Mathematics Paper 2 Questions - BSJE Mock Exams 2023

• The paper contains two sections. Section I and Section II.
• Answer all the questions in Section I and only FIVE questions from Section II
• Show all the steps in your working giving your answer at each stage in the spaces below each question.
• 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 1 (50MKS)

                                                                        Attempt all questions in this section

1. Calculate the standard deviation of the numbers to 3 decimal places.  (3mks)
   0.3, 0.7, 0.9, 0.13, 0.17, 0.11                                                                         
2. Maina wishes to buy a laptop which costs Ksh 100,000, he borrowed the money from a bank. The loan has to be paid at the end of 2years. The bank charges an inte4rest rate of 24% p.a. compounded quarterly. Calculate the total interest payable to the bank.          (3mks)                                                                                                                                     
3. A rectangular block has a square base whose sides are exactly 20cm; the height measured to the nearest millimeter is 10.4cm. calculate the percentage error in calculation of the volume       (3mks)
4. A particle moves in a straight line from a fixed point. The velocity v of the particle after t seconds is given by  V = t² − 2t + 6. Calculate the distance travelled by the particle during the fourth second.    (3mks)                                                                                               
5. Use the expansion of (x − y)⁴ to evaluate (9.7)⁴ correct to 4 significant figures.    (3mks)
6. By taking a small change in x as ∆x find the gradient function of y = 3x² + 4x + 3 at a point (2,4).  (3mks)                                       
7. Solve for ϴ in 3sinϴtan ϴ = 8 , for 0°≤ ϴ ≤ 360°            (3mks)                                                                                           
8. Solve the following simultaneous equations             (3mks)                                                                 
   x + y = 0 
   2x = (y + 1)  
9. Mark and John working together can do a piece of work in 3 days. Mark working alone takes two days less than John. How long does it take John to do the work alone?  ( give your answer to the nearest day)  (3mks)
10. In the figure below the circles center o and c touches internally at N and <POQ = 80°
                                                            
    If the radius of the larger circle is 12cm, calculate the radius of the smaller circle.       (3mks)               
11. A boat’s speed in still water is 4km/h. the boat cruises from A to B along a river flowing at an average speed of xkm/hr in the direction of A to B. if the distance AB is 5km and the boat takes 2 hours more on its return journey, determine x. hence find the total time taken for the whole journey.               (4mks)
12. Make n the subject of the formula; S = ⁿ/₂ [2a + (n − 1)d]                        (3mks)
13. Two points lie on the earth’s surface such that their positions on the surface of the earth are given as A(60°S,20°W)  and B60°S,60°E.  Find the shortest distance in nm to the nearest whole number.     (4mks)                                                                                             
14. Two boats R and P left town A at 9:00am. R sailed at a speed of 45km/h on a bearing of 225° and P sailed at a speed of 20km/h on a bearing of 140°. Calculate the distance between the two boats at 12:00 noon the same day, to 1 decimal place.                      (3mks)
15. A shear with x-axis invariant maps point A(4,3) onto A^I(10,3). Determine the shear matrix, hence find the image of a point Q (1,−2) under the same shear.      (3mks)                                       
16. Find the area enclosed by the curve y = 3x² + 8x  and the x-axis.                                   (3mks)
    
                                                                              SECTION 1I (50 MARKS)
                                              Attempt ANY FIVE questions in this section in the spaces provided
17. The first term of an arithmetic progression(AP) is equal to the 1^st term of a Geometric progression (GP), the 2^nd term of the AP is equal to the 4^th term of the G.P, while the 10^th term of the AP is equal to the 7^th term of the GP
    1.  
       1. Given that a is the first term and d is the common difference of the AP while r is the common ratio of the G.P, write two equations connecting the A.P and the G.P   (2mks)
       2.  Find the value of r that satisfies the progression         (4mks)
    2. Given that the 8^th term of the G.P is 1280, find the value of a and d            (2mks)
    3. Calculate the sum of the first 12 terms of the G.P                                      (2mks)
18. The table below shows the income tax rates in a certain year
    

    Income in Kenyan shillings p.m	Rate in Ksh  per pound
    Under Ksh12001	2
    From Ksh12001 but under Ksh22501	3
    From Ksh22501 but under Ksh33001	4
    From Ksh33001 but under Ksh43501	5
    From  Ksh43501 but under Ksh54001	6
    From Ksh54001 and over	7

    
    Mr Ole Kaelo earns A basic salary of Ksh 40,780. He is entitled to the following allowances per month: house allowance-Ksh 10,100, medical allowances- Ksh 3,850, commuter allowance-    Ksh 2,750 and a hardship allowance of Ksh 5,200. He is entitled to a personal tax relief of        Ksh 1,054 per month and a monthly insurance relief of 15% of the premium paid. He paid a monthly premium of Ksh 2,400 towards his life insurance.
    Calculate:
    1. Mr Ole Kaelo’s monthly taxable income in Ksh              (2mks)
    2. Ole Kaelo’s monthly Pay as You Earn.                    (5mks)
    3. Apart from income tax, the following monthly deductions were made from his salary;
       NIHIF – Ksh 3,000
       Bank loan – Ksh7,568
       WCPS – Ksh 880
       Calculate His net salary for that month                                                      (3mks)
19.  
    1. A die is biased so that when tossed, the probability of a number r showing up is given by       P( r) =Kr where K is a constant and r = 1,2,3,4,5,6 (the numbers on the faces of the die)
       1. Find the value of K                                                                                    (2mks)
       2. If the die is tossed twice, calculate the probability that the total score is 11.   (2mks)
    2. A tetrahedron with faces 1,2,3 and 4 is tossed together with a die and the difference noted down. Using x to represent the tetrahedron and y to represent the die,
       1. Draw a sample space showing all the possible outcomes                            (2mks)
       2. Find the probability such that P(y − x) ≥ 3                                                            (1mk)
       3. In a form 1 class there are 22 girls and 18 boys. The probability of a girl completing the secondary education course is ³/₅  whereas that of a boy is ²/₃ . Two students are picked at random. Find the probability that they are a boy and a girl and that both will not complete the course          (3mks)
20. In the figure below, the shaded region is bounded between the line y = 15x and the curve y = 3x²
                                                                                  
    1. Determine the coordinates of P             (2mks)
    2. By integration, determine the area of the shaded region                      (3mks)
    3. Estimate the area of the shaded region using trapezoidal rule with 5 strips     (5mks)
21. An aircraft took off from point E (x°S,20°E) at 0800h, local time. It flew due West to another point F(x°S,80°W), a distance of 6000km from E. After a stopover of 2hours 20minutes at point F, The aircraft took off and flew for 4hours 40minutes due north to a point G. the aircraft maintained an average speed of 1000km/h for the whole journey from E to F and also from F to G. (take π = ²²/₇ and the radius of the earth to be 6370km)
    1. Calculate to the nearest degree;
       1. Position of point F              (3mks)
       2. Position of point G               (3mks)
    2. Determine the local time at point G when the aircraft arrived                            (4mks)
22.  
    1. Fill in the table below for the function y=cos½x° and y = 1 + 2sin⁡(x° + 45°)    (2mks)
       

       x0	-90	-60	-30	0	30	60	90	120	150	180	210	240	270	300	330	360
       cos½x°
       1 + 2sin⁡(x°+ 45°)

    2. On the same axes, use the grid provided below to draw the graph of y = cos½x° and y = 1 + 2sin⁡(x°+ 45°), using a scale of 1 big square to represent 300 on the x-axis and 2 big square to represent 0.5 units on the y-axis                          (4mks)       
                                               
    3. Use your graph to:
       1. Solve − 1 + cos½x = 2sin⁡(x + 45)       (2mks)
       2. State the amplitude and period of each function       (2mks)
23. The diagram below shows a view of a rectangular tank of length 6m. one end of the tank is square ABCD of side 2m
                                                     
    If AP is a space diagonal of the tank, calculate to 2 decimal places;
    1.  
       1. Then length AP                       (2mks)
       2. The angle between line AP and plane ABCD       (2mks)
       3. The angle between line AP and line PD                                                    (2mks)
    2. Then shortest distance between plane APD and line BC                                       (4mks)
24. A factory has 160 litres of solution A, 110 litres of solution B and 150 litres of solution C. to prepare a bottle of of Syrup X, 200ml of solution A, 100ml of solutuion B and 300ml of solution C are needed. For a bottle of Y, 100ml of solution A 200ml of solution B and 300ml of C are needed. Syrup X sells at Ksh 60 per bottle and Syrup Y sells at Ksh 100per bottle.
    1. Form the inequalities to represent the above information          3mks)
    2. Graph the above inequalities on the grid provided       (5mks)
       
    3. How many bottle of each type of Syrup should the firm make in order to obtain maximum amount of money.        (2mks)