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Form 4 Mathematics Paper 1 Exam Revision Questions With Answers

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 π in your answer, calculate:
i. the capacity of the bucket in litres.
ii. the capacity of the tank in litres.
b) Calculate the number of buckets that must be drawn to fill the tank.

(10m 46s)

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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)#
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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.
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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.
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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#
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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.
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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.
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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.
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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.