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Form 4 Mathematics Paper 1 Exam Revision Questions With Answers Set 2

Given the column vectors P=#((-2),(3)), q=((12),(-8)), r=((6),(-9))# and that #a = 2p – 1/2q +2/3r# express a as a column vector and hence calculate its magnitude |a|.

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1. Without using a calculator or mathematical tables, evaluate: #(36 - 8x - 4 - 15 div -3)/(3x - 3 + -8 (6 - (-2)) #
2. Simplify; #(2y^2 - 3xy - 2x^2) / (4y^2 – x^2 ) #

3. A UK tourist comes to Kenya with £30, 000. He pays 20% commission at the airport and his expenses in Kenya amounted to Ksh. 90,000. How much money did he remain with in Ksh? (Take 1 UK £ = Ksh 70.50)
4. Solve for y in the equation 2 + log 3 + log y = log 5 + 1
5. Find the equation of the perpendicular bisector of the line segment passing through the points A (4,3) and B(2,7) giving your answer in the form y = mx + c
6. The figure below shows the angles of a polygon ABCDE. Obtain the size of each of the following angles, (i)
7. All prime numbers less than 10 are arranged in a descending order to form a number which forms a quotient of 1076 with a certain number. Calculate the number
8. In the figure below
9. Solve for t in the equation #2^(2t + 2) - 5(2 ^t) + 1 = 0#
10. When a certain number is divided by 30, 45, 54, there is always a remainder of 21. Find the least number.
11. A piece of metal has a volume of #20cm^3# and a mass of 300g. Calculate the density of the metal in kg/#m^3#.
12. A line L passes through point (-2,3) and (-1, 6) and is perpendicular to a line P at (-1, 6) a) Find the equation of L. b) Find the equation of P in the form ax + by = c.
13. The area of a sector of a circle of diameter 126cm is 4158#cm^2#. Calculate the angle subtended at the centre of the circle. (Take p =22/7 )
14. A Kenyan Company received US dollars 100,000. The money was converted into Kenya Shillings in a bank which buys and sell foreign Currencies as shown below. a) Calculate the amount of money in ksh, the Company received. b) The company charged the Kenya shillings calculated above into sterling pounds to buy Car in Britain. Calculate the cost of the car to the nearest sterling pounds.
15. A businessman sold a car at sh.900 000 after allowing his customer a 10% discount on the marked price of the car. In so doing he made a profit of 20%. a) Calculate: i. The marked price of the car. ii. The price at which the businessman had bought the car. b) If the businessman had sold the same car without giving a discount. Calculate the percentage profit he would have made.
16. A company saleslady sold goods worth sh. 1,600,000. From this sale she earned a commission of sh. 40,000. a) Calculate the rate of Commission. b) If she sold goods whose marked price was sh. 3 600,000 and allowed a discount of 2%, calculate the amount of commission she received.
17. A hollow metal pipe whose external and internal diameters are 6.3cm and 2.8cm respectively is 3.5m long. a) Calculate the volume of the metal used to make the pipe. b) The pipe is melted down and recast into a solid cylinder of height 1.75m. Calculate the radius of the cylinder to two decimal places. c) Given that the density of the metal above is 4.2g/#cm^3#, calculate the mass
18. Three business people Kamau, Gachui and Maina agreed to contribute Kshs. 1 210 000 to start a business. The ratio of Kamau’s contribution to Gachui’s contribution is 3 : 2 while that of Gachui to Maina is 1 : 3. a) Determine the ratio of Kamau’s contribution to Maina’s contribution. b) Determine the amount of money contributed by Kamau. c) They agreed to share their profits
19. A manufacturer sells bottle of fruit juice to a trader at a profit of 40%. The trader sells it for Ksh 84 at a profit of 20%. Find a) The trader’s buying price b) The cost of manufacture of one bottle
20. Factorise the expression#2x^2-x-6# and hence solve the quadratic equation
21. Given the column vectors #p=((12),( -9)), q=((-4),(8)), r=((-2),(1))# and that #b = 2/3 p+ 1/2 q-3r#, express b as a column vector and hence calculate |b|
22. Simplify #(x-5)/(x+5) - (7x-35)/(x^2-25)#
23. The figure below shows a circle centre 0 and radius 7 cm. The minor sector AOB is made up of triangle AOB and the shaded segment of the circle. a) Given that the area of the triangle is #12.25 cm^2#, find the value of ?. b) Calculate the area of the minor sector AOB and hence determine the area of the shaded region.
24. Four towns are situated in such a way that town Q is 240 km due east of town P. Town R is 100 km due south of town Q and on a bearing of 112° from P. The fourth town S is 225 km on a bearing of 202° from P. a) Draw a sketch diagram showing that positions of towns P, Q, R, and S. b) Find the size of angle PQR and angle RPS. c) Without using scale drawing, calculate to the nearest whole number
25. A stationary observer on a platform notices that a goods train 60 metres long takes 14.4 seconds to pass him completely. Find the speed of the goods train in km/h.
26. A Jua Kali artisan divided the cost of manufacturing an article into the cost of materials, labour and general expenses in the ratio 6 : 3 : 1. The cost of materials has now increased by 35%, the cost of labour also increased by 18% while general expenses has decreased by 15%. Calculate the percentage increase in the total cost of manufacturing the article.
27. Awino bought a number of blouses and a number of skirts at sh 150 per blouse and sh 300 per skirt and spent a total of sh 22 500. If she had bought twice as many blouses and half as many skirts at the same prices, she would have saved sh 4 500. a) Determine the number of each item she bought. b) Awino sold all the items at a profit of 30% per blouse and 40% per skirt. Calculate the total profit
28. The diagram shows a frustrum which represents a bucket with an open end diameter of 40 cm and a bottom diameter of 30 cm. The bucket is 30 cm deep and it is used to fill a cylindrical tank of radius of 1.2 m and height 1.35 m which is initially three-fifths full of water. a) Leaving p in your answer, calculate i. the capacity of the bucket in litres. ii. the volume of water required to fill
29. Simplify the following expression and then factorize completely #6x^2 - 11x - 6 - (2x - 3)^2#
30. Joshua is now three times as old as his son. In 10 years time he will be twice as old as his son. Find their present ages.
31. Solve the equation 4(3x - 2) + 3(5 - x) = 5(2x + 3) - 2(3x - 1)
32. The figure shows an equilateral triangle ABC in which all the dimensions are given in centimetres. a) Find the value of x. b) Use any two sides and an included angle or otherwise to calculate the area of triangle ABC.
33. Factorize completely the expression #80x^2 - 125y^2# and hence or otherwise find its value when x = 6 and y = 3.
34. The figure shows an octagon made from a cardboard measuring 30 cm by 20 cm. Four congruent triangles are removed from each corner of the cardboard as shown. a) Find the values of x and y. b) Hence calculate the area of the octagon.
35. The length of a rectangle is increased in the ratio 4: 3 while its width is decreased in the ratio 2 : 3. a) Given that the original rectangle measures 18 cm long and 15 cm wide, find the dimensions of the new rectangle. b) Determine the ratio in which the area of the rectangle changes and state whether it is an increase or a decrease.
36. The diagram below shows a right prism standing on a rectangular base ABCD. The prism is 12 cm long and its triangular faces are equilateral triangles of side 5 cm. A path ABDEF is marked on the diagram. a) Draw the sketch A of the net of the prism and label all the vertices. b) Find the length of the path ABDEF.
37. Halima bought a dress for sh 480 and sold it to a customer at a profit of 50%. When selling the dress she had allowed her customer a 10% discount on the marked price. Determine the marked price of the dress.
38. a) Use mathematical tables to find (i)the square of 28.6 (ii)the reciprocal of 14.66 b) Hence or otherwise evaluate to 4 significant figures #28.6^2/14.66#
39. Jane is paid a basic salary of sh 6 580 per month. She is also paid commission at the rate of 2 cents per shilling on all sales up to sh 100 000. On anything above this she is paid commission at the rate of 3.5 cents per shilling. During a certain month she sold goods worth sh 180 000. Calculate her total income for that month.
40. Given the ratio a : b = 3 : 2, find the ratio (a + 3b) : (8a -3b)
41. The figure shows a rectangular metal sheet measuring 1 m long and 80 cm wide. From each of the four corners of the rectangle a square of side x cm is removed. The remaining part is then folded along the dotted line to form an open cuboid. The metal is 2 mm thick, and its density is 2.5 g/#cm^3#. a) If A #cm^2# is the area of the unshaded part, show that #A = 8 000 - 4x^2 #
42. An artisan made some articles and sold them to a wholesaler at 20% profit. The wholesaler sold the articles to a hawker at a profit at 25%. The hawker finally sold the articles to his customers at a profit of 50%. a) A hawker paid sh 600 for an article. Calculate how much i. it had cost the artisan to make the article. ii. the customer paid for the article.
43. A bus left Nairobi on Thursday evening and travelled to Dar-es-salam according to the travel time table below and arrived there on Saturday morning. a) Determine the total i. travelling time for the whole journey ii. stoppage time in all stations iii. time taken for the whole journey b) Given that the average speed of the bus for the whole journey is 60 km/h, calculate the distance
44. The figure shows the frustum of a right pyramid standing on a square base PQRS of side 16 cm. The plane TUVW is parallel to the plane PQRS and halfway up the vertical height of original pyramid. Given that WV = VU = 8 cm and that the slant height of the original pyramid is 26 cm, calculate to the nearest whole number the total surface area of the frustrum.
45. Before planting fifty tree seedlings are uprooted from a nursery, their heights measured to the nearest centimetre and recorded in the given table. a) Copy and complete the table. b) State the class in which the mean and the median are most likely to be. c) Use the completed table to calculate the mean height of the seedlings.
46. In the figure PQRS is a square of side l cm. Points X and Y are on SR and SP respectively such that SX = #1/3#SR and SY= #2/3#SP. Show that the area of the shaded region is #7/18 l^2cm^2. # a) Show that the sum of the areas of triangles SXY and QXY is equal to half the area of the square. b) Given that the area of the shaded region is #56 cm^2#, find the value of l
47. Without using logarithms or a calculator, evaluate #sqrt((65.52×7.392)/(3.696×32.76))#
48. Simplify the expression 3(a - 3b + 2c) - 4(2a - 2b + c) - 5(b - c - 2a)
49. Evaluate #(243/32)^(-3/5)×(64/27)^(-2/3)×(144)^(1/2)#
50. Pamela has one-shilling coins, five-shilling coins, ten-shilling coins and twenty-shilling coins in the ratio 8 : 5 : 4 : 3. If she has a total of 60 coins, determine the amount of money she has.
51. The wheel of a racing car has a diameter of 1.1 metres and the car is moving at a speed of 180 km/h. Calculate to one decimal place, the number of revolutions the wheel makes per second. (Take p=3.142).
52. Without using logarithms or a calculator, evaluate #4/5 log_10 32+log_10 50-3 log_10 2#
53. Find the size of each exterior angle of a regular pentagon.
54. The straight line whose equation is 2y = 3x + 6 meets the x-axis and the y - axis at P and Q respectively. Write down the coordinates of p and Q.
55. A construction company employs 24 men to work 6 hours a day for 20 days in order to complete a certain job. For how many more hours per day must 30 men work in order to complete the same job in 12 days?
56. Given the column vectors P=#((-2),(3)), q=((12),(-8)), r=((6),(-9))# and that #a = 2p – 1/2q +2/3r# express a as a column vector and hence calculate its magnitude |a|.
57. By selling an article for sh 725 a trader usually makes a profit of 45%. During a clearance sale the trader reduced the price of the article by 20%. Find the percentage profit the trader made.
58. The actual area of an estate is 3 510 hectares. This estate is represented by a rectangle measuring 2.6 cm by 1.5 cm on a map whose scale is 1: n. Find the value of n.
59. A vertical pole 3 metres high casts a shadow 5 metres long on the ground. Calculate to the nearest whole number the angle of elevation of the sun.
60. Six men take 10 days to plant 480 trees. Find how many trees 9 men can plant in 8 days.
61. A group consisting of 60 adults and children hired a bus for sh 18 600 in order to take a trip. To raise the money each adult paid sh 400 and each child sh 250. Find the number of adults and the number of children in the group.
62. A cylindrical tank of diameter 1.4 m and height 1.2 m is two-thirds full of water. The tank is filled using a cylindrical bucket of diameter 35 cm and height 20 cm. Find the number of buckets required to fill the tank.
63. Four ropes measuring 12 m, 18 m, 24 m and 36 m are cut into pieces of equal length so that an exact number of pieces are obtained from each rope without wastage. Find the longest length of each piece.
64. The sum of the interior angles of an n-sided polygon is 720°. Find the value of n and hence deduce the name of the polygon.
65. Fig. (i) above shows a triangle ABC in which AB = a cm, BC = 12 cm and angle ABC = 30°. Fig. (ii) shows a rectangle PQRS in which PQ = 9 cm and QR = b cm. Given that the area of the triangle is two-thirds the area of the rectangle, find the ratio a : b.
66. A draper bought some shirts and some trousers from wholesaler A at sh 200 per shirt and sh 600 per trouser spending a total of sh 22 000. If he had bought the same number of items from wholesaler B he would have paid 25% more for a shirt and 15% less for a trouser and he would have spent sh 700 more. a) Determine the number of each item he bought. b) He sold all the items at a profit of 50%
67. A piece of wire can be folded into a rectangle whose dimensions are such that its length is 8 cm longer than its width. The area of the rectangle so formed is # 468cm^2#. a) Determine i)the dimensions of the rectangle ii)the perimeter of the rectangle b)The same piece of wire can also be folded into a circle. Taking p=22/7, find the radius of the circle and hence calculate its area.
68. A trailer left Mombasa at 10.15 a.m. and travelled towards Nairobi at an average speed of 30 km/h. At 1.35 p.m on the same day a lorry left Mombasa and travelled towards Nairobi at an average speed of 50 km/h on the same road. The distance between Mombasa and Nairobi is 500 km. Assuming that none of the vehicles stopped on the way, determine a) the time of day when the lorry overlook the trailer.
69. Four ships are at sea such that ship B is 520 km on a hearing of 210° from ship A. Ship C is due north of ship B and due west of ship A. The fourth ship D is 100 km on a bearing of 340° from ship A and 240 km a bearing of 070° from ship C. a) Draw a rough sketch showing the positions of ships A, B, C and D. b) Use your sketch to find the size of i. angle ADC. ii. angle ACB.
70. A trader sold an article at sh 3 600 after allowing his customer a 10% discount on the marked price of the article. In so doing he made a profit of 50%. a) Calculate i. the marked price of the article. ii. the price at which the trader had bought the article. b) If the trader had sold the same article without giving a discount, calculate the percentage profit he would have made.
71. Use logarithms to evaluate #73.48 ÷ {(0.7592)^3 × 0.08723} ^(2/5) #
72. Factorize and simplify as far as Possible. #(15x^2+xy-6y^2)/(5x^2-8xy+3y^2 )#
73. Twenty four men working at the rate of 10 hours a day take 18 days to complete a certain job. How long would 30 men working 9 hours a day take to complete the same job?
74. Solve the equation #3/(2x) =7/(5x-1)#
75. a)Using mathematical tables find i) the reciprocal of 2.754. ii) the square of 4.765. b) Hence or otherwise calculate to 3 significant figures the value of #4.7652 + 1/2.754#
76. Solve for x in the equation #25^x+5^(2x) =50#
77. A cylindrical tank of radius 2 m and height 1.5 m initially contains water to a depth of 50 cm. Water is added to the tank at the rate of 62.84 litres per minute for 15 minutes. Find the new height of water in the tank. (Take p = 3.142)
78. The straight line passing through the points A(-1, 2k) and B(k, -1) is parallel to the line whose equation is 2y + 3x = 8. Find the value of k and hence write down the coordinates of A and B.
79. Jerusha invested sh 50 000 at the rate of 10% per annum compound interest for #2 1/2# years. Kemuto invested sh 35 000 at the rate of 15% per annum simple interest for the same period. Whose investment earned more interest and by how much?
80. a) Find the greatest common divisor of the terms #8x^4y^3 and 2x^2y^5 # b) Hence factorize completely the expression #8x^4y^3 - 2x^2y^5#
81. Evaluate #7 1/2 ÷3/5 of 4 1/6 +8 1/3 times1 1/5#
82. The marked price of some goods was sh 480 000. A salesman sold these goods at a discount of #2 1/2 %#. He received sh 18 720 as commission on the sale. At what percentage rate of commission was he paid?
83. The base areas of two cylindrical containers are #135 cm^2# and #240 cm^2# respectively. a) Determine the area scale factor and hence the linear scale factor. b) If the capacity of the smaller container is #540 cm^3#, what is the capacity of the larger container in litres?
84. The longer parallel side of a trapezium is three times as long as the shorter parallel side. The perpendicular distance between the parallel sides is 15 cm. If the area of the trapezium is #180 cm^2#, calculate the length of its longer parallel side.
85. Three towns A, B and C are such that town B is 90 km on a bearing of 150° from town A and town C is 120 km on a bearing of 060° from town B. a) Draw a sketch showing the positions of the three towns. b) Calculate the distance between towns A and C.
86. The diagram below shows a trough which is 4 m long and has a triangular cross-section. All dimensions arc as shown on the diagram a) Calculate I. the cross-sectional area of the trough in square metres. II. the capacity of the trough in litres. b) The trough which is initially one third full of water is filled by a pipe which delivers water at the rate of 0.8 litres per second.
87. Banda and Chuma live in two towns 240 km apart. One day at 9.45 a.m. Banda left his town and drove towards Chuma's town at an average speed of 60 km/h. Chuma left his town at 10.50 a.m. on the same day and drove towards Banda's town at an average speed of 80 km/h. a) Find I. the distance from Banda's town to the place where the two met. II. the time they met
88. During the first year a poultry farmer had four times as many hens as ducks and one third as many turkeys as ducks. In the second year the number of ducks increased by 10%, the number of hens increased by 25% while the number of turkeys decreased by 20%. At the end of the second year the farmer had a total of 955 birds. Calculate to one decimal place the percentage increase in the total number of
89. The external measurements of a wooden box are 1.0 m long, 70 cm wide and 40 cm high. The wood used in making the box is 1.5 cm thick and has a density of 0.8 g/#cm^3#. The box contains 25 packets of tools. Each packet holds a dozen tools and each tool weighs 108.5 g. Calculate a) the volume of wood used in making the box. b) to 2 decimal places the mass of the empty box in kilograms
90. At the end of year 2000 a cooperative society realised a gross income of sh 33.5 million. The total expenses incurred by the society during that year was sh 20.75 million. Each year 60% of the net income is paid to members as dividends which is calculated per share. During the year 2001 the society's gross income increased by 30% while its total expenditure decreased by 20%.
91. The diagram below shows a rectangle measuring 15 cm by 12 cm from which two semi-circles each of a diameter 5.6 cm are removed. The figure represents the cross section of a wooden block 10.5 metres long. The wood making the block has a density of 0.8 g/#cm^3#. a) Calculate to one decimal place i. the cross sectional area of the block ii. the mass of the wood in the block in kilograms.
92. A school hired a number of buses and matatus to transport a group of students to Lake Nakuru. The number of matatus was three times the number of buses. The hire charges were sh 3 500 per matatu and sh 6 500 per bus. The total cost of hiring the vehicles was sh 17 000. Each matatu can carry 13 students while a bus can carry five times as many. a) Determine the number i. of buses hired
93. Without using logarithms or calculator, evaluate #root3((0.95 times 5.5 times 10.5)/(0.21 times2.09))#
94. Kamemia uses #1/3#of his farm for planting coffee, #1/4# for planting tea and #2/5#of the remainder for mixed farming. He still has 6 hectares of unused land. Find the size of Kamemia's land.
95. Mwaura is now twice as old as his daughter and four times as old as his son. In eight years time Mwaura's age will be equal to the sum of ages of his daughter and son. Determine Mwaura's present age.
96. The straight line passing through the point (-3, -4) is perpendicular to the line whose equation is 2y + 3x = 11, and intersects the x-axis and the y- axis at P and Q respectively. Determine the equation of the second line and hence write down the coordinates of P and Q.
97. Given the column vectors #p=((1),(2)), q=((-3),(-6)) and r= ((2),(-3))# and that #a=3p- 1/3q + r#, express a as column vector.
98. Mutisya left a will directing that his 18 hectare land he divided among his three sons Nyange, Kalweo and Mutua in the ratio 3 : 4 : 5. He also directed that his savings of 3 825 000 be divided among the three sons in the ratio 5 : 3 : 1. If the Kalweo sold his share of land at sh 80 000 per hectare, determine the total amount of money he received.
99. The cost of 3 pairs of trousers and 2 shirts is sh 2 400. The cost of two pairs of trousers and 3 shirts is sh 1 975. Find the cost of one pair of trousers and 4 shirts.
100. A cylindrical container of diameter 7 cm contains water to a depth of 5 cm. Ten spherical metal balls of radius 1.4 cm each are placed in the container. Find to the nearest whole number the new height of water in the container.