Mathematics Paper 2 Questions - KCSE 2022 Mock Exams Set 1

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INSTRUCTIONS TO CANDIDATES

  • Write your name and index number in the spaces provided at the top of this page.
  • Write your school name, sign and write the date of the examination in the spaces provided above.
  • This paper consists of Two sections: Section I and Section II
  • Answer ALL questions in Section I and any five questions from Section II
  • Show all the steps in your calculations, giving your answers at each stage in the spaces provided 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.
  • Candidates should check the question paper to ensure that all the pages are printed as indicated and no questions are missing.
  • Candidates should answer the questions in English.

For Examiners’ Use Only
SECTION I

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

Total

                                 

SECTION II

17

18

19

20

21

22

23

24

Total

 

Grand Total

 
                 

QUESTIONS

SECTION I (50 Marks)
Answer all the questions in this section

  1. Solve for x in the equation below without using a mathematical table or calculator. (4 marks)
    (log10⁡x)2= 3 - log10⁡x2
  2. The base of a right angled triangle is 4.1 cm and the height is 5.0 cm. Calculate the percentage error in the area of the triangle. (3 marks)
  3. Given that tan⁡ θ=1/√5 , θ is an acute angle, without using a calculator or mathematical tables, find sin⁡(90 - θ), leaving your answer in simplified surd form. (2 marks)
  4. Find the interest on Ksh. 200,000 for 2 years at 14% per annum compounded semi-annually. (3 marks)
  5. Make v the subject of the formula (3 marks)
    5 zddada
  6. A coffee trader buys two grades of coffee at Kshs. 80 and Kshs. 100 per packet. Find the ratio in which she should mix the two brands so that by selling the mixture at Kshs. 120 per packet, a 25% profit realized? (3 marks)
  7. A bakery prepares cakes for sale. It has 80 eggs and 10 cups of sugar for use. It bakes two cake types: P and Q. Type P cake requires 6 eggs and 2 cups of sugar while type Q cake requires 12 eggs and three-quarters cup of sugar. By letting type P cakes to be x and type Q cakes to be y, form all the inequalities that represent the above information. (3 marks)
  8. Find the radius and the centre of a circle whose equation is given by 3x2+3y2+6x-12y-12=0. (3 marks)
  9. The equation of a trigonometric function is y=2 cos⁡(bx-60)º . The period of the function is 1200.
    1. Determine the value of b (1 mark)
    2. Deduce the phase angle of the function. (1 mark)
  10. A point R is 2100 nautical miles to the east of another point Q (60ºN, 0º), find the position of P. (3 marks)
  11. An arithmetic progression is such that its first term is 200 and common difference 500. Given that Sn=80,100, find the value of n (4 marks)
  12.                                  
    1. Expand (3+x)5 in ascending powers of x up to the term in x3. (1 mark)
    2. Use the expansion in (a) above to approximate the value of 12 adada correct to 4 decimal places. (2 marks)
  13. P varies as the cube of Q and inversely as the square root of R. If Q is reduced by 20% and R increased by 21%, find the percentage change in P. (3 marks)
  14. Use tables of squares, reciprocals and square roots only to evaluate (4 marks)
    14 adada
  15. In the figure below, AD = 9 cm, AB = 11cm and angle BAD = 800. BD is the diameter of the semi-circle BCD.
    15 adadad
    Calculate the area of the semi-circle, correct to 2 decimal places. Use π = 3.142 (4 marks)
  16. Two regular polygons have sides n and n+3. Given that the ratio of the sum of their interior angles is 1: 2, calculate the value of n. (3 marks) 

SECTION II (50 Marks)
Answer any five question in this section

  1. The table below shows income tax rates in a certain year.

    Taxable Income

    (Ksh per month)

    Tax Rate

    (%)

    0 – 13 450

    10

    13 451 – 26 350

    15

    26 351 – 39 250

    20

    39 251 – 52 150

    25

    52 151 and above

    30

    In that year, the monthly earnings for Amilo were as follows: basic salary Ksh 35 500, house allowance – Ksh 12 600 and other allowances that amount to Ksh. 5 872 were exempted from taxation.
    Amilo contributes 12.5% of her basic salary to a pension scheme. She is entitled to a personal tax relief of Ksh 1 845 per month.
    Calculate:
    1. Amilo’s taxable income in Ksh per month. (2 marks)
    2. Amilo’s P.A.Y.E that month. (5 marks)
    3. Amilo’s net pay that month, given that the following are deducted monthly from her salary; NHIF – Ksh 1 000, Union dues – Kshs 455 and BBF – Ksh 200. (3 marks)
  2. A mode is in the shape of a polygon with vertices A, B, C, D and E such that; AB=4.4 cm, AE=10 cm, ED=5.2 cm and BC=7.9 cm. The bearing of B from A is 0300 and A is due east of E, while D is due north of E and angle EDC=1200
    1. Using a ruler and a pair of compasses only,
      1. Construct the accurate plan of the model. (4 marks)
      2. Measure DC. (1 mark)
    2. A foundation plaque is to be placed closer to CD than CB, nearer to D than to E and not more than 6 cm from A.
      1. Construct the locus of points equidistant from CB and CD. (1 mark)
      2. Construct the locus of points equidistant from E and D. (1 mark)
      3. Construct the locus of points 6 cm from A. (1 mark)
    3. Shade and label as R, the region within which the foundation plaque could be placed in the garden. (2 marks)
  3. The probability that it rains on a certain day is 0.8. If it rains the probability that a school bus will be stuck in a traffic jam is 0.7 but otherwise it is 0.4. If the bus is stuck in the jam, the probability that students using it to school will arrive late is 0.6, otherwise the probability of students using the bus to arrive late is 0.3.
    1. Draw a tree diagram to represent this information. Use the letters R, J and L to represent the events of rain, jam and late respectively (2 marks)
    2. Determine:
      1. The probability that it rains, the bus isn’t held in the jam but the students arrive late in school. (1 mark)
      2. The probability that students arrive in school on time. (3 marks)
      3. The probability that the students arrive in school late. (2 marks)
      4. The probability that the bus is held in the jam. (2 marks)
  4. The vertices of a triangle ABC are P'(-1,1), B^' (-5,4) and C'(-1,2) under a transformation whose matrix is 20 adada
    1. Find the coordinates of ABC (3 marks)
    2. On the grid provided, draw triangles ABC and A'B'C'. (2 marks)
      graph adad
    3. Triangle A''B''C'' is the image of triangle A'B'C' under a transformation represented by the matrix 19 c adadad
      1. Determine the coordinates of ΔA''B''C''. (2 marks)
      2. On the same grid, draw ΔA''B''C''. (1 mark)
    4. Another transformation T maps ΔA''B''C'' on to ΔA'''B'''C''' such that A'''(-1,-2), B'''(-5,-8) and C''' (-1,-4). Describe T fully. (2 marks)
  5. The figure below shows a frequency polygon representing the scores of Form 4 Green students in a History contest.
    21 adada
    1. Generate the frequency distribution table for the data under the headings given in the table below. (5 marks)

      x

      f

      d = x - 67
            5

      fd

      fd2

               
               
               
               
               
               
               
       

      Ʃf =

       

      Ʃfd =

      Ʃfd2

    2. Calculate the standard deviation of the marks. (3 marks)
    3. The mean weight of 11 babies in a clinic is 4.5 kg. If one more baby comes to the clinic, the total mass of the babies becomes 60 kg. Find the mass of the additional baby. (2 marks)
  6. In a triangle OAB, OA=12a, and OB=12b. P and Q are points on OA and OB respectively such that 3OP=OA and OQ=1/3 OB. M is the midpoint of AB.
    1. Express the following in terms of a and b
      1. OM (1 mark)
      2. PM (1 mark)
    2. OM and BP intersect at R such that PR=kPB and OR=hOM.
      1. Express PR in two ways (2 marks)
      2. Hence find the values of h and k (3 marks)
    3. Show that A, R and Q are collinear. (3 marks)
  7. The figure below represents a right pyramid with a vertex V and a rectangular base PQRS. VP=VQ=VR=VS=18 cm. PQ=16 cm and QR=12 cm. M and O are the midpoints of QR and PR respectively.
    23 adadad
    Calculate, correct to 2 decimal places;
    1. The length of the projection of the line VP on the plane PQRS (2 marks)
    2. The angle between the lines VP and the plane PQRS. (2 marks)
    3. the angle between planes VQR and VPS. (4 marks)
    4. The angle between the planes VQR and PQRS (2 marks)
  8. Two functions, x+y=4 and y=x2+2, intersec at C and D
    1. Determine the coordinates of C and D (4 marks)
    2. Using the trapezium rule with 6 trapezia, estimate the area bound by y=x2+2, the x-axis and the vertical lines through C and D. (4 marks)
    3. Find the exact area in (b) above. (3 marks)

SECTION II (50 Marks)
Answer any five questions in this section

  1. A straight line L1 has a gradient -1/2 and passes through the point P(-1,3). Another line L2 passes through the points Q(1,-3) and R(4,5)
    Find:
    1.                        
      1. The equation of Lin the form y=mx+c, where m and c are constants (2 marks)
      2. Hence find k given that S(0,k) (1 mark)
    2.                          
      1. The gradient of L2 (1 mark)
      2. The equation of L2 in the form ax+by=c, where a,b and c are integral values. (2 marks)
    3. The equation of a line L3 passing through a point T(0,5) and perpendicular to L2. (3 marks)
    4. Calculate the acute angle that L3 makes with the x-axis. (1 mark)
  2. Water flows through a cylindrical pipe of diameter 7 cm at a rate of 15 m per minute.
    1. Calculate the capacity of water delivered by the pipe in one minute in litres. Use π=22/7. (3 marks)
    2. A storage tank that has a circular base and depth 12 m is filled with water from this pipe and at the same rate of flow. Water begins flowing into the empty storage tank at 6.30 a.m. and is full at 1310 hours. Calculate the area of the cross-section of this tank in square metres. (4 marks)
    3. A school consumes the capacity of this tank in one month. The cost of water is Ksh. 100 for every 1 000 litres and a standing charge of Ksh 1 950. Calculate the cost of the school’s water bill for one month. (3 marks)
  3.                      
    1. Find A-1 given that 19 adadada (1 mark)
    2. An ICT firm bought 8 printers and 12 copiers for a total of Ksh 294 000. Had the firm bought 1 more printer and 3 more copiers, it would have spent Ksh 43 500 more.
      1. Form two equations to represent the information above. (2 marks)
      2. Hence, using A-1 in (a) above, calculate the cost of each item. (4 marks)
      3. A two-digit number is such the that difference between tens and ones digit is 1. If the digits are reversed, the sum of the two numbers is 165. Find the original number (3 marks)
  4. Two equal circles with centres P and Q and radius 8 cm intersect at points A and B as shown below.
    20 adadada
    Given that the distance between their centres is 12 cm, find, correct to 4 significant figures;
    1. The length of chord AB (2 marks)
    2. The area of the shaded region . Use π = 3.142 (8 marks)
  5. In the figure below, P, Q, R, S and T lie on the circumference of a circle centre O. Line UPV is a tangent to the circle at P. Chord ST of the circle is produced to intersect with the tangent at U. Angles UPT, RST and ORQ are 28º, 100º and 50º respectively.
    21 ssfsfs
    1. Determine the sizes of the following angles;
      1. RTP (3 marks)
      2. QTP (3 marks)
    2. Given that PQ = 6 cm, calculate correct to 1 decimal place, the radius of the circle. (4 marks)
  6.                                
    1. Two trains, A and B are such that they are 40 m long and 160 m long respectively. Their speeds are 60 km/hr and 40 km/hr respectively. The two trains are 100 m apart and moving towards the same direction in a pair of parallel tracks. Calculate the time in seconds it takes train B to completely overtake train A. (4 marks)
    2. The figure below (not drawn to scale) shows a velocity time graph for a robot in a robotic challenge.
      22 adada
      1. If the distance covered by the robot in the first 15 seconds was 180 metres, calculate the value of m (3 marks)
      2. Describe the movement of the robot between the 15th and 45th seconds. (1 mark)
      3. Calculate the deceleration of the particle in the last 20 seconds. (2 mark)
  7. A farmer wanted to make a trough for cows to drink water. He had a metal sheet of dimensions 240 cm by 120 cm and 1 cm thick. The density of the metal sheet is 2.5 g/cm3. A square of sides x cm is removed from each corner of the rectangle and the remaining part folded to form an open cuboid.
    1. Sketch the sheet after removing the squares from the four corners, showing all the dimensions. (2 marks)
    2. Calculate
      1. the value of x, to the nearest whole number, that maximizes the volume of the cuboid (5 marks)
      2. hence calculate the maximum volume of the box . (3 marks)
  8.                            
    1. Complete the table below for the curve y=x3-5x2+2x+9 for -2≤x≤5. (2 marks)
       x  -2  -1  0  1  2  3  4 5
       y      9          
    2. On the grid provided, draw the graph of y=x3-5x2+2x+9 for -2≤x≤5. (3 marks)
      24 adadad
    3. Use the graph in (b) above to find the roots to the following equations:
      1. x3-5x2+2x+9=0 (2 marks)
      2. x3-5x2+6x=-5 (3 marks)

 M


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