Mathematics Paper 2 Questions with Answers - Sukellemo Joint Mock 2020/2021

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SUKELLEMO JOINT MOCK
Kenya Certificate of Secondary Education
MATHEMATICS PAPER 2
2 ½ hours

Instructions to Candidates

  • This paper consists of TWO sections; Section I and II
  • Answer ALL the questions in Section 1 and only five questions from Section II.
  • Show all the steps in your calculation, giving your answer at each stage
  • 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.

SECTION I (50 MARKS). 

  1. Find the selling price of 6 kg of a mixture of maize flour and millet flour if 4 kg of maize flour costing sh. 60 per kg is mixed with 6 kg of millet flour costing sh. 45 per kg and a profit of 20% is realized. (3 marks)
  2. If x = 9.6, y= 3.60 and z = 5, Find the percentage error in the calculation of (x+y)/z, giving your answer to three significant figures. (3 marks)
  3. Solve the equation log2⁡ (x2-4) - log2⁡ (x+2) = -4 (3 marks)
  4. Form a quadratic equation whose roots are 2.5 + √3 and 2.5 - √3 giving your answer in the form ax2+bx+c=0 where a, b and c are integers (3 marks)
  5. Make b the subject of the formula. (3 marks)
    x =           a          
          √(a-b)(a+b)
  6. Express in surd form and simplify by rationalizing the denominator. (3 marks)
    3 sin⁡ 45° - 2 cos⁡ 30°
             tan⁡ 30°
  7. Triangle ABC is such that AB = 8 cm, BC = 11 cm and AC = 15 cm. calculate correct to 2 decimal places the;
    1. Angle ABC (2 marks)
    2. Radius of the circum circle (2 marks)
  8. In the figure below, the chords CD and AB intersect externally at T. DT = 4 cm, BT = 3 cm and CD = 5 cm. calculate the length AB. (3 marks)
    sukmathsp28
  9. Solve the following equation for 0≤ x ≤ 3600 2cos⁡x = sin2 x + 2 (4 marks)
  10. Given that A (4, -2, 6) and B (-3, 1, -2) and that a point N divides AB in the ratio -2:7. Find the vector ON in terms of i, j and k. (3 marks)
  11. The equation of a circle is given by 2/3 x2/3 y2  - 4x+ 2 2/3 y - 2 = 0. Determine the centre and the radius of the circle. (3 marks)
  12. Expand (1+2x/3)in ascending powers of x up to the fourth term. Hence use your expansion to evaluate (0.98)8 to three significant figures. (3 marks)
  13. Find the base radius of a cylindrical hole with maximum volume which can be drilled into a cone of height 16 cm and radius 12 cm as shown below. (3 marks)
    sukmathsp213
  14. ABCD is a regular tetrahedron. AB=BC=CA=AD=BD=CD= 8cm.
    1. Calculate the angle between line AD and plane ABC. (2 marks)
    2. Calculate the angle between planes ABD and ABC (2 marks)
  15. A contractor intends to transport 1,000 bags of cement using a lorry and a pick up. The lorry can carry a maximum of 80 bags while a pick up can carry a maximum of 20 bags. The pickup has to make more than twice the number of trips the lorry makes and the total number of trips has to be less than 30. The cost per trip is sh 2000 for the lorry and sh 900 for the pickup and the contractor wishes to minimize cost. Let x and y be the trips for the lorry and pickup respectively.
    1. State the objective function. (1 mark)
    2. Write down all the inequalities which govern the condition above. (2 marks)
  16. The first, second and fifth terms of an arithmetic sequence are the first three consecutive terms of a geometric sequence. Find the common ratio (3 marks)

SECTION II (50 marks)

  1. Using a ruler and a pair of compasses only for all constructions in this question.
    1. Construct triangle ABC in which AB = 6 cm, BC = 7 cm and angle ABC = 75o(3 marks)
    2. Find locus X such that AX = 3 cm (1 mark)
    3. On the same side of BC as A, construct the locus of P such that angle BPC = 120o(3marks)
    4. Show by shading the locus of Q inside triangle ABC such that angle BPC≥angle BQC. (1 mark)
    5. On the side of AB opposite C, construct the locus of T such that the area of triangle ATB = 6 cm2 (2 marks)
  2. The data below shows the heights of students in a class.
    Height  130 -139 140 -149 150-159  160-169 170-179 180-189 190-199
     Frequency  8  10  11  18  14 12 

    1. Using assumed mean of 164.5, calculate the mean and the standard deviation(5 marks)
    2. Draw a cumulative frequency curve on a grid and use it to estimate the median (5 marks)
  3. P varies directly as the cube of Q and inversely as the square root of R
    1. Given that P = 35 when Q = 8 and R = 144, find P when Q = 20 and R = 22 (5 marks)
    2. If Q decreases by 24% and R increases by 40% find the percentage change in P.(5 marks)
  4. The triangle ABC has vertices A (1, 2), B (2, 1) and C (2, 3). ABCis the image of ABC under transformation given by the matrix
      sukmathsp220
    1. What are the coordinates of ABC? Plot ABC and ABC on the same axis (3 marks)
    2. State the ratio of the areas of the two triangles and use the area of ABC to calculate the area of AII BII CII  (3 marks)
    3. If AII BII CII  is the image of ABC under transformation given by the matrix sukmathsp220c . Find the coordinates of AII BII CII. Plot AII BII CII and describe the transformation fully. (4 marks)
  5.    
    1. An industrialist has 460 litres of a chemical which is 75% pure. She mixes it with a chemical of the same type but 90% pure so as to obtain a mixture which is 78% pure. Find the amount of the 90% pure chemical used. (3 marks)
    2. Three machines A, B and C are set to work together. A working alone takes 6 hours to complete the work; B takes 8 hours while C takes 12 hours. All the three machines started working at the same time. 40 minutes later machine A broke down. B and C continued for another 1 hour before B ran out of fuel and therefore stopped working for 20 minutes while C continued. If B resumed working after 20 minutes, calculate the:
      1. Fraction of the work left after machine A broke down. (2 marks)
      2. Fraction of the work done by C working alone for 20 minutes (2 marks)
      3. Total time taken for the work to be completed. (3 marks)
  6. The table below shows taxation rates in Kenya
    Monthly taxable income (kshs p.m)  Tax rate % 
     1-9680  10
     9681-18800   15
     18801-27920   20
     27921-37040  25
     37041 and above  30
    A civil servant is provided with a house and pays a nominal rent of sh 6,260 per month. In addition the government gives him taxable allowances amounting to sh 16,000 per month. He is entitled to a personal relief of sh 1,520 per month. He has a life insurance policy for which he pays sh 1,200 per month and claims insurance relief at the rate of sh 3 per k£. The civil servant’s PAYE is sh 6,900.Apart from PAYE and insurance his other monthly deductions are WCPS 2% of basic salary, HELB loan sh 4,000 and cooperative shares sh 600. Calculate his:
    1. Taxable income per month. (6 marks)
    2. Basic salary per month (2 marks)
    3. Net monthly pay. (2 marks)
  7. The probability of James, Tyson and David passing an examination are 4/5,3/4 and 2/3 respectively. Find the probability that in one attempt:
    1. only one passes the examination. (2 marks)
    2. All the three passes the examination. (2 marks)
    3. Two pass the examination. (2 marks)
    4. None passes the examination. (2 marks)
    5. At least one passes the examination. (2 marks)
  8. When aero plane left town P (65º N,15º E) to another town Q (65º N,165º W) at a speed of 200 knots using the shortest route. Take π = 22/7 and radius of the earth R= 6370 km.
    1.  
      1. Calculate the distance travelled in nautical miles. (2 marks)
      2. Calculate the time taken to travel from P to Q in hours. (2 marks)
    2. Another plane left P at 1.30 pm local time and travelled to T (65ºN,60ºE) along a parallel of latitude. Calculate the:
      1. Distance between P and T to the nearest km (3 marks)
      2. Local time of arrival at town T if the plane flew at the speed of 470 km/h(3 marks)

 

 

MARKING SCHEME
SECTION I (50 MARKS). 

  1. Find the selling price of 6 kg of a mixture of maize flour and millet flour if 4 kg of maize flour costing sh. 60 per kg is mixed with 6 kg of millet flour costing sh. 45 per kg and a profit of 20% is realized. (3 marks)
    BP per kg = (4 x 60) + (6 x 45)
                                 4 + 6
    = Sh. 51 
    BP of 6kg = 51 x 6 = Sh. 306
    SP per kg = 120/100 x 306
    = Sh. 367.20

  2. If x = 9.6, y= 3.60 and z = 5, Find the percentage error in the calculation of (x+y)/z, giving your answer to three significant figures. (3 marks)
    maximum value = 9.65 + 3.605 = 2.9456
                                        4.5
    minimum value = 9.55 + 3.595 = 2.39
                                         5.5
    actual value = 9.6 + 3.60 = 2.64
                                 5
    absolute error = 2.9456 - 2.39 = 0.2778
                                       2
    percentage error = 0.2778/2.64 x 100% = 10.5%
                                     
  3. Solve the equation log2⁡ (x2-4) - log2⁡ (x+2) = -4 (3 marks)
    log2 x2 -4  = Log22-4
            (x+2)
    (x+2)(x-2) = 2-4
      (x+2)
    x-2 = 1/16
    x = 2 + 1/16
    x= 21/16
  4. Form a quadratic equation whose roots are 2.5 + √3 and 2.5 - √3 giving your answer in the form ax2+bx+c=0 where a, b and c are integers (3 marks)
    x = 2.5 + √3 , x = 2.5 - √3
    (x - 2.5 - √3)(x - 2.5 + √3) = 0
    x2 - 2.5x +x√3 - 2.5x + 6.25 - 2.5√3 - x√3 + 2.5√3 -3 =0
    = 4x2 - 20x + 13 = 0
  5. Make b the subject of the formula. (3 marks)
    x =           a          
          √(a-b)(a+b)
    x =       a        
          √(a2 -b2)
    x2        a2
            a2 -b2
    a2 = a2x2 - b2x2
    b2x2 = a2x2 - a2
    b2 = a2x2 - a2 /x2
    b2 = a2 a2
                    x2
    b = ±√( a2 - a2/x2)
  6. Express in surd form and simplify by rationalizing the denominator. (3 marks)
    3 sin⁡ 45° - 2 cos⁡ 30°
             tan⁡ 30°
    sin 45º = 1/√2 × √2/√2√2/√2
    Cos 30º√3/2
    Tan 30º = 1/√3 × √3/√3√3/3
    (3 √2/2 - 2 √3/)/√3/3
    √2 - 2√3
     2√3/3
    (9√2 - 6√3) √3
       2√3 (√3)
    9√6 - 18
        6
    3√6 - 6
        2
    or 3√6 - 3
           2
  7. Triangle ABC is such that AB = 8 cm, BC = 11 cm and AC = 15 cm. calculate correct to 2 decimal places the;
    1. Angle ABC (2 marks)
      b2 = a2 + c2 - 2ac Cos B
      152 = 112 + 82 - 2(11 x 8)Cos B
      225 = 185 - 176 Cos B
      176 Cos B = - 40
      Cos B = - 0.2273
      B = 103.14º
    2. Radius of the circum circle (2 marks)
            15          = 2r
      sin 103.14º
      R = 7.70cm
  8. In the figure below, the chords CD and AB intersect externally at T. DT = 4 cm, BT = 3 cm and CD = 5 cm. calculate the length AB. (3 marks)
    sukmathsp28
    AT.BT = CT.DT
    (AB + 3).3 = 9 x 4
    AB + 3 = 36/3
    AB = 12 - 3
    9cm
  9. Solve the following equation for 0≤ x ≤ 3600 2cos⁡x = sin2 x + 2 (4 marks)
    2 cos x = 1 - cos2x + 2
    cos2x + 2cosx - 3 = 0
    p = cos x
    p2 + 2p -3 = 0
    p2 + 3p -p -3 =0
    p(p+3) - 1(p+3) =0 
    p = 1 or -3
    cos x = 1
    x=0º, 360º
  10. Given that A (4, -2, 6) and B (-3, 1, -2) and that a point N divides AB in the ratio -2:7. Find the vector ON in terms of ij and k. (3 marks)
    sukmathsp2ans 10a                        sukmathsp2ans 10b

  11. The equation of a circle is given by 2/3 x2/3 y2  - 4x+ 2 2/3 y - 2 = 0. Determine the centre and the radius of the circle. (3 marks)
    3/2 x ( 2/3x2 + 2/3y2 - 4x + 2 2/3y - 2=0)
    = x2 + y2 - 6x + 4y -3 = 0
    x2 - 6x + 9 + y2 + 4y + 4 = 3 + 9 + 4
    (x - 3)2 ( y +2)2 = 42
    centre = (3,-2)
    Radius = 4 units
  12. Expand (1+2x/3)in ascending powers of x up to the fourth term. Hence use your expansion to evaluate (0.98)8 to three significant figures. (3 marks)
    1 + 8(2x/3) + 28(2x/3)2 + 56(2x/3)3
    = 1 + 16x/3 + 112x2/9 + 448x3/27
    (1 + 2x/3) = [ 1 + (-0.02)]
    2x/3 = -0.02
    x = -0.03
    0.988
    1 +  16/3(-0.03) + 112/9(-0.03)2
    + 448/27( -0.02)3
    1 - 0.16 + 0.0112 - 0.000448
    = 0.851
  13. Find the base radius of a cylindrical hole with maximum volume which can be drilled into a cone of height 16 cm and radius 12 cm as shown below. (3 marks)
    sukmathsp213
    r/12 = 16-h/16
    16r = 12(16 -h)
    4/3r 16 -h
    h = 16 - 4/3 r
    V = πr2h
    =πr2( 16 - 4/3 r)
    = (16πr2 - 4/3πr3)
    dv/dr = 32πr - 4πr2
    at max, dv/dr = 0
    4πr(8 -r) = 0
    r=0 or 8
    r=8cm
  14. ABCD is a regular tetrahedron. AB=BC=CA=AD=BD=CD= 8cm.
    1. Calculate the angle between line AD and plane ABC. (2 marks)
      Sin 60º = 4/AO
      AO = 4/Sin60º
      =4.619cm
      sukmathsp2ans14a
      Cosθ = 4.619/8
      θ = 54.73º

    2. Calculate the angle between planes ABD and ABC (2 marks)
      sukmathsp2ans14bi 
      h = √(82 - 42)
      = 6.928cm
      sukmathsp2ans14bii
      Cos θ = 2.309/6.928
      θ = 70.53º
  15. A contractor intends to transport 1,000 bags of cement using a lorry and a pick up. The lorry can carry a maximum of 80 bags while a pick up can carry a maximum of 20 bags. The pickup has to make more than twice the number of trips the lorry makes and the total number of trips has to be less than 30. The cost per trip is sh 2000 for the lorry and sh 900 for the pickup and the contractor wishes to minimize cost. Let x and y be the trips for the lorry and pickup respectively.
    1. State the objective function. (1 mark)
      2000x + 900y 
    2. Write down all the inequalities which govern the condition above. (2 marks)
      y > 2x
      x + y <30
      80x + 20y ≥ 1000, 4x + y ≥ 50
  16. The first, second and fifth terms of an arithmetic sequence are the first three consecutive terms of a geometric sequence. Find the common ratio (3 marks)
    a, a+d, a+ 4d
    a + d/a a + 4d/a+d
    a(a + 4d) = (a+d)2
    a2 + 4ad = a2 +2ad+d2
    2ad = d2
    2a=d
    common ratio (r)
    a+2d/a = a+2a/a
    =3a/a =3

SECTION II (50 marks)

  1. Using a ruler and a pair of compasses only for all constructions in this question.
    1. Construct triangle ABC in which AB = 6 cm, BC = 7 cm and angle ABC = 75o(3 marks)
    2. Find locus X such that AX = 3 cm (1 mark)
    3. On the same side of BC as A, construct the locus of P such that angle BPC = 120o(3marks)
    4. Show by shading the locus of Q inside triangle ABC such that angle BPC≥angle BQC. (1 mark)
    5. On the side of AB opposite C, construct the locus of T such that the area of triangle ATB = 6 cm2 (2 marks)
      sukmathsp2ans17
  2. The data below shows the heights of students in a class.
    Height  130 -139 140 -149 150-159  160-169 170-179 180-189 190-199
     Frequency  8  10  11  18  14 12 

    1. Using assumed mean of 164.5, calculate the mean and the standard deviation(5 marks)
      Class  d=x-A  fd  fd2 cf 
       130-139  134.5  -30  8  -240  7200  8
       140-149  144.5  -20  10  -200  4000  18
       150-159  154.5  -10  11  -110  1100  29
       160-169  164.5  0  18  0  0  47
       170-179  174.5  10  14  140  1400  61
       180-189  184.5  20  12  240  4800  73
       190-199  194.5  30  7  210  6300  80
      ∑f = 80
      ∑fd=40
      ∑fd= 24,800
      mean = A + ∑fd
                            ∑f
      164.5 + 40/80 =165
      standard dev = √(∑fd2/∑f) - (∑fd/∑f)2  
      =√(24800/80) - (40/80)2
      = 17.160
    2. Draw a cumulative frequency curve on a grid and use it to estimate the median (5 marks)
      sukmathsp2ans18

      median = ½ x 80 =40
      From cumulative frequency 
      curve, median = 165.5
  3. P varies directly as the cube of Q and inversely as the square root of R
    1. Given that P = 35 when Q = 8 and R = 144, find P when Q = 20 and R = 22 (5 marks)
      P=KQ3
           √R
      35 = K x 8 x 8 x 8
                   √144
      35=512k/12
      K= 105/128
      P =     105Q3
              128√R
      P=   105  x  203
             128 x √225
      = 105 x 8000m
              128 x 15
      =437.5 
    2. If Q decreases by 24% and R increases by 40% find the percentage change in P.(5 marks)
      Original value of P =KQ3
                                        √R
      new value = K x (0.76Q)3
                              √1.4R
      = 0.438976KQ
            1.1832√R
      = 0.3710 KQ3
                   √R
      change = (0.3710 -1) KQ3
                                          √R
      0.6290KQ3
              √R
      %change = -0.6290 x 100
      -62.90%
      P has decreased by 62.90%
  4. The triangle ABC has vertices A (1, 2), B (2, 1) and C (2, 3). ABCis the image of ABC under transformation given by the matrix
      sukmathsp220
    sukmathsp2ans20a
    1. What are the coordinates of ABC? Plot ABC and ABC on the same axis #(3 marks)
      sukmathsp2ans20b
    2. State the ratio of the areas of the two triangles and use the area of ABC to calculate the area of AII BII CII  (3 marks)
      Ratio of the 2 triangles = determinant of matrix
       (2 x 3) = (0 x 0) =6
      thus ratio = 6:1
      Area of ABC = ½ x 2 x 1= 1 square unit
      Area of AIBIC= 1 x 6= 6 square units
    3. If AII BII CII  is the image of ABC under transformation given by the matrix sukmathsp220c . Find the coordinates of AII BII CII. Plot AII BII CII and describe the transformation fully. (4 marks)
      sukmathsp2ans20c
      AII (-3,2)  BII (0,1) CII(-4,3)
      The transformation is a shear parallel to the x axis with the x axis invariant shear factor. The image of A(1,2) is A1(-3,2)
  5.    
    1. An industrialist has 460 litres of a chemical which is 75% pure. She mixes it with a chemical of the same type but 90% pure so as to obtain a mixture which is 78% pure. Find the amount of the 90% pure chemical used. (3 marks)
      75/100 x 460 = 345
      345 + 0.9x = 358.8 + 0.78x
      Let 90% pure chemical use x 0.12x=13.8
      345 + 90/100x = 75/100(460 + x)
      x = 115 litres
    2. Three machines A, B and C are set to work together. A working alone takes 6 hours to complete the work; B takes 8 hours while C takes 12 hours. All the three machines started working at the same time. 40 minutes later machine A broke down. B and C continued for another 1 hour before B ran out of fuel and therefore stopped working for 20 minutes while C continued. If B resumed working after 20 minutes, calculate the:
      1. Fraction of the work left after machine A broke down. (2 marks)
        1/6 + 1/8 + 1/12 = 3/8
        40/60 x 3/8 = 1/4
        = 1 - 1/4 = 3/4
      2. Fraction of the work done by C working alone for 20 minutes (2 marks)
        1hr    = 1/12
        20min
        20/60 x 1/12 = 1/36
      3. Total time taken for the work to be completed. (3 marks)
        B & C = 1/8 + 1/12= 5/24
        3/4 - 5/24 = 13/24, 13/24 - 1/36 = 37/72
        5/24 = 1hr
        37/72 = 37/72 x 24/5 = 27/15
        time= 40minutes + 1hr 20 mins + 2hrs 28mins
        =4hrs 28min
  6. The table below shows taxation rates in Kenya
    Monthly taxable income (kshs p.m)  Tax rate % 
     1-9680  10
     9681-18800   15
     18801-27920   20
     27921-37040  25
     37041 and above  30
    A civil servant is provided with a house and pays a nominal rent of sh 6,260 per month. In addition the government gives him taxable allowances amounting to sh 16,000 per month. He is entitled to a personal relief of sh 1,520 per month. He has a life insurance policy for which he pays sh 1,200 per month and claims insurance relief at the rate of sh 3 per k£. The civil servant’s PAYE is sh 6,900.Apart from PAYE and insurance his other monthly deductions are WCPS 2% of basic salary, HELB loan sh 4,000 and cooperative shares sh 600. Calculate his:
    1. Taxable income per month. (6 marks)
      Tax= net tax + relief = 6900 + 1520 + (3/20x1200) = Sh.8600
      8600 - 6440=2160
      9680 x 0.1 = 968
      9120 x 0.15=1368
      9120 x 0.20=1824
      9120 x 0.25=2280
                             6440
      y x 0.30 = 2160
      y = 7200 
      taxable income = 7200 +(9120 x 3)+ 9680
      = 44,240
    2. Basic salary per month (2 marks)
      Taxable income = (1.15 of BS) + allowance
      44240 = 1.15x + 16000 - 6260
      1.15x = 34500
      x= 30,000
    3. Net monthly pay. (2 marks)
      Total deductions = 6900 + 1200 (12/100 x 30,000) + 4000 + 600
      =13,300
      Net monthly pay = 44,240 - (0.15 x 30,000) - 13,300
      =26,440
  7. The probability of James, Tyson and David passing an examination are 4/5,3/4 and 2/3 respectively. Find the probability that in one attempt:
    1. only one passes the examination. (2 marks)
      (1/3 x 4/5 x 1/4) + (3/4 x 1/5 x 1/3) + (2/3 x 1/5 x 1/4)
      = 1/15 + 1/20 + 1/30= 3/20
    2. All the three passes the examination. (2 marks)
      4/5 x 3/4 x 2/3 = 2/5
    3. Two pass the examination. (2 marks)
      (4/5 x 3/4 x 1/3) + (3/4 x 2/3 x 1/5) + (4/5 x 2/3 x 1/4)
      = 1/5 + 1/10 + 2/15
      =13/20
    4. None passes the examination. (2 marks)
      1/5 x 1/4 x 1/3 = 1/60
    5. At least one passes the examination. (2 marks)
      1 - 1/60 = 59/60
  8. When aero plane left town P (65º N,15º E) to another town Q (65º N,165º W) at a speed of 200 knots using the shortest route. Take π = 22/7 and radius of the earth R= 6370 km.
    1.  
      1. Calculate the distance travelled in nautical miles. (2 marks)
        θ = 180º - (65 + 65) =50º
        D= 60x50 = 3000nm
      2. Calculate the time taken to travel from P to Q in hours. (2 marks)
        Time = d/s = 3000nm/200knots = 15hours
    2. Another plane left P at 1.30 pm local time and travelled to T (65ºN,60ºE) along a parallel of latitude. Calculate the:
      1. Distance between P and T to the nearest km (3 marks)
        Longitude difference = 60-15 = 45º
        Distance(km) = θ/3602πRcos α
        45/360 x 2 x 22/7 x 6370 x cos65
        =2115km
      2. Local time of arrival at town T if the plane flew at the speed of 470 km/h(3 marks)
        Time = 2115.2/470 = 4hrs 30 minutes
        45 x 4/60 = 3hrs
        time at T = 1:30  + 41/2 + 3hrs
        =9:00pm

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