KCSE MATHEMATICS QUESTIONS AND SOLUTIONS ~ Topically Analyzed
Form 1 Mathematics
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 20% of the profit accrued back into the business and to use 35% of the profits for running the business (official operations). The remainder was to be shared among the business partners in the ratio of their contribution. At the end of the year, a gross profit of Ksh 225 000 was realised. (a) Calculate the amount: (i) put back into the business; (ii) used for official operations (b) Calculate the amount of profit each partner got. (c) If the amount put back into the business was added to individuals’s shares proportionately to their initial contribution, find the amount of Chirchir’s new shares.
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Form 4 Mathematics
(a) On the grid provided, draw the graph of y = 4 -1/4x2 for -4 ≤ x ≤ 4
(b) Using trapezium rule, with 8 strips, estimate the area bounded by the curve and the x-axis. (c) Find the area estimated in part (b) above by integration. (d) Calculate the percentage error in estimating the area using trapezium rule. Form 2 Mathematics
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; (b) the size of ∠BCD; (c) the length of BD; (d) the size of ∠BDC. Form 4 Mathematics
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.
(i) Draw triangle A"B"C"
(ii) Describe fully the transformation represented by matrix T. (iii) Find the area of triangle A’B'C' hence find area of triangle A"B"C". Form 4 Mathematics
The distance covered by a moving particle through point O is given by the equation, s = t3 - 15t2 + 63t — 10.
Find: (a) distance covered when t = 2 (b) the distance covered during the 3rd second; (c) the time when the particle is momentarily at rest; (d) the acceleration when t = 5. Form 2 Mathematics
Two vertices of a triangle ABC are A (3,6) and B (7,12).
(a) Find the equation of line AB. (b) Find the equation of the perpendicular bisector of line AB. (c) Given that AC is perpendicular to AB and the equation of line BC is y = -5x + 47, find the coordinates of C. Form 2 MathematicsThe diagram below represents a rectangular swimming pool 25m long and 10m wide. The sides of the pool are vertical. The floor of the pool slants uniformly such that the depth at the shallow end is 1m at the deep end is 2.8 m. (a) Calculate the volume of water required to completely fill the pool. (b) Water is allowed into the empty pool at a constant rate through an inlet pipe. It takes 9 hours for the water to just cover the entire floor of the pool. Calculate: (i) The volume of the water that just covers the floor of the pool ( 2 marks) (ii) The time needed to completely fill the remaining of the pool ( 3 marks) Form 3 MathematicsThe points P, Q, R and S have position vectors 2p, 3p, r and 3r respectively, relative to an origin O. A point T divides PS internally in the ratio 1:6 (a) Find, in the simplest form, the vectors OT and QT in terms P and r ( 4 marks) (i) Show that the points Q, T, and R lie on a straight line ( 3 marks) Form 2 Mathematics
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. (b) Find the capacity of the tank, in litres, correct to the nearest litre. (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. Form 3 Mathematics(a) (i) Complete the table below for the function y = x3 + x2 – 2x (2 marks) (ii On the grid provided, draw the graph of y = x3 + x2 – 2x for the values of x in the interval – 3 ≤ x ≤ 2.5 (2 marks) (iii) State the range of negative values of x for which y is also negative (1 mk) (b) Find the coordinates of two points on the curve other than (0,0) at which x- coordinate and y- coordinate are equal (3 marks) Form 2 MathematicsA boat which travels at 5 km/h in still water is set to cross a river which flows from the north at 6km/h. The boat is set on a course of x0 with the north. (a) Given that cos x0 = 3/5 , calculate (i) The resultant speed of the boat ( 2 marks) (ii) The angle which the track makes with the north ( 2 marks) (b) If the boat is to sail on a bearing of 1350, calculate the bearing of possible course on which it can be set ( 4 marks) Form 4 MathematicsForm 4 MathematicsThe gradient of a curve at point (x,y) is 4x – 3. the curve has a minimum value of – 1/8 (a) Find (i) The value of x at the minimum point ( 1 mark) (ii) The equation of the curve ( 4 marks) (b) P is a point on the curve in part (a) (ii) above. If the gradient of the curve at P is -7, find the coordinates of P ( 3 marks) Form 4 MathematicsThe table below gives some of the values of x for the function y = ½ x 2 + 2x + 1 in the interval 0≤ x ≤ 6. (a) Use the values in the table to draw the graph of the function ( 2 marks) (b) (i) Using the graph and the mid – ordinate rule with six (6) strips, estimate the area bounded by the curve, the x- axis, the y- axis and the line = 6 (ii) If the exact area of the region described in (b) (i) above is 78cm2, calculate the percentage error made when the mid – ordinate rule is used. Give the answer correct to two decimal places ( 2 marks) Form 2 MathematicsThe distance between towns M and N is 280 km. A car and a lorry travel from M to N. The average speed of the lorry is 20 km/h less than that of the car. The lorry takes 1 h 10 min more than the car to travel from M and N. (a) If the speed of the lorry is x km/h, find x ( 5 marks) Form 2 Mathematics
The length of a room is 3 m shorter than three times its width. The height of the room is a quarter of its length. The area of the floor is 60 m2.
(a) Calculate the dimensions of the room. (b) The floor of the room was tiled leaving a border of width y m, all round. If the area of the border was 1.69m2, find: (i) the width of the border; (ii) the dimensions of the floor area covered by tiles. Form 3 Mathematics
(a) The 5th term of an AP is 82 and the 12th term is 103.
Find: (i) the first term and the common difference; (ii) the sum of the first 21 terms. (b) A staircase was built such that each subsequent stair has a uniform difference in height. The height of the 6th stair from the horizontal floor was 85 cm and the height of the 10th stair was 145 cm. Calculate the height of the 1st stair and the uniform difference in height of the stairs. Form 4 Mathematics
A trader stocks two brands of rice A and B. The rice is packed in packets of the same size. The trader intends to order fresh supplies but his store can accommodate a maximum of 500 packets. He orders at least twice as many packets of A as of B.
He requires at least 50 packets of B and more than 250 packets of A. If he orders x packets of A and y packets of B, (a) Write the inequalities in terms of x and y which satisfy the above information. (b) On the grid provided represent the inequalities in part (a) above (c) The trader makes a profit of Ksh 12 on a packet of type A rice and Ksh 8 on a packet of type B rice. Determine the maximum profit the trader can make. Form 3 Mathematics
Three quantities X, Y and Z are such that X varies directly as the square root ofY and inversely as the fourth root of Z. When X = 64, Y = 16 and Z = 625.
(a) Determine the equation connecting X, Y and Z. (b) Find the value of Z when Y = 36 and X = 160. (c) Find the percentage change in X when Y is increased by 44%. Form 4 Mathematics
The figure below represents a cuboid ABCDEFGH in which AB = 16 cm, BC = 12 cm and CF = 6 cm.
(a) Name the projection of the line BE on the plane ABCD.
Calculate correct to 1 decimal place: (i) the size of the angle between AD and BF; (ii) the angle between line BE and the plane ABCD; (iii) the angle between planes HBCE and BCFG. (c) Point N is the midpoint of EF. Calculate the length BN, correct to 1 decimal place. Form 3 Mathematics
The table below shows values of x and some values of for the curve y = x3 -2x2 -9x + 8 for -3 ≤ x ≤ 5. Complete the table.
(b) On the grid provided, draw the graph of y = x3- 2x2- 9x + 8 for -3 ≤ x ≤ 5 for Use the scale; 1 cm represents 1 unit on the x-axis 2 cm represents 10 units on the y-axis
(c) (i) Use the graph to solve the equation x2 - 2x3 -9x + 8 = 0. (ii) By drawing a suitable straight line on the graph, solve the equation x2 - 2x2- 11x + 6 = 0. Form 4 Mathematics
The vertices ofa rectangle ABCD are: A(0,2), B(0,4), C(4,4) and D(4,2). The vertices of its image under a transformation T are; A’(0,2) , B’(0,4) , C’(8,4) and D’(8,2).
On the grid provided, draw the rectangle ABCD and its image A’B’C’D’ under T. (ii) Describe fully the T. (iii) Determine the matrix of transformation. (b). On the same grid as in (a), draw the image of rectangle ABCD under a shear with line x =-2 invariant and A(0, 2) is mapped onto A”(0,0). Form 3 Mathematics
The income tax rates of a certain year were as shown in the table below:
In that year, Shaka’s monthly earnings were as follows: Basic salary Ksh 28600
House allowance Ksh 15 000 Medical allowance Ksh 3 200 Transport allowance Ksh 540 Shaka was entitled to a monthly tax relief of Ksh 1056. (a) Calculate the tax charged on Shaka’s monthly earnings. (b) Apart from income tax, the following monthly deductions were made; a Health Insurance fund of Ksh 500, Education Insurance of Ksh 1 200 and 2% of his basic salary for widow and children pension scheme. Calculate Shaka’s monthly net income from his employment. Form 4 Mathematics
The equation of a curve is given as y = 2x3 -9/2 x2 -15x + 3.
(a) Find: (i) the value of y when x = 2; (ii) the equation of the tangent to the curve at x = 2. (b) Determine the turning points of the curve. Form 3 Mathematics
An institution intended to buy a certain number of chairs for Ksh 16 200. The supplier agreed to offer a discount of Ksh 60 per chair which enabled the institution to get 3 more chairs.
Taking x as the originally intended number of chairs, (a) Write an expressions in terms of x for: (i) original price per chair; (ii) price per chair after discount. (b) Determine: (i) the number of chairs the institution originally intended to buy; (ii) price per chair after discount; (iii) the amount of money the institution would have saved per chair if it bought the intended number of chairs at a discount of 15%. |
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