September 24, 2022

 2017 Mathematics Alt B Paper 1 Section I (50 marks) 1. (a) Express 4732 in terms of its prime factors. (1 mark) (b) Find the smallest positive nmnber that must be multiplied by 4732 to make it a perfect square. (1 mark) 2. Three people Juma, Weru and Njeri went round a circular racing track, 3. 12km long. They all started from the same point and moved in the same direction. Juma walked at 48m per minute,Weru ran at 120m per minute while Njeri cycled at 156m per minute. If they started travelling at 0700 h, find the time when they were first together again. (3 marks) 3. Evaluate 4. Without using a calculator evaluate 5. Use logarithms to evaluate 6. The diagram below represents a cube of side 10cm from which a cuboid measuring xcm by xcm by 10cm is removed as shown. Write an expression in terms of x, for the surface area of the remaining solid. (3 marks) 7. A cylindrical tank 1.4m in diameter contains 3234 litres of water. Find the depth, in metres, of the water. (Take Tr =7). (3 marks) The figure below represents a quadrilateral ABCD in which angle DAB = 60°, angle BCD = 30° and BC = DC = 40 cm. Side AB = AD. 8. Calculate the area of the quadrilateral ABCD correct to 4 significant figures. (4 marks) 9. The area of a sector of a circle is 36.96 cmz. The sector subtends an angle of 135° at the centre of the circle. Find the radius of the circle. (Take π=22/7). (3 marks) 10. Evaluate the expression given that t= 5 and r = 27. (2 marks) 11. Two employees Njoka and Okoth contributed i and % of their salaries respectively to start a project. The contribution amounted to Ksh 16 000. If Njoka contributed 2 and Okoth % of their salaries, the contribution would have been Ksh 30 O00. Calculate each person‘s salary. (3 marks) 12. Solve x – 8⋜-x⋝ 4 — 3x and represent the integral values of x on a number line. (4 marks) Figure ABCDEF is a regular hexagon. Line AE and BF intersect at G. size of angle F GE. (3 marks) 14. Using a ruler and a pair of compasses only, construct triangle PQR in which PQ = 8cm, A RPQ = 60° and L PRQ = 75°. Measure PR. (4 marks) 15. The marked price of a _TV set is Ksh 36 000. A dealer sold the set and allowed a 12% discount on the marked price and still made a 25% profit on the cost price. Find the cost price of the set.(3 marks) 16. Figure A’B’C’D’ is the image of ABCD under a rotation. By construction, detennine the centre P and the angle of rotation. (3 marks) SECTION II (50 marks) 17. A saleslady eams a monthly salary of Ksh 60 000. She gets a commission of 4% on the value of goods she sells above Ksh 250 O00 but less than Ksh 400 000. On goods sold above Ksh 400 000, she gets a commission of 7.5%. (a) In a certain month, she sold goods worth Ksh 525 O00. Calculate her total earnings that month. (4 marks) (b) In another month, she earned a total of Ksh 94 500. Find the value of goods that she sold that month. (6 marks) 18. Lines y + 2x = 4 and 3x — y = 1 intersect at point T. (a) Find the equation of line L] which passes through point T and (3,—2). (5 marks) (b) A line L2 passes through (5,4) and is parallel to L]. Find the equation of L, in the form y = mx + c where m and c are constants. (2 marks) (c) Another line L3 is perpendicular to L1 at T. Find the equation of L3 in the fonn ax + by = c where a, b and c are integers. (3 marks) 19. A car travelled from town A to town B. The car started from rest at A and moved with a constant acceleration for 2 minutes and attained a speed of 1.2 km/minute. lt then maintained this speed for a further 10 minutes before decelerating at a constant rate for another four minutes. The car finally came to rest at B. (a) On the grid provided, draw a speed-time graph for the car. (4 marks) (b) Use the graph to calculate: (i) the distance, in metres, the car travelled during its deceleration; (2 marks) (ii) the distance, in kilometres, covered by the car in the whole journey; (2 marks) (iii) the average speed, in km/h, for the whole journey. (2 marks) 20. The figure below is a square of side x cm. The square is divided into four regions A, B, C and D. Regions A and C are squares. Square C is of side ycm. Regions B and D are rectangles. (a) Find the total area of the following regions in terms of x and y in factorised form: (i) A and C; (1 mark) (ii) B and D; (2 marks) (ii) A, B, C and D. (b) Find the total area of B and D in terms of x given that y = 2cm. (c) Factorise 25csup>2 – 16. (d) Evaluate Without using mathematical tables: (i) 50242-49762 (ii) 8.962 -1.042 21. The figure below represents a right pyramid VEFGH mounted on a cuboid ABCDEFGH. LineAB =6cm,DA= 8cm andAF =BG =CH=DE=3cm. LineVE=VF=VG=VH= 13cm. Calculate, correct to 2 decimal places: (a) the surface area of the rectangular faces; (b) the surface area of the triangular faces. (c) the total surface area of the solid. 22. The figure below is a solid which consists of a frustum of a cone, a cylinder and a hemispherical top. The internal radii of the frustum are 42 cm and 2l cm. The vertical height of the original cone was 40 cm and the height of the cylinder is 30 cm Calculate: (a) the

2018 Mathematics Paper 2 Section I (50 marks) 1. Given that 2 log x2+log = k log x, find the value of k. (2 marks) 2. A variable P varies directly as t3 and inversely as the square root of s. When t = 2 and s = 9, P = 16. Determine the equation connecting P, t and s, hence find P when s = 36 and t=3. (4 marks) 3. Asia invested some money in a financial institution. The financial institution offered 6% per annum compound interest in the first year and 7% per annum in the second year. At the end of the second year, Asia had Ksh 170 130 in the financial institution. Determine the amount of money Asia invested. (3 marks) 4. The figure below represents a wedge ABCDEF. EF 10 cm, angle FBE 45° and the angle between the planes ABFE and ABCD is 20°. Calculate length BC, correct to l decimal place.(3 marks) 5. Simplify √ 54 + ∛ 3/ √ 3 (2 marks) 6. In the figure below, AB is a tangent to the circle, centre O and radius 6 cm. The arc AC subtends an angle of 60° at the centre of the circle. Calculate the area of the shaded region, correct to 1 decimal place. (4 marks) 7. Use completing the square method to solve 3×2 + Sx — 6 = 0, correct to 3 significant figures. (3 marks) 8. Three workers, working 8 hours per day can complete a task in 5 days. Each worker is paic Ksh 40 per hour. Calculate the cost of hiring 5 workers if they work for 6 hours per day tc complete the same task. (3 marks) 9. The table below represents a relationship between two variables x and y.   x 1 2 3 4 5 6 y 3.5 4.5 8.0 8.5 11 13 (a) On the grid provided draw the line of best fit.(3 marks) (b) Use the graph to find the value off when x — 0.(1 marks) 10. State the amplitude and the phase angle of the curve y = 2sin x — 30° .(2 marks) 11. The mass, in kilograms, of 9 sheep in a pen were: 13, 8, 16, 17, 19, 20, 15, 14 and 11. Determine the quartile deviation of the data. (3 marks) 12. The position of two points C and D on the earth’s surface are (8°N, l 0OE) and (8°N, 30°E) respectively. The distance between the two points is 600 nm. Determine the latitude on which C and D lie. (3 marks) 13. In the figure below OP — p, OR = r, PQ:QR = 1:2 and PS = 3PR. Express QS in terms of p and r. (4 marks) 14. In a certain firm there are 6 men and 4 women employees. Two employees are chosen at random to attend a seminar. Determine the probability that a man and a woman are chosen. (3 marks) 15. Under a transformation T = (4 —3) (2 3) , triangle OAB is mapped onto triangle OA’B’ with vertices O(0,0), A’(18,0) and B'(18, 6). Find the area of triangle OAB. (3 marks) 16. Find the value of k if Section II Answer any 5 questions from this section 17. The 5th and 10th terms of an arithmetic progression are 18 and —2 respectively. (a) Find the common difference and the first term. (4 marks) (b) Determine the least number of terms which must be added together so that the sum of the progression is negative. Hence find the sum. (6 marks) 18.Complete the table below for the equation y=x2-4x+2 (2 marks)   x 0 1 2 3 4 5 y (b) On the grid provided draw the graph y = x2 — 4x + 2 for 0≤ x ≤ 5. Use 2 cm to represent 1 unit on the x-axis and 1 cm to represent 1 unit on the y-axis. (3 marks) (c) Use the graph to solve the equation, x2 — 4x + 2 = 0 (2 marks) (d) By drawing a suitable line, use the graph in (b) to solve the equation A-2 — 5x + 3 = O. (3 marks) 19. (a) The table below shows the frequency distribution of heights of 40 plants in a tree nursery. (a) State the modal class. (1 marks) (b) Calculate: (i) the mean height of the plants; (3 marks) (ii) the standard deviation of the distribution. (4 marks) (c) Determine the probability that a plant taken at random has a height greater than 40 cm. (2 marks) 20. (a) Using a ruler and a pair of compasses only, construct: (i) a parallelogram ABCD, with line AB below as part of it, such that AD = 7 cm and angle BAD = 600; (3 marks) ii) the locus of points equidistant from AB and AD; (1 mark) (iii) the perpendicular bisector of BC. (1 mark) (b) (i) Mark the point P that lies on DC and is equidistant from AB and AD. (1 mark) (ii) Measure BP. (1 mark) (c) Describe the locus that the perpendicular bisector of BC represents. (1 mark) (d) Calculate the area of trapezium ABCP. (2 marks) 21. The table below shows some values of the curves y = 2 cos x and y = 3 sin x. (a) Complete the table for values of y = 2 cos x and y = 3 sin x, correct to 1 decimal place. (2 marks) On the grid provided, draw the graphs of y = 2 cos x and y = 3 sin x for 0° x 360°, on the same axes. (5 marks) (c) Use the graph to find the values of x when 2 cos x — 3 sin x = 0 (2 marks) (d) Use the graph to find the values of y when 2 cos x = 3 sin x. (1 marks) 22. The figure below is a model of a watch tower with a square base of side 10 cm. Height PU is

Kenya Certificate of Secondary Education 2018 Mathematics Paper 1 Section I (50 marks) Answer all the questions in this section. 1. Without using a calculator, evaluate:(3 marks) 2. Given that 6 2n-3= 7776, find the value of n. (3 marks) 3. The base of a right pyramid is a rectangle of length 80 cm and width 60 cm. Each slant edge of the pyramid is 130 cm. Calculate the volume of the pyramid. (3 marks) 4. In the figure below ABCDEF is a uniform cross section of a solid. Given that FG is one of the visible edges of the solid, complete the sketch showing the hidden edges with broken lines. 5. The lengths of three wires were 30m, 36 m and 84m. Pieces of wire of equal length were cut from the three wires. Calculate the least number of pieces obtained. (4 marks) 6. A two digit number is such that, the sum of its digits is 13. When the digits are interchanged, the original number is increased by 9. Find the original number. (4 marks) 7. (a) Using a ruler and a pair of compasses only, construct a quadrilateral PQRS in which PQ = 5 cm, PS = 3 cm, QR = 4 cm, PQR = 135° and SPQ is a right angle. (2 marks) (b) The quadrilateral PQRS represents a plot of land drawn to a scale of l:4000. Determine the actual length of RS in metres. (2 marks) 8. Given that OA = 3 and OB = . Find the mid point M of AB. 9. Two towns R and S are 245 km apart. A bus travelling at an average speed of 60 km/h left tow: R for town S at 8.00 a.m. A truck left town S for town R at 9.00 a.m and met with the bus c 11.00a.m. Determine the average speed of the truck. (4 marks) 10. In the parallelogram WXYZ below, WX = 10 cm, XY = 5 cm and WXY = 150°. Calculate the area of the parallelogram. (3 marks) 11. Without using mathematical tables or a calculator, evaluate sin 30°-sin60 °/tan60°(3 marks) 12. Use matrix method to solve: 5x + 3J = 35 3x — 4y — —8 (3 marks) 13. Expand and simplify. (2x + 1)2 + (x — 1)(x — 3).(2 marks)   14. Use mathematical tables to find the reciprocal of 0.0247, hence evaluate 𢆳.025/0.1247 correct to 2 decimal places.(3 marks) 15. A Kenyan businessman intended to buy goods worth US dollar 20 000 from South Africa Calculate the value of the goods to the nearest South Africa (S.A) Rand given that 1 US dollar = Ksh 101.9378 and 1 S.A Rand = Ksh 7.6326. (3 marks) 16. A photograph print measuring 24cm by 15 cm is enclosed in a frame. A uniform space of width x cm is left in between the edges of the photograph and the frame. If the area of the space i‹ 270cm’, find the value of x. (3 marks) Section II (50 marks) Answer any five questions from this section. 17. A school water tank is in the shape of a frustum of a cone. The height of the tank is 7.2 m and the top and bottom radii are 6m and 12 m respectively. (a) Calculate the area of the curved surface of the tank, correct to 2 decimal places. (4 marks) (b) Find the capacity of the tank, in litres, correct to the nearest litre. (3 marks) (c) On a certain day, the tank was filled with water. If the school has 500 students and each student uses an average of 40 litres of water per day, determine the number of days that the students would use the water. (3 marks) 18. Two vertices of a triangle ABC are A (3,6) and B (7,12). (a) Find the equation of line AB.(3 marks) (b) Find the equation of the perpendicular bisector of line AB.(4 marks) (c) Given that AC is perpendicular to AB and the equation of line BC is y = -5x + 47, find the co-ordinates of C. (3 marks) 19. The distance covered by a moving particle through point O is given by the equation, s = t3 – 15t2 + 63f — 10. Find: (a) Distance covered when f = (2 marks) (b) The distance covered during the 3rd second;(3 marks) (c) The time when the particle is momentarily at rest;(3 marks) (d) The acceleration when t — 5.(2 marks> 20. The diagram below shows triangle ABC with vertices A(— 1, —3), B(1, — 1) and C(0,0), and line M. (a) Draw triangle A’B’C’ the image of triangle ABC under a reflection in the line M. (2 marks) (b) Triangle A“B“C“ is the image of triangle A’B’C’ under a transformation represented by the matrix T = (1 2) (0 1) (i) Draw triangle A”B”C“ (3 marks) (ii) Describe fully the transformation represented by matrix T. (3 marks) (iii) Find the area of triangle A’B’C’ hence find area of triangle A“B“C”. (2 marks) 21. The figure below shows two triangles, ABC and BCD with a common base BC = 3.4 cm. AC = 7.2 cm, CD = 7.5 cm and ABC = 90°. The area of triangle ABC = Area of triangle ∠BCD. Calculate, correct to one decimal place: (a) The area of triangle ABC;(3 marks) (b) The size of ∠BCD; (3 marks) (c) The length of BD;(2 marks) (d) The size of ∠BDC.(2 marks) 22. (a) On the grid provided, draw the graph of y = 4-1/4x2for -4 ≤ x ≤ (2 marks) (b) Using trapezium rule, with 8 strips, estimate the area bounded by the curve and the z-axis. (3 marks) (c) Find the area estimated in part (b) above by integration. (3 marks) (d) Calculate the percentage error in estimating the area using trapezium rule. (2 marks) 23. Three business partners Abila, Bwire and Chirchir contributed Ksh 120 000, Ksh 180 000 and Ksh 240 000 respectively, to boost their business. They agreed to put

Kenya Certificate of Secondary Education 2019 Mathematics Paper 2 Section I (50 marks) Answer all the questions in this section. 1. Simplify √5+3∕ √5-2 . Give the answer in the form a + b∕c where a, b and c are integers. (2 marks) 2. Two types of flour, X and Y, cost Ksh 60 and Ksh 72 per kilogram respectively. The two types are mixed such that the cost of a kilogram of the mixture is Ksh 70. Calculate the ratio X:Y of the mixture.(3 marks) 3. A quantity P varies inversely as the square of another quantity L. When P = 0.625, L = 4. Determine P when L — 0.2. (3 marks) 4. An arc of a circle subtends an angle of 150° at the circumference of the circle. Calculate the angle subtended by the same arc at the Centre of the circle. (2 marks) 5. Solve the equations: x + 3y = 13 x² + 3y² = 43 (4 marks) 6. A bag contains 6 red counters and 4 blue counters. Two counters are picked from the bag at random, without replacement. (a) Represent the events using a tree diagram. (1 mark) (b) Find the probability that the two counters picked are of the same colour. (2 marks) 7. Find the coordinates of the turning point of the curve y=x²-14x+10 (3 marks) 8. OAB is a Sector Of a circle of radius r cm. Angle AOB = 60°. Find, in its simplest form, an expression in terms of r and z for the perimeter of the sector.(2 marks) 9. In a mathematics test, the scores obtained by 30 students were rccordcd as shown in the toblc below.          Score (x) 59 61 65 K 71 72 73 75       No. of students 2 3 5 6 7 4 2 1 The score K with a frequency of 6 is not given. Given that ⅀fd∕⅀f = — 1.2 where d —— x – 69, and using an assumed mean of 69, determine score K. 10. Determine the amplitude and the period of the function y = 3 sin(2x + 40°). (4 marks) 11. The figure ABCDEFGH represents a box. The top lid of the box is opened such that the height OT is 35cm. Calculate the: (a ) angle the top lid makes with the plane FGHE; (b) length BE, correct to 2 decimal places. 12. The table below’ shows income tax rates in a certain year.   Monthly Income in ksh Tax rate  in each shilling (%) 0  – 10164 10     10165 – 19740 15      19741 – 29316 20      29317 – 38892 25       38893 and above 30 In that year,mawira earned a salary of 41000 per month.calculate mawira’s income tax per month given that a monthly tax relief of ksh 1162 was allowed (3 marks) 13. The position vectors of points A,B and C are OA =[3/4],OB = [1/2] and OC =[7/-1] show that A,B and C are collinear (3 marks) 14. The vertices of a triangle PQR are P(-3, 2), Q(0, — 1) and R(2, — 1). A transformation matrix maps triangle PQR onto triangle P’Q’ R’ whose vertices are P'(—7, 2), Q'(2, — 1) and R’(4, – 1). Find M°’, the transformation that maps P’Q’R’ onto PQR.(4 marks) 15. Solve for x in log(7x—3) + 2 log 5 = 2 + log(z + 3).(4 marks) 16. The length of a shadow’ of a mast was measured at intervals of 1 hour and recorded as shown in the table below.   Time( hr) 0 1 2 3 4 5 Length( m) 18.7 8.7 5.0 2.9 1.3 0 (a) On the grid prov’ided, draw the graph of length against time. (2 marks) (b) Determine the rate of change of the shadow’ length at = 2 (2 marks) SECTION II (50 MARKS) Answer any five questions 17.The first term of an Arithmetic Progression(AP) is equal to the first term of a Geometric Progression (GP). The second team of the AP is equal to the fourth term Of the GP while the tenth term of the AP is equal to the seventh term of the GP. (a) Given that a is the first term and d is the common difference of the AP while r is the common ratio of the GP, write the two equations connecting the AP and the GP. (2 marks) (b) Find the value of r that satisfies the progressions.(4 marks) (c) Given that the tenth term of the GP is 5120, find the values of a and d.(2 marks) (d) Calculate the sum of the first 20 terms of the AP.( 2 marks) 18. Mbaka bought some plots at Ksh 400000 each. The value of each plot appreciated at the rate of 10% per annum. (a) Calculate the value of a plot after 2 years. (2 marks) (b) After some time t, the value of a plot was Ksh 558400. Find i, to the nearest month. (4 marks) (c) Mbaka sold all the plots he had bought after 4 years for Ksh2 928200. Find the percentage profit Mbaka made, correct to 2 decimal places. (4 marks) 19. The figure KLMN below is a scale drawing of a rectangular piece of land of length KL = 80m (a) On the figure, construct (i) The locus of a point P which is both equidistant from points L and M It and from lines KL and LM. (3 marks) (ii) the locus of a point Q such that ∠KQL = 90°. (3 marks) (b) (i) Shade the region R bounded by the locus of Q and th Locus of poinis equidistant from KL and LM. (3 marks) (ii) Find the area of the region R in m². (Take ℼ= 3.142). (3 marks) 20. A ship left point P(l0°S, 40°E) and sailed due East for 90 hours at an average speed of 24 knotS to a point R. (Take I nautical mile (nm) to be 1.853 km and radius of the earth to be 6370 km) (a) Calculate the distance

Kenya Certificate of Secondary Education 2019 Mathematics Paper 1 Section I (50 marks) Answer all the questions in this section. 1. Without using mathematical tables or a calculator, evaluate: 5.4∕ 0.025 x 3.6 (3 marks) 2. Express 1728 and 2025 in terms of their prime factors. Hence evaluate:(4 marks) ∛1728 √ 2025 3. Juma left his home at 8.30a.m. He drove a distance of l40km and arrived at his aunt’s home at 10.15 a.m. Determine the average speed, in km/h, for Juma’s journey.(3 marks) 4. Expand and simplify: 4 (q – 6) + 7 (q — 3).(2 marks) 5. In the trapezium PQRS shown below, PQ = 8 cm and SR = 6 cm. If the area of the trapezium is 28 cm², find the perpendicular distance between PQ and SR. (2 marks) 6.Given that = ∛ 94=3n find the value of n.(3 marks) 7. Three villages A, B and C rife Such that B is 53 km on a bearing of 295° from A and C IS 75 kin east of B. (a) Using a scolc of 1 cm to represent 10 km, draw a diagram to show the relative positions of villages A, B and C. (2 marks) (b) Determine the distance, in km, of C from A. (2 marks) 8. A retailer bought a bag of tea leaves. If the retailer were to repack the tea leaves into smaller packets of either 40 g, 250g or 350 g, determine the least mass, in grams, of the tea leaves in the bag.(3 marks) 9. Given that sin 2x = cos (3z — 10°), find tan z, correct to 4 significant figures. (3 marks) 10. A tourist converted 5820 US dollars into Kenya Shillings at the rate of Ksh 102.10 per dollar. While in Kenya, he spent Ksh450 000 and converted the balance into dollars at the rate of Ksh 103.00 per dollar. Calculate the amount of money, to the nearest dollar, that remained.(3 marks) 11. Given that b = (2) (4) C,= (3) (2)   and a = 3c — 2b, find the magnitude of a, correct to 2 decimal (4 marks) 12. Using a ruler and a pair of compass only, construct a rhombus PQRS such that PQ = 6cm and dSPQ = 75°. Measure the length of PR. 13. Solve the inequality 2x — 1 ≤ 3x + 4 < 7 — x. (3 marks) 14. Given that A = ( 2 3 4 4 ) B, ( X 1 2 3 ) and that AB is a Singular matrix, find the value ofx.(3 marks) 15. A trader bought two types of bulbs A and B at Ksh 60 and Ksh 56 respectively. She bought a total of 50 bulbs of both types ct a total of Ksh 2872. Determine the number of type A bulbs that she bought. (3 marks) 16. A bus plies between two towns P and R via town Q daily. On each day it departs from P at 8.15 a.m. and stops for 40 minutes at Q before proceeding to R. On a certain day, the bus took 5 hours 40 minutes to travel from P to Q and 3 hours 15 minutes to travel from Q to R. Find, in 24 hour clock system, the time the bus arrived at R. (3 marks) SECTION II (50 marks) 17. A rectangular water tank measures 2.4 m long, 2 m wide and 1.5 m high. The tank contains some water up to a height of 0.45 m. (a) Calculate the amount of water, in litres, needed to fill up the tank (3 marks) (b) An inlet pipe was opened and water let to flow into the tank at a rate of 10 litres per minute. After one hour, a drain pipe was opened and water allowed to flow out of the tank at a rate of 4 litres per minute. Calculate: (i) the height of water in the tank after 3 hours;(4 marks) (ii) the total time taken to fill up the tank. (3 marks) 18. (a) A line, L, posies through tho points (3,3) and (5,7). Find the equation of L, in the form y = mx+c where m and c arc constonti. (3 marks) (b) Another line L2 is perpendicular to L, and passes through (—2, 3). Find: (i) the equation of L2; (ii) the x-intercept of L2. (c) Determine the point of intersection of L, and L2. (3 marks) 19. A triangle ABC wÎth Vertices A (—2,2),B (1,4)and C (-1,4) is mapped on to triangle A’B’C by a reflection in the line y=x+1. (a) On the grid provided draw (i) triangle ABC (3 marks) (ii) the line y = x + 1; (2 marks) (iii) triangle A’B’C’. (2 marks) (b) Triangle A”B”C“ is the image of triangle A’B’ C’ under a negative quarter turn (0,0). On the same grid, draw triangle A“B“C“. (3 marks) 20.The figure below is a right pyramid VEFGHI with a square base of 8cm and a slant edge of 20cm Points A B C and D lie on the slant edges or the pyramid such that VA = VB = VC = VD = I0 cm and plane ABCD is parallels to the base EFGH. (a) Find the length of AB. (2 marks) (b) Calculate to 2 decimal places (i) The length of AC (2 marks) (ii) The perpendicular height of the pyramid VABCD (2 marks) (c) The pyramid VABCD was cut off. Find the volume of the frustrum ABCDEFGH correct to 2 decimal places (4 marks) 21. The heights of 40 athletes in a county athletics competition were as shown in the table below. Height,cm Frequency 150 -159    2 160 – 169    8 170 – 179   10 180 – 189   x 190 -199   6 200 – 209   2 (a) Find the value of X. (1 mark) (b) State the modal class. (1 mark) (c) Calculate: (i) The mean height of the athletes;(4 marks) (ii) The median height, correct to 1 decimal place,of the athletes (4 marks) 22. The figure below represents a triangular 8ower garden ABC in which

Kenya Certificate of Secondary Education 2020 Mathematics Paper 2 Section I (50 marks) Answer all the questions in this section.

2020 Mathematics Paper 1 Section I (50 marks) Answer all the questions in this section. 1. Without using mathematical tables or a calculator, evaluate –3(6 + –2) – –12 ÷ 4 + 5 –4 x –6 + –3 x 5(3 marks) 2.Express 5.5 as a mixed number (1 mark) 3. 3. Without using a calculator or mathematical tables, evaluate: (3 marks) 493/2x{256/2401}¾ 4. A rectangular floor of a room measures 5.4 m long and 4.2 m wide. The room is to be covered with square tiles. Calculate the minimum number of square tiles that can be used to cover the floor.(4 marks) 5. Simplify 2x² – xy – 6y² x² – 4xy + 4y²(3 marks) 6. The line 2y+x= 1 is perpendicular to a line L. Line L passes through point (2,−1), Determine the equation of L in the form y = mx + c, where m and c are constants. (3 marks) 7. The sum of the interior angles of a regular polygon is 12600. Find the size of each exterior angle of the polygon.(3 marks) 8. Using the grid provided below, solve the simultaneous equations. x – 4y = -5 –x + 2y = 1 (3 marks) 9. Given that sin(θ+30°)=cos2θ, find the value of cos(θ+40°). (3 marks) 10. In the figure below, AB and CD are arcs of two concentric circles, centre O. Angle AOB = 60° and AD=BC=7cm. Given that the perimeter of ABCD is 28 2/3 cm, find OA, the radius of the inner circle. (3 marks) 11. Using a ruler and a pair of compasses only, construct a parallelogram ABCD in which AB = 6cm, BC = 5cm and angle ADC = 150°.(3 marks) 12. A Kenyan Non-Governmental Organization (NGO) received a donation of 200 000 US dollars. The money was converted into Kenyan shillings in a bank which buys and sells foreign currency as follows:   Buying (Ksh) Selling (Ksh) 1 USD Dollar 102.40 102.50 100 Japanese Yen 92.80 93.30 (a) Calculate the amount of money, in Kenya Shillings, the NGO received. (1 mark) (b) The NGO used 90% of the donation to buy a machine from Japan. Calculate the cost of the machine to the nearest Japanese Yen. (3 marks) 13. A chord of a circle, 7 cm long subtends an angle of 60° at the centre of the circle. Determine the area correct to 2 decimal places, of the major segment of the circle. (4 marks) 14. A vertical electric pole was erected 6.4m from the foot of a vertical fencing pole on the same horizontal level. The fencing pole is 2m high. The angle of elevation of the top of the electric pole from the top of the fencing pole is 30°. Determine the height of the electric pole correct to 1 decimal place. (2 marks) 15. In a 4×400 m relay competition, a team of athletes each completed their round in 45 sec, 43 sec, 44 sec and 45 see respectively. If the race started at 1:35:31 p.m., find the time when the team completed the race. (3 marks) 16. The figure below shows a quadrilateral ABCD and a mirror line M1M2. (a) Draw quadrilateral A’B’C’D’, the image of ABCD, under a reflection in the mirror line M1M2.(2 marks) (b) State the type of congruency between ABCD and A’B’C’D’.(1 mark) SECTION II (50 marks) Answer any five questions from this section in the spaces provided. 17. Three business partners, Kosgei, Kimani and Atieno contributed Ksh 1,750,000 towards an investment. Kosgei contributed 20% of the money. Kimani and Atieno contributed the remainder in the ratio 3:5 respectively. (a)(i) Calculate the amount of money Kimani contributed.(2 marks) (ii) Find the ratio of the contributions by the three partners.(2 marks) (b) The money earned a compound interest at a rate of 8% per annum. After 3 years, the partners withdrew the interest and donated 10% to a charitable organisation. The partners then shared the remainder in the ratio of their contributions. Calculate the amount of money, to the nearest shilling, that each partner received.(6 marks) 18. A solid consists of a conical part, a cylindrical part and a hemispherical part. All the parts have the same diameter of 12cm. The height of the cylindrical part is 15cm and the slanting height of the conical part is 10 cm. (Take π = 3.142). Calculate the: (a) height of the solid;(2 marks) (b) surface area of the solid, correct to 1 decimal place,(4 marks) (c) volume of the solid, correct to 1 decimal place. (4 marks) 19. The average speed of a pick-up was 20km/h faster than the average speed of a lorry. The pick-up took 45 minutes less than the lorry to cover a distance of 180 km. (a) If the speed of the lorry was x km/h: (i) Write expressions in terms of x for the time taken by the lorry and the pick-up respectively to cover the distance of 180 km(2 marks) (ii) Determine the speed of the lorry and that of the pick-up.(5 marks) (b) The distance between towns A and B is 240 km. On a certain day the pick-up started from town A at 8.30 a.m. and the lorry started from town B at the same time. Determine the time that the lorry and the pick-up met. (3 marks) 20. A forest is enclosed by four straight boundaries AB, BC, CD and DA. Point B is 25km on a bearing of 315° from A, C is directly south of B on a bearing of 260° from A and D is 30km on a bearing of 210° from C. (a) Using a scale of 1:500,000, represent the above information on a scale drawing (3 marks) (b) Using the scale drawing, determine the: i. distance, in kilometres, of D from A: (2 marks) ii. bearing of A from D (1 mark) (c) Calculate the area, correct to 1 decimal place, of the forest in square kilometres. (4 marks) 21. The position vectors of points A and B are OA={2/4}

2022 Mathematics Paper 2 Section I (50 marks) Answer all the questions in this section. 1. An empty tank of capacity 18480 litres is to be filled with water using a cylindrical pipe of diameter 0.028 m. The rate of flow of water from the pipe is 2 m/s. Find the time in hours it would take to fill up the tank. (Take π = 22/7). (3 marks) 2. The first term of a Geometric Progression (G.P) is 2. The common ratio of the G.P is also 2. The product of the last two terms of the G.P is 512. Determine the number of terms in the G.P. (3 marks) 3. The expression ax2 – 30x + 9 is a perfect square, where a is a constant. Find the value of a. (2 marks) 4. Make x the subject of the formula y = bx OVER √cx2 – a5 (3 marks) 5.The figure below shows a circle and a point outside the circle’. Using a ruler and pair of compasses, construct a tangent to the circle from P. (4 marks) 6. Four quantities P Q R and S are such that P varies directly as the square root of Q and inversely as the square of the difference of R and S. Quantity Q is increased by 44% while quantities R and S are each decreased by 10%. Find the corresponding percentage change in P correct to 1 decimal place. (4 marks) 7. The figure below represents a prism ABCDEFGH of length 6 cm. The cross section BCFG of the prism is a trapezium in which GF = 11 cm, BC = 8 cm, BG = 5 cm and ∠GFC = ∠BCF = 90° Calculate correct to 1 decimal place the angle between the line FA and the plane GFEH. (3 marks) 8. The cash price of a gas cooker is Ksh 20 000. A customer bought the cooker on hire purchase terms by paying a deposit of Ksh 10 000 followed by 18 equal monthly instalments of Ksh 900 each, Annual interest, compounded quarterly, was charged on the balance for the period of 18 months. Determine, correct to 1 decimal place, the rate of interest per annum. (4 marks) 9. The table below shows the values oft and the corresponding values of h for a given relation   t 1 2 3 4 5 6 7 8 h 8 4 2.7 2 1.6 1.3 1.1 1 a. On the grid provided, draw a graph to represent the information on the table given. (2 marks) b. Use the graph to determine, correct to 1 decimal place, the rate of change of h at = 3. (2 marks) 10. The equation of a trigonometric wave is y = 4 sin (ax – 70)o. The wave has a period of 180o. a. Determine the value of a. (1 mark) b. Deduce the phase angle of the wave. (1 mark) 11. A point Q is 2000 nm to the West of a point P(40°N, 155°W). Find the longitude of Q to the nearest degree. (3 marks) 12. A box contains 3 brown balls and 9 green balls. The balls are identical except for the colours. Two balls are picked at random without replacement. a. Draw a tree diagram to show all the possible outcomes. (1 mark) b. Determine the probability that the balls picked are of different colours. (2 marks) 13. The figure below shows triangle XYZ. Using a ruler and a pair of compasses locate point M is 2cm from line YX and is equidistant from lines YX and YZ. Measure length YM (3 marks) 14. The position vectors of points P, Q and Rare OP = 61-23 +3k, OQ = 121 – 5j + 6k and OR =8i-3j+4k. Show that P, Q and R are collinear points. (3 marks) 15. In a transformation an object of area x cm2 is mapped on to an image whose area is 13x cm2 Given that the matrix of the transformation is find the possible values of x. (3 marks) 16. Find the area enclosed by the curve y=x2+2x the straight lines x = 1,x= 3 and the x-axis. (3 marks) SECTION II(50 mks) Answer only five questions in this section in the spaces provided. 17. Pump P can fill an empty water tank in 7½ hours while pump Q can fill the same tank in 11¼ hours. On a certain day, when the tank was empty, both pumps were opened for 2½ hours. a. Determine the fraction of the tank that was still empty at the end of the 2½ hours.(4 marks) b. Pump P was later opened alone to completely fill the tank. Determine the time it took pump P to fill the remaining fraction of the tank.(2 marks) c. The two pumps P and Q are operated by different proprietors. Water from the full tank was sold for Ksh 15 750. The money was shared between the two proprietors in the ratio of the quantity of water supplied by each. Determine the amount of money received by the proprietor of pump P. (4 marks) 18. A rectangular plot measures 50 m by 24 m. A lawn, rectangular in shape, is situated inside the plot with a path surrounding it as shown in the figure below. The width of the path in x m between the lengths of the lawn and those of the plot and 2x m between the widths of the lawn and those of the plot. a. Form and simplify an expression in x for the area of the: i. lawn; (2 marks) ii. path.(1 mark) b. The area of the path is 1½ times the area of the lawn. i. Form an equation in x and hence solve for X (4 marks) ii. Determine the perimeter of the lawn. (3 marks) 19. In the figure below, points A, B, C, D and E lie on the circumference of a circle centre O. Line FAG is a tangent to

2021 Mathematics Paper 1 Section I (50 marks) Answer all the questions in this section. 1. Evaluate 1⅘ ÷ ⅔ of 2¼ -3/10 OVER 5/6 + 22/39 + 1 2/11 2. Two bells ring at intervals of 35 and 42 minutes respectively. The bells ring together at 8.48 a.m. Determine the time when the bells will ring together again. (3 marks) 3. Complete the figure below to show a rotational symmetry of order 3 about O. (3 marks) 4. Solve 5/3 -2x <1 – 2/3x ≤ 2 – x. Hence list the integral values that satisfy the inequalities. (3 marks) 5. The size of two interior angles of an irregular polygon each measures 90°. All the other remaining interior angles each measure 150°. (3 marks) Determine the number of sides of the polygon. 6. In a race Kipsang maintained an average speed of 5 m/s. When he was 310 m to the finishing line,Mutunga was 50 m behind him. However, Mutunga finished the race 10 m ahead of Kipsang. Determine Mutunga’s average speed. (3 marks) 7. Simplify (4 + 2y)²– (2y – 4)²(2 marks) 8. A table is sold at Ksh 4 500 and a chair at Ksh 2 000. A salesman earns a commission of 8% on every table and 5% on every chair sold. On a certain week, he sold 3 more chairs than tables and his total earnings were Ksh 3 980. Determine the number of chairs he sold that week. (3 marks) 9. A translation T maps A(-6, 2) onto A'(3,5). a. Determine the translation vector T. b. A point P'(-4, 2) is the image of P under T. Determine the coordinates of P. (2 marks) 10. The cost of one litre of Petrol is Ksh 110. John’s vehicle covers 12 km on one litre of petrol. He used Ksh 2 805 on petrol to travel from town A to town B. Jane’s vehicle consumes 12.5 litres of Petrol for every 100 km travelled. Calculate the amount of money that Jane would use to travel from town A to B on the same road.(3 marks) 11. Solve for θ sin (2θ -15) = cos 3θ(2 marks) 12. Line AB drawn below is a side of a trapezium ABCD. a. Using a ruler and pair of compasses only, complete trapezium ABCD in which AB is parallel to DC,∠BAD= 67.5°, AD = 5 cm, BC = 5.5 cm and ∠ABC is acute. (3 marks) b. Measure the length of DC. (1 mark) 13. Ali left Mombasa for Nairobi on Tuesday at 2.30 a.m. He arrived in Mtito Andei after 3 hours 12 minutes. He stayed in Mtito Andei for 36 hours and then left for Nairobi. He took 5 hours 25 minutes to arrive in Nairobi. Determine the day and time in the 12 hour system Ali arrived in Nairobi. (3 marks) 14. The height of a cone is 12 cm. A frustrum whose volume is one eighth the volume of the cone is cut off. Determine the height of the frustrum. (3 marks) 15. Solve the equation 8x+1 – 23x-1 = 120. (4 marks) 16. A curve is given by y=2x³- 3x² – 12x + 12. a. Find the gradient function of the curve. (1 mark) b. Determine the equation of the normal to the curve at the point (1,-1), in the form y = mx + c, where m and c are constants. (3 marks) SECTION II (50 marks) Answer only five questions in this section in the spaces provided. 17. A factory packs fruit jam in cylindrical tins of radius 5 cm and height 15 cm. The tins are then packed into rectangular cartons each measuring 60 cm long, 30 cm wide and 30 cm high. a. Determine the maximum number of tins that can be packed in one carton. (2 marks) b. An empty carton and an empty tin weighs 560 g and 300 g respectively. The jam packed in one tin weighs 990 g. A pick-up which can carry a maximum of 600 kg is used to transport the jam. Determine the maximum number of cartons the pick-up can carry. (4 marks) c. The factory delivered a pick-up full of cartons of jam to a retailer. The factory sells one carton to a retailer for Ksh 2 880. The retailer sells each tin at Ksh 110. Calculate the percentage profit made by the retailer. (4 marks) 18.a. The length of each side of an equilateral triangle ABC is 10 cm. Calculate the area of the triangle, correct to 2 decimal places. (2 marks) b. Triangle ABC in 18(a) forms the base of a solid triangular pyramid VABC. The perpendicular height of the pyramid is 15 cm. Calculate the volume of the pyramid. (2 marks) c. The pyramid VABC in 18(b) above is recast into a cone of base radius 3.5 cm. Calculate, correct to 2 decimal places: i. the height of the cone; (2 marks) ii. the surface area of the cone. (4 marks) 19. Elimu School bought 25 textbooks and 35 exercise books for Ksh 13 500 from bookshop A. From the same bookshop Soma School bought 21 textbooks and 38 exercise books and spent Ksh 1 300 less than Elimu School. Take x to represent the price of a textbook and y to represent the price of an exercise book. a. Form two equations representing the above information. (2 marks) b. Use matrix method to determine the price of each item. (5 marks) c. In bookshop B, the cost of a textbook was 5% less and that of an exercise book was 5% more than in bookshop A. Kasuku School bought the same number of textbooks and exercise books as Elimu School in bookshop B. Calculate the difference in the amount spent by Kasuku School and Elimu School (3 marks) 20. The figure below is a quadrilateral ABCD in which AB=8 cm, BC= 6 cm, CD = AD, ∠ABC = 70° and ∠ADC = 50°. a. Calculate, correct to one decimal place:

5.4.3 Physics Paper 3 (23213) PART A (d) (i) See graph (5 marks) Scale and axis Plotting Line 250—350 (ii) Slope = 2 ; \d (42 18) X 10 (iii) K d (1) = -4.2 >< 10* S‘cm’2; (3 marks) = / 41:<10 = 963 S“cm’2; 1. PART B (c) 1=0.1m b=0.0l m m=0.06 kg 0/W (g) p = Lg“ (0.11 + 0.011) = 2.02 >< 10*‘ (i) (I) (II) t=75s T=7.5s 2 c 2. PART A G = 1.42 >< 10 unit not required. (b) V0 = 3.0V PART B (h) (i) (ii) W k Vo1tage(V) 2.5 2.25 2.0 1.75 1.5 1.25 Time(s) 1.7 2.6 3.9 4.8 6.5 7.9 (i) see graph (ii) t% =6.4 S R I 6.4 X10“ 0.693 >< 2200 = 4200 Q LI = 47.4 cm = 0.68 N _ 0.474 >< 0.05 >< 10 I ‘ 0.35 (I) L2=28 cm _ 0.28 X 0.05 ><10 1- 0 35 (11) w = 0.4 N T1 = 26°C Accept (18 – 32°C) (i) L3 = 28.5 cm (ii) T2 = 83°C Accept (60 – 95°C) (iii) W3 = 0.285 X 10 = 0.41 K = (0.68 — 0.4)— (0.68— 0.41) (0.68 — 0.41) (83— 26) = Milli 0.27 >< 57 = 6.5 >< l0“‘K”  

5.4.2 Physics Paper 2 (232/2) SECTION A 1 angle of incidence = angle of reflection = 0 (1 mark) 2 larger hole acts as many small holes (1 mark) many overlapping images of same object (1 mark) 3 Within the magnet, N and S poles of the dipoles cancel out but at the end of the poles they don’t. (1 mark) 4 (a) 2V (1 mark) (b) 1.6V (1 mark) 5. Object at the intersection of incident ray; (1 mark) Incident rays; (2 marks) 6. Ray totally reflected by face AC (1 mark) i= 6O’hence r= 60′ (1 mark) 7 .a= 1 ar1db=0 (1 mark) x = neutron (1 mark) 8 .NS/NP=Np/ VP (1 mark) 9. Each lamp on full voltage (1 mark) Failure of one lamp does not affect the others (1 mark) 10. X rays ionise air molecules between plates Ions move to plates of opposite sign 11. Sun being hotter produces short wavelength infrared waves which penetrate glass; buming wood produces long wavelength infrared waves which do not penetrate glass. 12. K=E- T 13 Arsenic shares 4 of its 5 electrons with germanium. the extra electron is free for conduction. SECTION B 14 (a) fl= l0cm b i) to produce a magnified real image ii) to produce a magnified virtual image of the 1*‘ image. c) i) move A so that the object is slightly outside fik ii) fiove B so that the real image is within fa. d)i) m= A “‘ 16 = 1 2 ii) m = 27f = 7 15 Negative charges fiow from earth to cap. Negative charge neutralizes the positive. b) i) C = 2,uF (1 mark) (ii) Q=¢V (1 mark) =2×4 = spc (lmark) (m) Q= 8/1C (1 mark) ‘ (C) radical field; Correct dirrection; (2 marks) 16 a) (i) Energy = QV (1 mark) (ii) Power = 15. = 2 (1 mark) (iii) I : Q (rate of flow of charge) (1 mark) P = %.V P = I.V (1 mark) (b) Power = VI = 20 x 60 (1 mark) 240 x I = 12O0W (1 mark) I : 1200 240 = SA (l mark) 4A < 5A hence fuse will blow. (1 mark) 17 (a) (i) Thermionically by cathode (1 mark) (ii) causing fluorescence on screen (1 mark) (iii) (i) control brightness of fluorescence (1 mark) (ii) to focus the electron beam (1 mark) (b) 1 wavelength = 2 cm (1 mark) = 4 X10_3S (1 mark) f = % (1 mark) = 1 (1 mark) 4><10″ = 250HZ (1 mark) 18 a) – curved waves – converging before focus (1 mark) – diverging after focus. (1 mark) (b) (i) O cm – trough and crest interference (2 marks) (ii) +10 – crest and crest interference (2 marks) (c) (i) Waves produced are reflected at the fixed ends. (1 mark) Incident and reflected waves interfer constructively at antinodes. (1 mark) and destructively at nodes. (1 mark) (ii) A = %>< 1.5 = 1m (1 mark) 19 b) coil moves to and fro (1 mark) force on coil varies direction as current varies in direction. (l mark) (i) dilute sulphuric acid (1 mark) (ii) (I) Zinc ions go into acid leaving electrons on the plate (1 mark) (II) Give up electrons to discharge hydrogen Ions. (l mark) (m) Electrons flow from zinc plate to the copper plate. (I mark)

5.4.1 Physics Paper 1 (232/1) 1. 5.32 cm (1 mark) 2. – magnitude of the force – The perpendicular distance between the force and the pivot. (1 mark) 3. Patmosphere = Pmercury + pair enclosed; Pair = 760 – 600: = 160 mm Hg; (3 marks) 4. (a) F = Ke; 20 = 0.5 K; K = 40 Ncm” (2 marks) (b) F = 40 x 0.86 = = 34.4 N; (1 mark) 5. – Weight of object in air – Weight of object when fully immersed in fluids (2 marks) 6. Upthrust = weight in air – weight of object in fluid. (1 mark) 7. Wood is a poor conductor of heat; hence heat is used to bum paper, while most heat is conducted away by copper: hence paper takes long to bum. (2 marks) 8. Clockwise moments = anticlockwise moments; 0.l8x = l(50-x)+0.12(l00-x) 0.l8x = 50-x+l2-l2x O.l8x = 62- 1.l2x 7.30x = 62 x = 47.69 cm; (3 marks) 9. Air is compressible; so the transmitted pressure is reduced; (2 marks) 10. The high velocity of the gas causes a low pressure region; Atmospheric pressure is higher; Pressure difference draws air into the region; (3 marks) 11. Water molecules have a high adhesion forces; With glass molecules and hence rise up the tube while mercury molecules have greater cohesion; Forces within than adhesion with glass hence do not rise up. (2 marks) 12. Allow for expansion; Water expands on cooling between 4° C and 0° C; 13 Diffusion of the ink molecules; SECTION B 14 (a) – increasing the angular velocity; – Reducing the radius of the path; (b) (i) Tension in the string; (ii) Arrow to centre of circle; (iii) Direction of motion of object changes and causes the velocity to change with time; (iv) F = mv/r 0.5 * 8*2/2 = l6N; (c) (i) V2 = uz + 2as; O€u’—2>< 10>< 100 u= ‘/ 2000 44.72 ms”; (ii) V = u + at ; O = 44.72 – 10 X t t= 4.472 Total time = 2 X 4.472 = 8.94s ; temperature; (b) (i) E=Pt; =6O><5 X60; = 13000]; (ii) Mass of water = 190 – 130 = 60g; mlf = Pt. 60 = . ——1OO0l, 60><60><5, 1,= 3 >< 105 J/Kg; 15 (a) Quantity of heat required to convert 1 kg of ice at 0° C to water without change in temperatures (iii) Heat from the surrounding melts the ice; (1 mark) 16 a) F = Ma; F = 2 >< 5 = ION; friction force = 12 – 10 = 2N; (3 marks) (b) (i) OA – the ball bearing decelerates; as the upthrust increases to a maximum; (2 marks) AB – ball attains terminal velocity; when upthrust = weight; (2 marks) (c) (i) VR = 2 (1 mark) (ii) To change direction of effort; (1 mark) (iii) Efficiency =—1‘”% >< 100; so=%><1o0% MA=l.6; ..1.6_5O0 ” L=500X 1.6   =800 N; (3 marks) 17 (a) (i) F = mg = 10 X 10 = 100 N ; Additional pressure = = 1 Ncm ’ new reading = 10 + 1 = ll N; (4 marks) (ii) Pressure has increased; because, when the volume reduces, the collisions between the gas molecules and walls of the container increases; (2 marks) (b) (i) Pressure = ll Ncm’2 _ (1 mark) (11) Q = i; Tr T1 L = L- 300 T1 ‘ T1=i3°‘)l:)‘ 11 = 330k; T2 = 57° C (4 marks) 18 (a) (i) (I) – Reading decreases on spring balance; (II) – Reading on weighing balance increases. (ii) As the block is lowered, upthust increases; and hence it apparently weighs less; (b) (i) Upthrust – weight in air – weight in water = 2.7 – 2.46 = 0.24 N; Reading in weighing balance = 2.8 + 0.24 = 3.04 N; (ii) Relative density = weight in air; upthrust = 11 0.24 = 11.25 ; Density = R.d X density of water = 11.25 >< 1000 = 11250 kgm‘; c) The hydrometer sinks more; The density of(t.he water is reduced;