2(2×3 + (10−1)2)
             =120
  4. The table below shows the frequency distribution of marks scored by students  in a  test.
     Marks   1-10  11-20   21-30   31-40   41-50 
     Frequency     2     4     8     4     2
     C.F     2     6     14     18      20
    1. On the grid provided, draw a cumulative frequency curve for the data.    (3 marks)
      MathsPP2KJPreMAns20a
    2. Use your graph to determine;
      1. The pass mark if only 6 students passed the exam.                        (2 marks)
        20 − 6 = 14
                   = 30.5
      2. The upper quartile mark                                                                     (1 mark)
        ¾ × 50
        ¾ × 20
        = 31.5 marks
        = 15th value
    3. Find the percentage change if the upper quartile in b(ii) above was found by calculation.          (3 marks)
      = 30.5 + (15 − 14) 10
                           4
      = 30.5 + 2.5
      = 33
      (33 − 31.5)100 = 4.76%
          31.5
  5. A gold urn contains 3 red balls and 4 white balls and a silver urn contains 5 red balls and 2 white balls. A die is rolled and if a 6 shows, balls will be selected at random from the gold urn. Otherwise balls are selected from the silver urn.
    1. Find the probability of selecting a red ball.          (3 marks)
      1/6 × 3/7 or 5/6 × 5/7 
      (1/14 + 25/42) = 2/3
    2. If two balls are selected at random without replacement,
      1. Draw a tree diagram to represent this information.                                (3 marks)
        MathsPP2KJPreMAns21b
      2. Find the probability that two balls are white.                               (2 marks)
        P(SWW) or P(GWW)
        = (5/6 × 2/7 × 1/6) + (1/6 × 4/7 × 3/6)
          5  1    =  11 
        126    21    126
      3. Find the probability that there is at most one white ball from the silver urn.   (3 marks)
        p(0) or (p(1)
        P(SRR) or P(SWR) or P(SRW)

        5/6 × 5/7 × 4/6) + (5/6 × 2/7 × 5/6)  + (5/6 × 5/7 × 2/6)
        = 25/63 + 25/126 + 25/126 
        = 50/63
  6.  
    1. Using a ruler and a compass only construct triangle ABC where AB=7cm, Angle CBA=82.5° and BC=5cm   (4marks)
      MathsPP2KJPreMAns22a
    2.  
      1. Locate a point T inside the triangle which is equidistant from points A and B and also equidistant from lines AB and AC                     (3 marks)
      2. Measure TB                      (1 mark)
        (3.7 ± 0.1 )cm
    3. By shading the unwanted region show the area inside the triangle where P lies if it is nearer to point B than to point A and also nearer to the line AB than line AC.       (2 marks)
      MathsPP2KJPreMAns22c
  7.  
    1. Complete the table for y = Sin x + 2 Cos x.          (2 marks)
       0  30  60  90  120  150  180  210  240  270  300
      Sin x  0.00  0.50  0.87   1.00  0.87  0.50  0.00  −0.50  −0.87  −1.00  −0.87
      2 cos x  2.00  1.73  1.00  0.00  −1.00  −1.73  −2.00  −1.73  −1.00  0.00  1.00
      y  2.00  2.23  1.87  1.00  −0.13  −1.23  −2.00  −2.23  −1.87  −1.00  0.13
    2. Draw the graph of y = Sin x + 2 cos x.          (3 marks)
      MathsPP2KJPreMAns23b
    3. Solve sin x + 2 cos x = 0 using the graph.                     (2 marks) 
      114 ± 2 , 294 ± 2
    4. Find the range of values of x for which y < −0.5          (3 marks)
      132° − 280°
  8. A triangle ABC with vertices at A (1,-1) ,B (3,-1) and C (1, 3) is mapped onto triangle A1B1C1 by a transformation whose matrix is .Triangle A¹B¹C¹ is then mapped onto A¹¹B¹¹C¹¹ with vertices at A¹¹ (2, 2), B¹¹ (6, 2) and C¹¹ (2,−6) by a second transformation.
    1. Find the coordinates of A¹B¹C¹                 (3 marks)
      MathsPP2KJPreMAns24i
    2. Find the matrix which maps A¹B¹C¹ onto A¹¹B¹¹C¹¹. (3 marks)
      MathsPP2KJPreMAns24ii
    3. Determine the ratio of the area of triangle A¹B¹C¹ to triangle A¹¹B¹¹C¹¹. (1 mark)
      MathsPP2KJPreMAns24iiideterminant = 4
    4. Find the transformation matrix which maps A¹¹B¹¹C¹¹ onto ABC    (3 marks)
      MathsPP2KJPreMAns24iv

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