Mathematics Paper 2 Questions and Answers - Form 3 Mid Term 2 Exams 2021

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Instructions

  • Answer all the questions in section A and ONLY Five in section B.
  • Show all the calculations in the spaces provided
  • KNEC mathematical tables and non-programmable calculators may be used.

SECTION A

  1. Use logarithm tables to evaluate; [4 Marks]
    F3 MT2 Math PP2 Q1 2021
  2. A number is such that when it is divided by 3,7,11 or 13, the remainder is always 1. Find the number. [2 Marks]
  3. A square has an area of 144m2. Calculate its perimeter. [2 Marks]
  4. Factorise  2x2 − x − 6 hence solve the quadratic equation [3 Marks]
    2x2 − x − 6 = 0
  5. List all integral values of that satisfy the combined inequality; Represent the solutions on a single number line. [4 Marks]
  6. A body accelerates at 5m/sto reach a velocity of 60m/s in 5 seconds. Calculate the initial velocity.   [2 Marks]
  7.                                                                                        →     →      →
    Draw a triangle STR and put arrows on its side to show that TS + SR = TR [ 2 Marks]
  8. A point P(2,5) is translated to P’(1,6)
    1. Find the translation vector [2 Marks]
    2. The image of (3,0) under the same translation. [2 Marks]
  9. Solve for in [3 Marks]
  10. The sum of interior angles of a regular polygon is 3240°. Find the size of each exterior angle.  [3 Marks]
  11. Write 1936 and 1728 in terms of its prime factors hence evaluate;
    F3 MT2 Math PP2 Q11 2021
  12. Use reciprocal tables to evaluate; [4 Marks].
      16    +     24   
    2.674     0.1396
  13. Evaluate; [3 Marks]
    F3 MT2 Math PP2 Q13 2021
  14. Use substitution method to solve; (3 Marks)
    2x + 3y = 1
    3x − 2y = 8
  15. The straight line joining the points P(a,7) and Q(13,a) is parallel to the line whose equation is 3y+2x=9. Find the value of a. [3 Marks]
  16. The ratio of the areas of two circles is 16:25
    1. What is the ratio of their radii. [2 Marks]
    2. If the smaller circle has a diameter of 28cm, find the radius of the larger circle.
      [2 Marks]

SECTION B
Answer any 5 Questions

  1. The marks of 30 girls in a class were recorded as follows.
    220 250 204 230 210 227 221 252
    200 228 208 225 200 202 240 228
    212 225 252 216 212 226 227
    240 248 203 201 251 242 216
    1. Construct a frequency table with a class width of 5 Marks beginning with 199 marks. [3 Marks]
    2. What is the modal class? [1 Mark]
    3. Estimate the mean [3 Marks]
    4. Estimate the median [3 Marks]
  2. The initial velocity of a body is 30m/s. the body accelerates uniformly to a velocity of 60m/s in 6seconds. It moves at this constant velocity for 5 seconds before decelerating in 3 seconds.
    1. Using the graph paper provided, draw a velocity time graph to illustrate the information above.   [4 Marks]
    2. Calculate the initial acceleration [2 Marks]
    3. Calculate the total distance covered. [4 Marks]
  3. The diagram below shows two circles that share a common chord XY which is 13cm long.
    Calculate;
    F3 MT2 Math PP2 Q19 2021
    1. <XO1Y [1 Mark]
    2. < XO2Y[1 Mark]
    3. The area of the sector O1XBY [2 Marks]
    4. The area of the sector O2YAX [2 Marks]
    5. The area of the shaded part [4 Marks]
  4.  
    1. The diagram below shows a triangle OAB
      F3 MT2 Math PP2 Q20 2021
      Points M and K are on AB and OA respectively such that;
      AM:MB=2:3 and K is the mid point of OA.
      Express the following vectors interms of a and b.
      1.  →
        AB [1 Mark]
      2.  →
        OM [2 Marks]

      3. BK [2 Marks]
    2. The co-ordinates of P and Q are (6,10) and (8,14) respectively. Calculate;

      1. PQ [1 Mark]
      2. The mid-point of line PQ [2 Marks]
    3. Given that F3 MT2 Math PP2 Q20c 2021. Another vector P is such that p=2a − b + 4c
      Evaluate /P/ correct to 2 decimal places. [2 Marks]
  5. The diagram below shows a frustrum that was cut from a right cone.
    F3 MT2 Math PP2 Q21 2021
    Calculate;
    1. The highest of the cone [2 Marks]
    2. The volume of the frustrum [4 Marks]
    3. The surface area of the frustrum [4 Marks]
  6. A line L1 has the equation 3x + 4y = 12
    1. Calculate
      1. The gradient of line L1[2 Marks]
      2. The coordinates of P and Q where the line cuts the x-axis and y-axis respectively [4 Marks]
      3. Another line L2 is perpendicular to L1 and passes through (-4,5). Determine the equation of line L2 in the form [4 Marks]
  7. Form all inequalities that define Region R [10 Marks]
    F3 MT2 Math PP2 Q23 2021
  8.  
    1. Two trains T1 and T2 travelling in opposite directions on parallel trucks are just beginning to pass each other. Train T1 is 72m long and is travelling at 108km/hr and T2 is 78m long travelling at 72km/hr. Find the time in seconds it takes the two trains take to completely pass one another.    [3 Marks]
    2. A rally car travelled for 2 hours 40 minutes at an average speed of 120km/hr. the car consumes an average of 1 litre of fuel for every 4 kilometres.
      A litre of fuel costs sh. 64. Calculate the amount of money spent on the fuel.
      [4 Marks]
    3. Mwangi and Otieno live 40km apart. Mwangi starts from his home at 7.30am and travels towards Otieno at 16km/hr. Otieno starts from his home at 8.00am and cycles at 8km/hr towards mwangi. At what time do they meet? [3 Marks]


MARKING SCHEME

  1.  
    Math F3 MT2 PP2 ans 1 2021
    6.123 × 100
    =6.123
  2.  
    Math F3 MT2 PP2 ans 2 2021
    3 × 7 × 11 × 13 = 3003 + 1
                      =3004
  3. √144 = 12cm → one side
    Perimeter = 12 × 4 = 48m
  4. 2x2 − x − 6
    P = −12     S = −1  F = −4,3
    2x2 − 4x + 3x − 6
    2x(x −2) +3(x−2)
    (2x + 3)(x −2)
    (2x+3)(x−2) = 0
    x = −1½ or 2
  5. 3x − 2 = 5x+8
    −2x = 10
        x ≥ −5
    5x + 8 ≤ 3x+15
    2x ≤ 7
     x = 3½
    Math F3 MT2 PP2 ans 5 2021
    Integral values 
    = -4, -3, -2,-1,0, 1, 2, 3
  6. a= v−u 
           t
    5 = 60 − u 
               5
    25 = 60−u
      u = 60−25
    initial velocity = 35m/s   (check units)
  7.  
    Math F3 MT2 PP2 ans 7 2021
    NB Tr is a resultant vector (check the two arrows)
  8.  
    1.  
      Math F3 MT2 PP2 ans 8a 2021
    2.  
      Math F3 MT2 PP2 ans 8b 2021
  9. 9x + 32x = 54
    32x + 32x = 54
    let 32x = y
    2y = 54 
      y = 27
    32x = 33
    2x = 3 
      x =1½
  10. Sum of interior angles
    180(n−2) = 3240
       180           180
    n − 2 = 18
    n = 20 sides
    exterior = 360 =18°
                     20
  11.  
    Math F3 MT2 PP2 ans 11a 2021
    1728 = 26 × 33
    Math F3 MT2 PP2 ans 11b 2021
    1936 = 24 × 112
    Math F3 MT2 PP2 ans 11c 2021
    = 3/11
  12.   16     = 16 ×   1   
    2.674             2.674
    0.3739 ×16
      = 5.984
       24        24 ×    1     
    0.1369            0.1369
    7.163 × 24
    =171.912
    5.983 + 171.912 
    =177.896
  13. Numerator
    5/2 − 2/3 = 15 − 4 = 11/6
                          6
    ½ of 7/27/4     3/2 × 11/ 
    7/411/418/4 = 4½
    Denominator
    ¾ of 5/215/8 × 2/115/
    9/2 × 4/156/5 
    =11/5
  14. 2x + 3y =1 
    2x = 1 − 3y
               2
    x ( 1 − 3y
            2
    3 ( 1 − 3y ) − 2y = 8
            2
    3(1−3y) −4y = 16
    3 − 9y − 4y =16
    −13y =13
    y = −1
    x = 1 − (3x −1) = 1+3 = 4 = 2
                2                2      2
  15. 3y + 2x = 9
    3y = −2x + 9
    y = −2x + 3
            3
    Gradients are equal to (parallel lines)
    = −2/3
     a − 7−2 (cross multiply)
    13 − a      3
    3(a−7) = −2(13−a)
    3a − 21 = −26 + 2a
        a = −5
  16.  
    1. ASF = 16:25
      LSF = √16:√25
            = 4:5
       smaller:bigger
    2. 4/528/x (cross multiply)
      4x = 5 × 28 
      4          4
      x = 35cm diameter
      Radius = 35/2 = 17.5cm
  17.  
    1.  marks   F    x   fx 
       199-203
       204-208
       209-213 
       214-218
       219-223
       224-228
       229-233
       234-238
       239-243
       244-248
      249 - 253
       5
       2
       3
       2
       2
       7
       1
       0
       3
       1
       4  
       201 
       206
       211
       216
       221
       226
       231
       236
       241
       246
       251
       1005 
       412
       633
       432
       442
       1582
       231
         0
       723
       246
       1004
                     f=30        Σfx=6710
    2. Modal class = 224-228
    3. Mean = Σfx = 6710 = 223.67
                   Σf       30
    4. 30th = 15th + 16th = 15.5th
       2                 2
      223.5 + (15.5 −14 × 5)
                            7
      223.5 + (1.5 × 5)
                        7
      = 224.57marks
  18.  
    1.  
      Math F3 MT2 PP2 ans 18a 2021
    2.  60 − 30 = 30 = 5m/s2 (check units)
            6          6
    3. Total distance = area under the graph
      ½ × 90 × 6 = 270m
      5 × 60 = 300m
      ½ × 3 × 60 = 90m
      270+300+90=660m
  19.  
    1.  
      Math F3 MT2 PP2 ans 19a 2021
      sin−1 (6.5) = sin−1 0.8125
                 8             54.34  × 2
      <XO1Y = 108.68°
    2. sin−1 6.5/10   sin−1 0.65 = 40.54 × 2
      <XO2Y  = 81.082   
    3. πrθ/360
      3.142 × 8 ×8 = 108.68 = 60.71cm2 
                                360
    4. 3.142 × 10 × 10 = 81.08
                                    360
                               = 70.765cm2
    5. ½ × 8 × 8 sin 108.68
      60.71 − 30.31cm2
              = 30.4cm2
      ½ × 10 × 10 sin 81.08
           = 49.4cm2
      70.765 − 49.4
          =21.365cm2
      30.4 + 21.365
         =51.765cm2
  20.  
    1.  
      1. → = → + →
        AB   AO+OB
        = −a + b
        = b − a
      2. →  = → +→
        OM   OA  AM
        = a2/5 (b − a)
        =a2/5b − 2/5a
        =3/5a2/5b
      3. →      →      →
        BK = BO + OK
        =−b + ½a
    2.  
      1.         →       →
        PQ = OQ − OP 
        =Math F3 MT2 PP2 ans 20bi 2021
      2. 6 + 8  , 10+14
           2           2 
        (7, 12)
    3.  
      Math F3 MT2 PP2 ans 20c 2021   
      =√337
      =18.36units
  21.  
    Math F3 MT2 PP2 ans 21 2021
    25 = x + 15
    10       x
    25x = 10x + 150
    15x = 150
       x = 10cm
    1. 10 + 15 = 25cm
    2. Volume of bigger cone
      1/3 × 3.142 × 25 × 25 × 25
      =16364.58cm3
      Volume of smaller cone
      1/3 × 3.142 × 10 × 10 × 10
      =1047.33cm3
      frustrum = 16364.58 − 1047.33
                   = 15.317.25litres
    3. Slant height (L) = √(252 + 252)
                             =35.36cm
                        (l) = 102 + 102
                            =14.142cm
      Curved area 
      3.142 × 25 × 35.36  = 2777.53cm2
      3.142 × 10 × 14.142 = 444.34cm2
                                        2333.19cm2
       3.142 × 10 × 10 = 314.2cm2
      3.142 × 25 × 25 = 1963.75cm2
                      Total  = 4611.14cm2
  22.  
    1. 3x + 4y = 12
      4y = −3x + 12
      y = −¾x + 3
      Gradient = −¾
    2. Cuts the x=axis at y=0
      3x + 4y = 12
      3x = 12
        x = 4
      P(4,0)
      Cuts the y-axis at x=0
      4y = 12
       y = 3
      Q(0,3)
    3. Gradient of L2= ¾x = 1
                     x = 4/3
      y−5 = 4   (cross multiply)
      x+4    3
      3y − 15 = 4x + 16
      3y − 4x = 31
       3y = 4x + 31
        y = 4/3x + 31/3
  23. L1
    (−2,0)(0,4)
    4 − 0 = 4 = 2
    0−−2    2 
       y     = 2
    x + 2
    y = 2x + 4
    y − 2x = 4
    1 − 4 = − 3 ≤ 4
    y − 2x ≤ 4
    L2
    (0,2)(2,0)
    0 − 2 = −2 = −1
     2−0       2 
     y−2 = −1
       x 
    y − 2 = −x
    x+y = 2
    3 > 2
    x+y > 2
    L3
    (0,4)(4,0)
    0 − 4 = −4 = −1
     4−0       4 
     y−4 = −1
       x 
    y − 4 = −x
    x+y = 4
    3 ≤ 4
    x+y ≤ 4
    L4
    y = 4
    y ≤ 4
    1 ≤ 4
  24.  
    1. Distance = 72 + 78
                   = 150m
      Relative speed = 108 + 72 = 180km/hr
                            = 10/36 × 180 = 50m/s
      T = D/S150/50 = 3seconds
    2. D = 120 × 8/3 = 320km
      320 = 80 litres × 64 = Shs 5120
        4
    3. ½ × 16km/hr = 8km
      40 − 8 = 32km
        32   = 32 = 1hr 20minutes
      16+8    24
        8:00
      +1:20
        9:20am             

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