Mathematics Paper 2 Questions and Answers - Mokasa 1 Joint Pre Mocks Exams 2023

• The paper consists of two sections.  Section I and Section II.
• Answer ALL the questions in Section I and any FIVE from Section II.
• Marks may be given for correct working even if the answer is wrong.
• Negligence and slovenly work will be penalized
• Non programmable silent electronic calculator and KNEC Mathematical tables may be used except where stated otherwise.

Section I (50 marks)

Answer all questions in this section

1. The cash price of a phone is Ksh. 46,000. Linda bought the phone on hire purchase terms by paying a deposit of Ksh16,000 and the balance by 12 equal monthly installments of Ksh 3600. Find the compound rate of interest per month. (3 marks)
2. Given that cos 285° = √6 − √2 , simplify     1                                        (3 marks) 
                                             4                    cos75°
3. Expand and simplify (2 – ¹/₄x)⁶ up to the fourth term. Hence use your expansion to solve (1.96)⁶ correct to 3 decimal places. (4 marks)
4. Solve the equation (2+3x) + 32 = 2 + (2x+6)      (3 marks)   
5. Make x the subject of the formula; (3 marks)
   w = 20 − 1+x
                  1−x
6. The ends of the diameter of a circle has coordinates (6,−5) and (−2,3). Find the equation of the circle in the form x² + y² + bx + ay + c = 0. (4 marks)
7. The length and width of a rectangular floor are given as 24.2m and 7.2m respectively. The dimensions are given with 2.5% and 4% of error respectively. Find the range within which the area of this floor lie. (3 marks)
8. Find the period, amplitude and the phase angle of y = ¹¹/₂ Cos ( 3x + 28°). (3 marks)
9. The second, sixth and the eighth terms of an arithmetic progression corresponds to the first three consecutive terms of an increasing geometric progression. If the first term of the A.P is −36, the common difference of the A.P is d and the common ratio of the G.P is r. find the value of d and r. (4 marks)
10. A circle has two chords PQ and RS which intersect internally at point O. Given that PO= 8cm , OQ= 6cm  and RO = 4.5cm, find the measurement OS. (2 marks)
11.                                                                           →                 →          
    The position vectors of points U, V and W are OU=2i-j+3k, OV=6i-3j+9k and OW = 18i – 9j + 27k.  Show that U, V and W are collinear points.        (3 marks)           
12. The values below represent the height of different trees in a school.
    34,23,65,11,12,42,33,49,40,28,69,41,37,19,24,47, Calculate the quartile deviation of the height of the trees.                     (3 marks)
13. The table below shows the values of x and y. Draw the curve on the graph below hence find the gradient of the curve when x = 2 (3mks)
    

    X	1	2	3
    Y	3	2	1.7

14. In the figure below, PQRS is a cyclic quadrilateral. Point O is the centre of the circle. Angle PQO=35° and angle PSO=42°
    
    Calculate the size of angle QRS.        (2 marks)
15. Use logarithm tables to evaluate          (4 marks)
    √0.0024 × 32.48²
        321
16. T is a transformation represented by the matrix  5x 2 x -3  . Under T, a square of area 10cm² is mapped onto a square 110cm². Find the values of x. (3 marks)

Section II ( 50 marks)

Answer only five questions in this section.

17. The vertices of triangle A¹B¹C¹ are A¹(1,3,) B¹(3,−4) and C¹(5,−6). Triangle A¹B¹C¹ is the image of triangle ABC under a transformation whose matrix is (1 0 −2 1).
    1. Determine the coordinates of triangle ABC. (3 marks)
    2. On the grid provided, draw the objects and the image. (2 marks)
    3. Describe fully the transformation which maps ABC onto A¹B¹C¹. (2 marks)
    4. The triangle ABC undergoes a transformation given by the matrix (−1 0 0 −1)  , to give the image ABC. Draw the image on grid hence describe the transformation. (3 marks)
18. The table bel.w shows the masses of form three students in a class 
    

    Mass	Freq,
    30 - 44	2
    45 - 49	8
    50 - 54	15
    55 - 59	18
    60 - 64	8
    65 - 69	4
    70 - 74	1

    Using an assumed mean of 57, calculate
    1. The mean.          (4 marks)
    2. The standard deviation.          (6 marks)
19.  
    1. Using a ruler and a pair of compass only, construct triangle ABC in which AB = 8cm, and   AC=8cm and <BAC = 60°     (2 marks) 
    2. Determine the locus L₁  of points equidistant from A and B       (1 mark)
    3. Determine the locus L₂ of points equidistant from AC and AB      (1 mark)
    4. Determine the locus P of points such that <APB = 140°            (2 marks)         
    5.  Determine the locus Q of points such that CQ=5cm             (1mark)
    6. A point W moves inside the triangle such that AL₁≤BL₁, < BAL₂ ≤ <CAL₂ and CQ≥ 5cm. Shade the region W.       (3 marks)
20. Complete the table below, giving your values correct to 1 decimal places. (2 marks)
    

    x°	0	30	60	90	120	150	180
    3 sin x	0.0			3.0
    2 − Cos α			1.5			2.9

    1. On the grid provided using the same axes, draw the graphs of y=3sin x  and y=2-cos x  for 0°≤x≤360°. Scale Y-axis- 2cm rep 1 unit. X-axis, 1cm represent 10°         (5 marks)
    2. Use the graph in (b) above to solve the equations;
       1. 4 − 2cos x = 4     (1 mark)
       2. 3 sin x + cos x = 2 (2 marks)
    3. Determine the amplitude of
       1. 3 Sin x. (1 mark)
       2. 2 – Cos x. (1 mark)
21. Triangle OPQ is such that OP = p and OQ = q .Point R divides OP in the ratio 1:3 and a point S divides PQ in the ratio 5:2.OS and RQ meet at T.
    1. Express in terms of p and q.                                                            :
       1. OS                            ( 1 mark)
       2. RQ                    (1 mark)
    2. Given that OT=kOS. Express in terms of k, p and q.                                             (1 mark)
    3. Given also that RT=hRQ, express OT in terms of h, p and q.                          (1 mark)
    4. By expressing OT in two value of h and k.                                         (5 marks)
    5. State the ratio in which S divides OT.                                                               (2marks)
22.  
    1. The ratio of the cost of commodity X to that of commodity Y is 2:3 and the ratio of the cost of Y to the cost of commodity Z is 6:1. If the total cost of the three commodity is sh. 1100;
       1. Find the cost of x. (2 marks)
       2. Express the cost of Z as a percentage of the cost of Y. (2 marks)
    2. A factory requires 100 workers to perform a certain job. After they have worked for 15 days the factory employs extra 26 persons for 6 days so that the job can be completed in time. How many workers would the factory have required at the beginning in order to complete the job in 12 days. (3 marks)
    3. Tap A fills a tank in 20 minutes and tap B can empty the same tank when full in 25 minutes. Both taps are turned on at the same time for 10 minutes after which tap B is turned off. How long will it take tap A to fill the remaining part of the tank. (3 marks)
23. The table below shows tax rate in 2003.
    

    Income (sh p.m)	Tax rates in %
    1 - 8270  8271 -15790  15791-23310  23311-30830  30831-38350  38351-45870  45871-53390  Over 53390	5   10   15   20   25   35   45   50

    Mrs Odundo earns a monthly salary of shs. 23,520, a monthly House allowance of sh. 10,000, a medical allowance of sh.3,018, a commuter allowance of sh.916 and A non-taxable allowance of sh. 4,500. He is also entitled to a personal relief of sh. 1,600 p.m.
    Calculate:
    1. Calculate his taxable income per month.    (2 marks)
    2. Calculate his net monthly tax. (5 marks)
    3. He also has the following to pay. Nhif of sh. 1500, Nssf of sh. 2,500, wcps of sh. 4,500. Calculate his net pay.      (3 marks)
24. A student has a probability of ²/₃ of waking up on time. If he wakes up on time there is a probability of ⁷/₁₀ that he will catch the bus and be on time to school. If he oversleeps there is a probability of ²/₅ that he will catch the bus. If he catches the bus the probability that he will reach school on time is ⁷/₈, if he misses the bus there is only a probability of a ¹/₄ that he will be on time for school. Using the tree diagram or otherwise
    1. Determine.
       1. The probability that he catches the bus. (3 marks)
       2. The probability that he is late for the school. (2 marks)
       3. The probability that he oversleeps and is on time for school. (2 marks)
    2. Science club is made up of 6 boys and 8 girls. The club has three officials. Find the probability that the club officials has more boys than girls (3 marks)

MARKING SCHEME

1.  46000 − 16000 = 40000
   3800 × 12 = 43,200
   43,200 = 40,000 (1.1^r/₁₀₀)¹² 
   40,000    40,000
   ¹²√1.08 = ¹²√1 = 0.01r)¹² 
   1.006 = 1 + 0.01r
   0.006 = 0.01r
   r = 0.6%
2.      1     
   √6 − √2
         4
        4      × √6 + √2
   √6 − √2    √6 + √2
   4√6 + 4√2
       6−2
   √6 + √2
3.   
   
4.  
   
5.   
   
6.  
   
7.  2.5  × 24.2 = 0.605
   100
     4   × 7.2 = 0.288
   100
   
   185.73984 − 163.08864
8. Period = 360 = 120°
                   3
   amplitude = ¹¹/₂ = 5.5
   Phase angle = 28°
9.  
   
10.  
    
    8 × 6 = 4.5x
    x = 10²/₃ 
11.  
    
12. 11, 12, 19, 23, 24, 28, 33, 34, 37, 40, 41, 42, 47, 49, 65, 69
    23 + 24 = 23.5
         2
    42 + 47 = 44.5
         2
    44.5 − 23.5 = 10.5
            2
13.  
    
    (1, 2.6)
    (2.8, 1.4)
    2.6 − 1.4 = 1.2 = 0.6667
      2.8 − 1     1.8
14.  
    
    180 − 77 = 103°
15.  
    
16. (5x x −3) − (2x) = ¹¹⁰/₁₀ 
    −15x − 2x = 11
    − 17x = 11
        x = −¹¹/₁₇ 
17.  
    1. 
        
       
       
    2.  
       
    3. Shear, y-axis Invariand A(1,5) → A' (1,3)
    4.  
       
       Rotation, Centre (0,0) angle 180° or enlargment, E.S.F = −1, centre (0,0)
18. 
    

    Mass	Freq,	x	x − 57	ft	t2	ft2
    30 - 44	2	37	−20	−40	400	800
    45 - 49	8	47	−10	−80	100	800
    50 - 54	15	52	−5	−75	25	375
    55 - 59	18	57	0	0	0	0
    60 - 64	8	62	5	40	25	200
    65 - 69	4	67	10	40	100	400
    70 - 74	1	72	15	15	225	225
    	56			−100		2,800

    1. The mean
       A = Sft ⁻¹⁰⁰/₅₆
          = 55.21
    2. The standard deviation
       
       50 − 3.189 = √46.811
       = 6.842
19.  
    
20.  
    1. 
        
    2.  
       

       x°	0	30	60	90	120	150	180
       3 sin x	0.0	1.5	2.6	3.0	2.6	1.5	0
       2 − Cos α	0	1.1	1.5	2	2.5	2.9	3

    3.  
       1. 2 - 2 Cos x = 2
          30°, 150°
       2. − 2 − 3 = 5
21.  
          
    
    1.  
       1. OS = ⁵/₇q + ²/₇p
       2. RQ = q − ¼p
    2. k(⁵/₇q + ²/₇p) = ⁵/₇kq + ²/₇kp
    3. h(q − ¹/₄p) = hq − ¹/₄hp
    4.  
       
    5. 13 : −6
22.  
    1.  
       1. 2:3
             6:1
          6:6:1
          ⁶/₁₁ × 1100 = 600
       2. ¹/₁₁ × 1100 = 100
          100 × 100 = 6²/₃%
          600
    2. 100 × 15 = 1500
       126 × 6 = 756
       1500 + 756 = 2256 = 188
                                12
    3.  
       
23.  
    1. 23,520 + 10,000 + 3,018 + 916 = 37,454
    2. 8270 × 0.05 = 413.5
       7520 × 0.10 = 752
       7520 × 0.15 = 1128
       7520 × 0.20 = 1504
       6624 × 0.25 = 1656
                               5453.5
                               1600.0−
                                3853.5
    3. Gross income = 37,454 + 4,500 = 41,954
       deductions = 3853.5 + 1500 + 2500 + 4500 = 12353.5
       41,954 − 12,353.5 = 29,600.5
24.  
    1.   
       1. 
          
          (²/₃ × ⁷/₁₀ × ⁷/₈) + (²/₃ × ³/₁₀ × ¹/₄) + (¹/₃ × ²/₅ × ⁷/₈) + (¹/₃ × ³/₅ × ¹/₄)
                                        = ⁵/₈ 
       2. ²/₃ × ⁷/₁₀ + ¹/₃ × ²/₅ = ³/₅
       3. (¹/₃ × ²/₅ × ⁷/₈) + (¹/₃ × ³/₄ × ¹/₄) = ⁴³/₂₄₀ 
    2.  
       
       (⁶/₁₄ × ⁵/₁₃ × ⁴/₁₂) + (⁶/₁₄ × ⁵/₁₃ × ⁸/₁₂) +  (⁶/₁₄ × ⁸/₁₃ × ⁵/₁₂)  +  (⁸/₁₄ × ⁶/₁₃ × ⁵/₁₂)
                             = ⁵/₁₃