Mathematics Paper 2 Questions and Answers - Form 3 Term 3 Opener Exams 2023

SECTION A

1. Use logarithm tables to evaluate; [4 Marks]
                                ³ √146.34²×0.0063
                                          cos 54
2. A number n 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
   2x² − x − 6=0      [3 Marks]
5. List all integral values of x that satisfy the combined inequality; Represent the solutions on a single number line. [4 Marks]
6. A body accelerates at 5m/s² to 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 X(3,0) under the same translation. [2 Marks]
9. Solve for x in [3 Marks]
   9^x + 32^x = 54
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;
                                  ³ √1728
                                     √1936
12. Use reciprocal tables to evaluate; [4 Marks]
                  16    +    24
               2.674       0.1396
13. Evaluate; [3 Marks]
    ½ of 3½ + 1½ (2½ – ²/₃)
             ¾ of 2½ ÷ ½
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

17. 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]
18. The initial velocity of a body is 30m/s. the body accelerates uniformly to a velocity of 60m/s in 6 seconds. 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]
19. The diagram below shows two circles that share a common chord XY which is 13cm long. Calculate;
                            
    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]
20.  
    1. The diagram below shows a triangle OAB
                               
       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 in terms 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 a = (³ ₄) b (² ₁) and c = (³ _–4 ). Another vector P is such that p = 2a – b + 4c. Evaluate ΙpΙ correct to 2 decimal places. [2 Marks]
21. The diagram below shows a frustrum that was cut from a right cone.
                              
    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]
22. 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 y = mx + c [4 Marks]
23. Form all inequalities that define Region R   [10 Marks]
                                
24.  
    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. 
   
2. 
   
3. 
   √144    = 12cm → one side
   Perimeter = 12 x 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
   − 5 ≤ x ≤ 3½
   
   
6.  a = v - u
             t
   5 = 60 − u
              5
   25 = 60 - u        u = 60 − 25
        Initial velocity = 35 m/s
                            (check units)
7. 
    
8.  
   1. (¹₆) − (²₅) = (⁻¹₁)
   2. (³₀) + (⁻¹₁) = (²₁)
9. 
   9^x + 3^2x = 54
   3^2x + 3^2x = 54
   let 3^2x = y
   2y = 54   y = 27
   3^2x = 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. 
    
    (26 x 33) ^1/₃     =  2² x 3 =  12 3
    (2⁴ x 11²)^½          2² x 11     44  11
           = ³/₁₁
12. 
     16     =   16 x   1
    2.674             2.674
                0.3739 x 16
             = 5.984
      24                24 x     1
    0.1396                 0.1396
                       7.163 x 24
                   = 171.912
    5.984 + 171.896
                   = 177.896
13. Numerator
     5 − 2      15 − 4  =  11
     2    3            6         6
    ½ of ⁷/₂ = ⁷/₄          3¹ x 11
                                2     6₂         
    ⁷/₄ + ¹¹/₄ =  ¹⁸⁹/₄₂ = 4½
    Denominator
    ¾ of ⁵/_{2 =} ¹⁵/₈ x ²/₁ = ¹⁵/₄
    
    ⁹³/₂₁ x ⁴²/₁₅₅    = ⁶/₅  = 1¹/₅
14. 
    2x + 3y = 1
    2x = 1 − 3y
     2         2
    3(1 − 3y) − 2y = 8
           2
    3(1 − 3y)x2 − 2yx2 = 8 x 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
    3         3      3
    y = − 2x + 3
              3   
    Gradient are equal (parallel lines)
    = − 2 
          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. 
       

                                                                                         SECTION B

17.  
    1.  
       

       Marks	F	x	fx
       199 - 203	5	201	1005
       204 - 208	2	206	412
       209 - 213	3	211	633
       214 - 218	2	216	432
       219 - 223	2	221	442
       224 - 228	7	226	1582
       229 - 233	1	231	231
       234 - 238	0	236	0
       239 - 243	3	241	723
       244 - 248	1	246	246
       249 - 253	4	251	1004
       	∑f		∑fx
       	30		6710

    2. Modal class = 224 − 228
    3. Mean = ∑fx = 6710 = 223.67
                    ∑f       30
    4. 
       30th = 15th + 16th = 15.5th
         2               2
       223.5 + (15.5 − 14 x 5)
                             7
       223.5 + (1.5 x 5)
                       7
                 = 224.57 marks
18.  
    1. 
       
       • check scale
       • units on both
       • axis
    2. 60 − 30  =  30  = 5m/s²
            6            6           (check units)
    3. 
       Total distance - area under the graph
       1/21 x 90 x 63 = 270m
       5 x 60 = 300m
       ¹/₂₁ x 3 x 60³⁰ = 90 
       270 + 300 + 90  = 660m
19. 1.  
       
       Sin⁻¹ (6.5) =    Sin⁻¹ 0.8125
                     8         54.34 x 2
            ∠XO₁Y    =   108.68°
       
    2. Sin⁻¹ 6.5   Sin⁻¹ 0.65 = 40.54 x 2
                      ∠ XO₁Y = 81.08°
    3. πr²  0
             360
       3.142 x 8 x 8 x 108.68 = 60.71cm²
                                        360
    4. 3.142 x 10 x 10 x 81.08
                                     360
                           = 70.765cm²
    5. 
       ¹/₂₁ x 8⁴ x 8 Sin 108.68
                      = 60.71 − 30.31cm²
                      = 30.4cm²
       ½ x 10 x 10 Sin 81.08
                      = 49.4
       70.765 − 49.4
                     = 21.365cm²
       30.4 + 21.365
                     = 51.765cm²
20.  
    1.  
       1. →         →      →
          AM   =  AO + OB
                  = 
       2. →           →     →
          OM   =   OA + AM
                   =   a + ²/₅ (b - a)
                   =   a + ²/₅b - ²/₅a
                   =   
       3. →          →       →
          BK    =   BO +  OK
                   = − b + ½a
    2.  
       1. →          →       →
          PQ    =   OQ − OP
                   =   (⁸₁₄) − (⁶₁₀) = (²₄)
       2. 
          6 + 8 , 10 + 14
              2           2
            (7, 12)
    3.  
       P = (⁶₈) − (²₁) + (¹²₋₁₆)
       
       P = (¹⁶₋₉)   √16² +(−9)²
          = √337
          = 18.36 units
21. 
    
    25 = x + 15
    10        x                25x = 10x + 150
                                   15x = 150
                                        x = 10cm
    
    
    1. 10 + 15 = 25cm
    2. Volume of bigger cone
       ¹/₃ x 3.142 x 25 x 25 x 25
                   = 16364.58cm³
       Volume of smaller cone
       ¹/₃ x 3.142 x 10 x 10 x 10
                   = 1047.33cm³
       Frustrum = 16364.58 − 1047.33
                       = 15317.25 litres
    3. 
       slant height (L)   = √25² + 25²
                                          =   35.36cm
                            (L) =   10² + 10²
                                        =   14.142cm
       Curved area
       3.142 x 25 x 35.36   = 277.53cm²
       3.142 x 10 x 14.142 = 444.34cm²
                                           2333.19cm²
        3.142 x 10 x 10        =  314.2cm²
        3.142 x 25 x 25        = 1963.75cm²
                     Total            = 4611.14cm²
22.  
    1.  
       1. 3x + 4y = 12
          4y = −3x + 12
           4       4       4
          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 L_{2 =} 
                ¾x = +1    x = 4
                                      3
          y − 5 = 4
          x + 4    3
          3y − 15 = 4x +16
           3y − 4x = 31
          3y = 4x + 31
           3      3      3
          y =  4x + 31
                 3      3
23.  
    L1  = (−2, 0) (0,4)     x    y
                                 2    1
    4 − 0 = 4 = 2
    0−−2    2
       y   = 2           y = 2x +4
    x + 2                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      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      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 x 180 = 50m/s
       T = ^D/_S =   ¹⁵⁰/₅₀ = 3 seconds
    2. 
       D = 12040 x 8 = 320km
                           3₁
       320 = 80 litres x 64 = Shs 5120
          4
    3. ½ x 16km/hr = 8km
                              40 − 8 = 32km
           32  =  324 = 1hr 20 minutes
       16 + 8    243
          8.00
       + 1.20
          9.20 a.m