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Form 4 Mathematics Paper 1 Exam Revision Questions With Answers

Factorise completely the expression #75x^2 - 27y^2# and hence or otherwise find its value when x = 4 and y = 5.

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1. Without using logarithms or a calculator evaluate: #sqrt((0.8064 ×6.048)/(1.008×0.1344))#
2. Without using mathematical tables evaluate: #(sin60°·tan30°·cos 60° + sin30°·cos45°·sin 45°)/(sin90°·cos45°·sin45°- sin60°· cos30°· sin30°)#

3. The figure below shows a solid cone of diameter 21cm and height 8cm. Calculate to one decimal place: a) The slant height of the cone. b) The total surface area of the cone. (Take #pi=22/7#)
4. Simplify #(x+4)/(x-4)-(5x+20)/(x^2-16)#
5. The figure below shows a circle center O. AOB is a sector of the circle and angle AOB = 72° as shown. Given that the area of sector AOB is 5 pcm2, find the radius of the circle and hence calculate the area of the shaded part.
6. A piece of wire can be folded into a rectangle whose dimensions are such that its length is 2cm longer than its width. The area of the rectangle so formed is #120cm^2#. a) Determine: i) the dimensions of the rectangle ii) the perimeter of the rectangle b) The same piece of wire can be folded into a circle. Taking #pi = 22/7#, find the radius of the circle and hence calculate its area.
7. The cost of manufacturing a radio is divided into the cost of materials, labour, overheads, and advertising in the ratio 6: 3: 2:1. During a certain year the cost of materials increased by 40%, the cost of labour increased by 15%, overheads decreased by 12% while advertising remained the same. Calculate to one decimal place the percentage increase in the total cost of manufacturing the radio.
8. Lesoi bought 6 cows and 15 goats at an auction and spent a total of sh 97, 500. His friend Solei bought one cow less and five goats more than Lesoi and spent sh 5000 less. a) If the two friends bought each animal at the same price, determine the price of each animal at the auction.
9. The figure below shows a frustum which represents a bucket with an open end diameter of 30 cm and a bottom diameter of 20cm. The bucket which is 42cm deep is used to fill an empty cylindrical tank of diameter 1.8m and height 1.2m. a) Leaving p in your answer, calculate: i. the capacity of the bucket in litres. ii. the capacity of the tank in litres.
10. Expand the expression #(5x-7) (x+3)#. Hence without using calculator or long multiplication, use the result above with an appropriate substitution for x to find the value of #83times21#.
11. A bookshop owner bought 12 dozen exercise books at sh 80 per dozen and sold all of them at sh 10 for each exercise book. Find the percentage profit he made.
12. The straight line passing through the points A (5, k) and B(k,6) is perpendicular to the line whose equation is 4y-3x=12. Find the value of k.
13. Given that #p=4, q=1 and r=-3, evaluate (2pqr^2+pqr-pr)/(4qr-2pr-2pqr)#
14. The diagonals of a rhombus measure 15.5 cm and 11.7 cm respectively. Calculate to one decimal place the area of the rhombus.
15. Find the range of values of x which satisfies the following inequalities simultaneously. #4x-9<=6+x# #8-3x<=x+4#
16. The figure shows a circular fish pond of radius 6 meters surrounded by a pavement of uniform width. The width of the pavement is x meters as shown.
17. On a large-scale farm there are six more pigs than cows, four times as many goats as pigs and two thirds as many sheep as goats. a) If there are y cows, write down a simplified expression in y for the total number of animals. b) Find the total number of animals given that there are 112 sheep.
18. At the end of his stay in Kenya, a French tourist had 3420 French francs which he decided to change into Euros. Given the exchange rate was: 1 French franc = Ksh 11.25 1 Euro = Ksh 72.50 Calculate the number of Euros he received if the bank charged him a 2% commission.
19. In a form one class the ratio of boys to girls is 3:4. The mean mass of the girls is 35 kg and the mean mass of the whole class is 38kg. Given that there are 42 students in the class, calculate the mean mass of the boys.
20. The figure below shows a solid glass prism whose dimensions are shown. a) Calculate i. to 4 significant figures, the cross-sectional area of the prism. ii. the volume of the prism. b) The prism is completely immersed in water held in a cylindrical container of radius 14 cm. Calculate to one decimal place the increase in height of the water container.
21. The figure below shows a circle centre O and radius R cm. The minor arc AB subtends an angle of 105° at the centre of the circle and the corresponding sector AOB has an area of # 528cm^2#. #(Take pi=22/7)# a) Find the radius R of the circle. b) The sector is removed and folded into a cone, calculate: i. the area of the cone so formed. ii. to 2 decimal places the height of the cone.
22. The diagram below shows a cone of base radius 42cm from which a small cone of base radius 28cm has been removed to form a frustum PQRS. The height of the frustum is 20cm as shown. Calculate to 3 decimal places the total surface area of the frustum in square metres.
23. Mary is now four times as old as her daughter. Four years ago the product of their ages was 40. Find their present ages.
24. Before storage green maize is dried, treated and then packed in 90 kg bags. The mass of green maize decreases in the ratio 3:5 when dried. Determine the mass of green maize that must be dried to produce 5 such bags when packed.
25. Simplify the following quadratic expression by expanding, collecting terms and then factorizing completely. #15x^2-9x-5-(3x-2)^2#
26. Find the gradient of the line joining the points P(-1,4) and Q(-6,7). Hence find the equation of the line passing through P and perpendicular to the line PQ.
27. Starting from noon the minute hand of a clock moved so that the clock is showing quarter to one. a) Find the angle through which the minute hand has moved. b) Calculate the area of the sector described by the minute hand given that its length is 7 cm.
28. A solid cone of base radius 12cm and height 20cm is completely immersed in water held in a cylindrical container of radius 14cm. a) Leaving p in your answer, calculate the volume of the cone. b) Calculate to one decimal place the increase in height of the water of the container.
29. Given that log 2= 0.3010, log 3= 0.4771 and log 5= 0.6990. Evaluate log 1800 without using logarithm tables or a calculator.
30. From a point P the angle of elevation of the top of a building is 25.3°. From another point Q which is nearer to the base R of the building, the angle of elevation of the top of the building is 47.8°. Points P, Q, and R are on the same straight line on horizontal ground. Given that the distance between P and Q is 25 metres, calculate to 2 decimal places the distance between Q and R.
31. A trailer is loaded with a mixture of bags of maize and bags of dairy meal. One bag of maize has a mass of 90kg, and one bag of dairy meal has a mass of 70kg. The trailer is loaded with 300 bags whose total mass is 23.4 tonnes. Find the number of bags of each type it is carrying.
32. Solve for a in the equation: #(5a-3)/6-(5-a)/3=(2a+4)/4+(a-3)/2#
33. Mwongera bought a wrist watch at sh 1000 and marked it at a price such that after allowing his customer a 12% discount, he would make a profit of 32%. Find the price at which the watch was marked.
34. Convert the recurring decimal #0.dot40dot5#into a fraction in its simplest form.
35. The figure below shows a trapezium ABCD in which AB is parallel to CD and angle ABC=75.7°. AB=18cm, CD=11cm and BC= 9.5cm. Calculate to one decimal place the area of the trapezium.
36. Yusuf bought some rice at sh 30 per kg. He packed two-fifths of the rice in 2 kg packets which he sold at sh 85 per packet. He packed the other three-fifths in 3 kg packets and sold them sh 120 per packet. He sold all the rice in this way and made a profit of sh 4400. a) Determine the amount of rice bought. b) Calculate to one decimal place i. the percentage profit he made.
37. The diagram represents a solid hemispherical dome of diameter 42cm. The dome is painted on all faces at a cost of sh 500 per square metre and has a mass of 48.5kg. Calculate: a) The total surface area of the dome. b) The cost of painting the dome. c) The volume of the material making the dome. d) The density of the material in kg/m3.
38. A room measures 6m long and 4m wide. A carpet whose area is #15m^2# is laid in the middle of the room leaving a margin of uniform width all-round the room. a) Taking x m to represent the width of the margin, express the dimensions of the carpet in terms of x. b) Write down a simplified expression for the area of the carpet in terms of x.
39. A manufacturer made an article and sold it to a wholesaler at a profit of 25%. The wholesaler sold the article to a retailer at a profit of 40%. The retailer finally sold the article to a customer at 60% profit. a) Find how much a customer paid for an article that cost the manufacturer sh 600 to make. b) A customer paid sh 2100 for another article.
40. The model of a closed cylindrical tank is made in such a way that it is similar in shape to the actual tank to be constructed. The capacity of the model is #770cm^3# while that of the proposed tank is #49.28m^3#. Find: a) The volume scale factor. b) The linear scale factor and hence calculate the height of the tank in metres given that the height of the model is 5cm.
41. Factorise completely the expression #75x^2 - 27y^2# and hence or otherwise find its value when x = 4 and y = 5.
42. The figure shows a rectangle PQRS whose dimensions are given in centimetres. Find the value of x and hence calculate the area of the rectangle.
43. A racing car consumes petrol at the rate of 1 litre for every 4 ½ km. The car also requires 3 litres of engine oil each time the oil is changed, and this happens after every 1 200 km. This year's Safari Rally will cover a route 3 600 km long. Petrol costs sh 55.50 per litre and engine oil sh 400 per litre.
44. The base of a parallelogram whose area is #200 cm^2# is 10 cm longer than its height h. a) Form a quadratic equation in h for the area of parallelogram. b) Solve the quadratic equation and hence state the dimensions of the parallelogram.
45. A square and a triangle share the same base and the area of the triangle is three-quarters the area of the square. Find the height of the triangle given that the area of the square is #64cm^2#.
46. Simplify the expression 4(x - 2y+ z) - 5(x - y+ z) + 3(x - y - 2z)
47. Tuva marked an article at sh 900 and sold it to a customer at a discount of 12%. Find the percentage profit he made if he had bought the article at sh 600.
48. Given that a = 5, b = -3 and c = -1, find the value of: #(2ab^2-3ab^2 c^2-4ab^2 c)/(a^2 bc-2abc^2-4abc)#
49. Evaluate #(125/8)^(1/3) ×(25/16)^(-3/2) ×(625/64)^(1/2) #
50. In the figure PQRS is a rhombus in which angle PSR = 70° and PQ is produced to T such that PQ = QT. Giving reasons for your answer, find the size of a) angle QTR b) angle PRT
51. Given the column vectors #((15),(-5)) ((-3),(6)) ((-8),(4))# and that q= #4/5# a - #1/3#b + #3/4#c express q as a column vector and hence calculate to one decimal place its magnitude |q|
52. Given the ratio a:b= 3:4, find the ratio (6a-b) :(3a+3b)
53. Giving your answer in the simplest form, express #(2x +y)^2 – (3x-y)(2x+y) #in terms of x only, given that #y=x-3.#
54. Given that tan x° = ¾ , write down in fraction form the values of sin x and cos x.
55. Four towns are such that town B is 30 km due east of town A. Town C is 40 km due south of B and town D is 60 km due west of town C. a) Draw a sketch diagram showing the positions of towns A, B, C and D. b) Calculate to the nearest whole number i. the distance of town D from town B ii. the bearing to town C from town A.
56. A triangle whose base is 6 cm longer than its height has an area of #80 cm^2#. a) Taking h as the height of the triangle, write down a simplified quadratic equation for the area of the triangle. b) Hence determine the dimensions of the triangle.
57. The scale of a map is 1:400 000. On this map two towns are separated by a line 11.25 cm long. a) Find the distance between the two towns in kilometres. b) A game reserve is represented by a square of side 4.5 cm on this map. Find the area of the game reserve in hectares.
58. At the end of the day a kiosk owner had four times as many ten-shilling coins as twenty-shilling coins, seven times as many five-shilling coins as twenty-shilling coins and twice as many one-shilling coins as ten-shilling coins. After counting her money she found that she had a total of sh 515. Determine the number of coins she had.
59. It takes 20 men 10 days to lay 300 metres of pipes. Find how many days it would take 15 men to lay 270 metres of pipes.
60. A retailer usually makes a profit of 50% by selling an article at sh 540. If he reduces the price of the article by sh 54, calculate the percentage profit he will now make.
61. The diagram shows a cylindrical tank surmounted with a hemispherical dome. The cylindrical part has the same diameter as the hemispherical part. The cylindrical part is 1.5 m high and has a diameter of 2.8 m. a) Calculate to 2 decimal places i. the area of the hemispherical part. ii. the curved surface area of the cylindrical part iii. the total surface area of the tank
62. The figure shows a rectangle ABCD in which AB = x cm and BC = 2x cm. Points P and Q are on AD and CD respectively such that PD = 6 cm and DQ = 2 cm. a) Show that the area of the shaded region is #(5x — 6) cm^2#. b) Given that the area of triangle ABP is 40 cm2, find the value of x and hence calculate the area of the shaded region.
63. Factorise completely #(a - 2b) (2a + b) - (a-2b)^2#
64. Simplify the following expression by reducing it to a single fraction #(2x-5)/4-(1-x)/3-(x-4)/2#
65. The figure shows a circle 0 and AOC as a diameter. Angle A0B = 140° and angle ODA = 40°. Giving reasons for your answer, find the size of a) angle ADB. b) angle OBD. c) angle BDC.
66. Konga bought 4 pens and 10 exercise books at a total cost of sh 220. His friend Ndolo bought 3 pens and 12 exercise books at the same prices and spent sh 10 less. Determine the cost of each item.
67. The figure shows a parallelogram ABCD in which the base AB = (h + 5) cm and height CF = h cm. Given that the area of the parallelogram is #84 cm^2#, find the value of h.
68. You are given the points P (-3 , -2), Q(-6 , 3) and R(-12 , 13). a) Express PQ and QR as column vectors. b) Use the result in (a) above to explain why the points P, Q and R are collinear.
69. An open cylindrical water reservoir of radius 10.5 metres has a curved surface area of #198 m^2#. The reservoir is to be cemented on all its inside surfaces at a cost of sh 400 per square metre. a) Taking p=22/7, determine (i) the height of the reservoir. (ii) the cost of cementing the reservoir.
70. The above diagram represents a trough which is 6 metres long and whose cross-section is a semicircle of diameter 1.4 metres. a) Calculate i. the cross-sectional area of the trough in #cm^2#. ii. the capacity of the trough in litres b) The trough which is initially empty is filled by a diesel pump which pumps water at the rate of 1.54 litres per second.
71. A piece of wire can be folded into a circle whose area is #1386 cm^2#. Taking p=22/7, a) find the radius of the -circle and hence calculate its circumference. b) The wire in (a) above can also be folded into a rectangle whose length is 6 cm longer than its width. Determine the dimensions of the rectangle and hence calculate its area.
72. The diagram represents a solid frustum with base radius 21 cm and top radius 14 cm. The frustrum is 15 cm high and is made of a metal whose density is 2 g/#cm^3#. a) Calculate i. the volume of metal in the frustrum ii. the mass of the frustrum in kg. b) The frustrum is melted down and recast into a solid cube. In the process 20% of the metal is lost.
73. After allowing his customer a 5% discount on the marked price a sales agent sold a second hand bus at sh 1 140 000. The owner of the bus received sh 1 003 200 from the sales agent after the latter had deducted his commission. a) Determine the marked price of the bus. b) Calculate the percentage commission the agent received c) By selling the bus this way the owner incurred a loss of 25%.
74. The figure shows a rectangular metal sheet ABCD in which AB = 90 cm and BC = 70 cm. The sheet is 1.8 mm thick and is made of metal whose density is 2.5 g/#cm^3#. A square of side t cm is removed from each corner as shown and the remaining part folded along the dotted line to form an open cuboid. a) Given that the area of the unshaded part is #A cm^2#, write down an equation for A in terms of t.
75. Simplify# ((4x+2y)^2-(2y-4x)^2)/((2x+y)^2-(y-2x)^2 )#
76. The figure shows an equilateral triangle ABC in which all the dimensions are given in centimetres. Determine the value of x and hence calculate the area of the triangle
77. A solid metal sphere of radius 7.5 cm is melted down and recast into a solid cylinder of height 15 cm. In the process 4% of the metal is lost. Calculate a) in terms of p the volume of metal used to make the cylinder. b) the radius of the cylinder.
78. A truck which consumes diesel at the rate of 1 litre for every 5 km uses 72 litres of diesel for a certain journey. A bus uses 45 litres of diesel for the same journey. Find the distance the bus covers for every litre of diesel.
79. Solve for y in the equation #8^((2y-1))×32^y=16^((y+1))#
80. The distance between two towns P and Q is 480 km. At 10.28 a.m a bus left town P and travelled towards town Q at an average speed of s km/h. A car left town Q at the same time and travelled towards town P along the same road. The two vehicles met at 1.40 p.m. Given that on average the car travelled 30 km/h faster than the bus, determine the value of s and hence state the speed of the car.
81. The sum of the interior angles of an n-sided polygon is 1080°. Find the value of n and hence deduce the name of the polygon.
82. The wheel of a bicycle is rotating at the rate of 130 revolutions per minute. If the speed of the bicycle is 18.4 km/h, calculate to the nearest whole number, the diameter of the wheel in centimetres. (Take p = 3.142).
83. A straight line passes through the points P (5 , 3) and Q(0 , 6). Find equation of the line in the form y = mx + c and hence find its double intercept equation.
84. Form the quadratic equation whose roots are #y= 3/5 and y= -1#
85. Find the values of x which make the following inequality true. #9-3/4 x<2/3 x+4 3/4#
86. Omondi working alone takes 10 days, Simiyu 15 days and Muiruri 12 days to complete digging a shamba. Omondi and Simiyu started working together and after 3 days Muiruri joined them. Determine how many more days it took the three men to complete digging the shamba.
87. The diagram represents a water reservoir comprising of a hemispherical part surmounted on top of a cylindrical part as shown. The two parts have the same radius and the cylindrical part is 1.5m high.(Take p = 22/7 for all calculations below) a) Given that the reservoir has a total surface area of #528m^2#, find the radius of the reservoir. b) Calculate the capacity of the reservoir in cubic metre
88. a) Calculate to 2 decimal places the areas of the following triangles. (i) (ii) b) Without using mathematical tables or calculator, evaluate (i) #(tan?30° · sin?30° + tan? 60° · cos? 30° )/(cos? 30° ·tan? 60°· -sin? 30° · cos? 60° ) # (ii)# (tan? 60° · sin? 60°- sin?45° · cos ?45°) /(cos? 30° · tan? 60°· +sin? 30° · tan? 45° ) #
89. The figure shows two circles centres C and D of radii 8 cm and 10 cm respectively. The two circles subtend angles ? and ß respectively at their centres and intersect at A and B as shown. a) Given that the area of triangle #ACB = 30.07 cm^2# and that of triangle #ADB = 43.30 cm^2#, calculate the size of the i. angle marked ?. ii. angle marked ß.
90. In order to start a business, three businesswomen, Amina, Balinda and Chini contributed sh 25 000, sh 35 000 and sh 45 000 respectively as business capital. They also had to pay sh 15 000 more as rental fee for business premises. The rental fee was shared equally among the partners. The three partners agreed to put 25% of the annual profits back in the business and share the rest in the ratio
91. Without using logarithms or a calculator, evaluate # root(3)((4.536×0.8064×v20.25)/(0.2268×1.134))#
92. Factorise #5x^2 -9x + 4#
93. A South African businessman exchanged 6 000 Rands and received Ksh 56 160 after the bank deducted its commission. Given that the exchange rate was 1 SA Rand = Ksh 9.75, calculate the percentage commission the bank charged him.
94. The base radius of a cone is decreased by 10% while its height is increased by 40%. Find the percentage change in its volume and state whether it is an increase or a decrease.
95. A school used 25% of the money raised in a harambee for renovating classrooms. 35% for building a new laboratory, three-fifths of the remainder for buying laboratory equipment and the rest for installing electricity. If the school bought laboratory equipment worth sh 360 000, calculate the total amount raised at the harambee and hence determine the amount spent for installing electricity.
96. Five is subtracted from a certain number, the result doubled and then divided by 3. If the answer is 10, find the number.
97. A rectangular water cistern measuring 2.5 m long, 1.8 m wide and 1.2 m high is initially half full of water. Water is pumped into the cistern at the rate of 1.25 litres per second. Calculate the time in minutes it takes to fill the cistern.
98. On a map whose scale is 1 : 200 000 a coffee estate is represented by a square of side 5.5 mm. Calculate the area of the estate in hectares.
99. The figure shows a solid cone of diameter 10 cm and slant height 13 cm. Leaving p in your answer, calculate. a) the total surface area of the cone. b) the volume of the cone.
100. In a meeting there are 200 more women than men. One-third of the men and two-fifths of the women are elderly people. If there are 300 elderly people in the meeting. find how many young people attended the meeting.