UNIVERSITY OF SWAZILAND SGCSE MODEL ANSWERS MATHEMATICS

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1 PPER List all the factors of. {,,, 4, 6, } Work out (a). +., (a).4 [] (b) 5. 8 (b) (a) Find 7. (a) [] (b) Simplify 4b 5b 6 8 (b) 0b 4 (a) There are 4 hours in a day. Lungile works 6 hours each day. (i) Write the time Lungile works in a day as a fraction of a day. Give your answer in simplest form. (ii) 6 (a) (i) 4 4 Express the time Lungile works in a day as per percentage of a day. (a)(ii) 00% 5% 4 (b) Express 0% as a fraction in its simplest form. 0 (b) people watched a soccer game. (a) Express the number of people correct to significant figures. (a) [] (b) Express your answer to part (a) in standard form. 4 (b).5 0 []

2 6 Given that a = and b =, find (a) b a, (a) ( ) = ( ) ( ) = 9 [] (b) a b. (b) [] 8 7 List all the subsets of {a, b}. { }, {a}, {b}, {a, b} 8 Solve the simultaneous equations y = x x y = 6. x (x) = 6; x = 6; x = ; y = 6 [] 9 The figure below is formed by a square and four identical isosceles triangles. The figure represents the net of a solid. (a) (b) Write down the name of the solid. (a) square based pyramid [] Draw all the lines of symmetry on the diagram. (b)

3 0 Rearrange this formula to make T the subject. x = T + c x = 6T + c x c = 6T x c T 6 [] Solve (a) x 5 = 7, (a) x = x = 4 x (b) +. (b) x + 6 < x < 4 BC is a triangle. C B (a) Draw the locus of point 5 cm from B and inside the triangle. (b) Draw the locus of point cm from B and inside the triangle. []

4 C B For the function f: x x, find the output set of the input {, 0, }. {,, } 4 BC is a right angled triangle. B ˆC 90, CB ˆ x and CB ˆ x. xº C xº B Find the size ofc ˆ B. x + x + 90 = 80 5x = 90 x = 8 [] 5 The probability that it will rain on a particular day is. 5 What is the probability that it will not rain on that day? [] 5 5 4

5 6 Q P R Construct the bisector of angle PQR. 7 bus covers a distance of 40 km in hour 45 minutes. (a) Show that hour 45 minutes is 4 7 hours (a) h 45m = h [] (b) Calculate the average speed of the bus. D 40 4 (b) T 7 7 S km/h 4 5

6 8 N 0º N 6º B N C Calculate the bearing of (a) C from B, (a) 60 (6 + 60) = 64º (b) B from C. (b) 80 (6 + 60) = 84º 9 The diagram below shows the points and B. B (a) Express B as a column vector. B [] 6 6

7 (b) C = B. Mark and label the point C on the diagram. C B [] 0 rrange the following in order of size, starting with the smallest. (0.) (0.) 0. ( 0.) ( 0.) ( 0.), (0.), 0. Here is a list of numbers. Find (a) (b) (c) (d) the range, the mode, the median, the mean [] [] 4 [] [] 9 9 7

8 (a) You are given that f (x) = Find the values of (i) f(), (ii) f( ).. (x 5) PPER [] ( 5) ( ( ) 5) 8 (b) You are also given that g(x) = 7x 4. Find g (x). 4 g (x) x [] 7 ship sails from port to port B on a bearing of 065 for 40 km. It then sails to port C on a bearing of 50 for 00 km. (a) Using a scale of cm to 0 km, draw the two stages of the journey. [] [] B C (b) Measure angle BC. 97º [] (c) Find the distance C in kilometres. 80 km 8

9 y 0 x (a) (b) Workout the gradient of the line. m Write down the equation of the line. y x [] 4 teacher s annual salary is E His salary is taxed at 0%. (a) How much tax does he pay per year? E (b) The teacher receives an increase in salary of %. (i) (ii) What will be his new annual salary? E Calculate the teachers new tax per year. E [] (iii) The amount that remains after tax deduction is called a take-home salary. Calculate the teacher s monthly take-home salary. E (a) Write 050 as a product of its prime factors [] (b) Given that = ( ), B = (. 0 4 ) evaluate, 9

10 giving your answers in standard form, (i) (ii) B, B, (iii) B [].8 0 [] [] 6 Solve the following equations. (a) x 7 = 4 5 x = 9 (b) x + x 6 = 0 x = x = [] 7 (a) Factorise each of the following. (i) 6x 4 8(x ) [] (ii) 4 p (p )(p + ) [] 4 5x y (b) Simplify as far as possible 5x. x y [] 4 8 You are given that =, B = x 0 5 (a) Work out (i) B, and C = 7 y. 4 0

11 5 4 4 (ii) B. 7 6 x 4x (b) If B = C, find the values of x and y. x =, y = 6 9 The longer side of a rectangle is cm more than the shorter side. The perimeter of the rectangle is 40 cm. You are given that b is the length of the shorter side. (a) Write an equation for the perimeter in terms of b. b + (b + ) = 4 (b) Solve the equation to find the value of b. 4 b On the grid below, triangles and B have been drawn. Triangle has vertices (, ), (, 4) and (, ). Triangle B has vertices (, ), (, ) and (, 4).

12 y 5 4 B x (a) Describe the single transformation that maps onto B. Reflection in y-axis (b) (i) Draw the image of triangle after reflection on the line y =. Label it R. (ii) Draw the image of after a translation with vector. Label it T. 4 y T B R x -5 The distance between town and town B is 00km. bus leaves town 8.00 am and travels at a constant speed to reach town B at 0.0 am.

13 It stops for hours at town B. It then travels back to town at a constant speed. The bus travels at 50km/h on the return journey. (a) Draw a distance-time graph showing the journey of the bus (b) t what time does the bus arrive back at town? 400hrs or :00 pm [] The equation of the line passing through the points (, 4) and (, 8) is given by ax + by =. (a) Write down two equations in terms of a and b. a + b = ; a + 8b = (b) Solve the two equations to find the value of a and the value of b. a =, b = [] In a football stadium, 7 of the people were female. The total number of people who were in the stadium was 000. (a) How many males were in the stadium altogether? 000 [] In the football stadium the ratio of children to adults was :7. (b) Find the number of adults

14 Frequency 4 group of 40 pupils were asked to state the number of siblings (brothers and sisters) they had. The information is shown in the table below. Number of siblings Frequency (a) On the grid below, draw a bar chart to show this information. [] Number of siblings (b) (c) What is the modal class of siblings? 9 [] One of these pupils is selected at random. Find the probability that this pupil has (i) (ii) less than siblings, or more siblings. 4

15 The figure below shows a regular pentagon, BEFG, and a rhombus, BCDE. The points, B and C lie on a straight line. F G E D B C (a) Work out the size of each interior angle of the pentagon. 0 [] (b) Find (i) angle BCD, 0 [] (ii) angle DEF, 0 (iii) angle EDF. 0 [] 6 Ten pupils in a Form 5 class had the following marks for tests in Maths and History. Student B C D E F G H I J Maths mark History mark (a) On the grid below plot the scatter diagram for these marks. [] 5

16 History Maths (b) State the type of correlation. Negative correlation [] (c) Draw a line of best fit. 6

17 7 The figure below represents a geometric instrument from a set square BC and a protractor. The protractor has two semi-circles with the same centre. The radius of the small semicircle is 4 cm. The radius of the large semicircle is 4.5 cm. B = cm and BC = 9 cm. cm 9 cm 4.5 cm 4 cm The unshaded part is hollow. (a) Calculate the area of the hollow part..5 cm [] (b) Hence find the area of the shaded part cm [] 7

18 PPER (a) Theresa bought a camera in Botswana for P year later its value had been reduced by 7%. What was its new value? P095 (b) Gift bought a video camera. He later sold it for P00.00 making a 65% profit. How much did Gift pay for his camera? P7.7 (a) Simplify x x x writing your answer as a single fraction. 5x x x [] (b) Simplify x x x 6x. (c) h k = h + k = x x [] (i) (ii) Write these equations in matrix form. h k Find the inverse of the by matrix. [] 5 (iii) Hence solve the matrix equation to find the value of h and the value of k. h =, k = 0 8

19 (a) Find the gradient of this line. x 7y = 5 gradient = 7 [] (b) Rearrange this equation to make z the subject. x = z 7y 5y z 5xy 7y z [] x 4 survey was carried out to determine the number of people who owned a car (C), a television set (T) or a bicycle (B). total number of 650 people took part in the survey. 55 people had a car, 54 people had a television set and 8 people had a bicycle. 480 people had a television set and a car. 50 people had a television set and a bicycle. 5 people had a car and a bicycle. 0 people had a television set, a car and a bicycle. (a) Copy and complete this Venn diagram to show this information. [] 55 ξ C B T 650 C B T 0 9

20 (b) Find the number of people who had (i) (ii) a television set and a car but not a bicycle, 70 [] at least two of the items. 55 (c) Work out n(c T B)'. 0 [] (d) Find the probability that one of the 650 people, chosen at random, had only one of these items. 0 5 the whole of this question of graph paper. Forty teachers were asked how much they spent on air time per week. The following table shows their responses. mount spent (Ea) 0 a 5 5 a 0 0 a 5 5 a a 45 Number of Teachers (a) Using a scale of cm to 5 units on each axis, draw a frequency polygon to represent this information. [] (b) Calculate an estimate of the mean amount spent on air time by a teacher each week.. 75 [] 0

21 6 solid cone shaped candle of height 0cm and radius cm is melted down and cooled to form a cylindrical candle of the same height. [Volume of a cone = πr h ] 0 cm 0 cm cm (a) Work out the volume of the cone shaped candle. 06 cm (b) What is the radius of the cylindrical candle? cm [] (c) Calculate the total surface area of the cylindrical candle. 77cm [4] 7 The probability that a head teacher is in the office at noon on any school day is 0.6. The outcome that he is in the office is (P), and that he is not in the office is (). P 0.6 P

22 (a) Copy and complete the probability tree diagram to show all the possible outcomes for three consecutive days. [] 0.6 P P P 0.6 P P P P 0.4 PPP 0.6 PP 0.44 PP 0.44 P PP 0.44 P P (b) Use your tree diagram to calculate the probability that (i) the head teacher is in the office at noon on all three days, 0.6 [] (ii) the head teacher is in the office at noon on at least one of the first two days, 0.84 (iii) the head teacher is in the office at noon on at least two of the three days [] (c) What is the probability that the head teacher is in the office at noon on the fifth day? 0.6 [] 8 (a) K,L,M and P are points on the circumference of a circle. KM and LP intersect at X. L X M K P

23 (i) (ii) Show that triangles KLX and PMX are similar. K ˆ P ˆ (angles in the same segment) Lˆ Mˆ (angles in the same segment) Xˆ Xˆ (vertically opposite angles) LX = 6 cm, MX = 0.4 cm and MP = 5 cm. Calculate KL. KL. 5 (b) The diagrams below show the ends of two prisms. The prisms are similar. 40 cm 60 cm (i) The area of the smaller prism is 4 m. Calculate the area of the larger prism m (ii) The volume of the larger prism is 8 9 m. Calculate the volume of the smaller prism. m 9 Two boys Sifiso and lpheous were asked to paint lab stools and chairs. Sifiso painted lab stools only and lpheous painted chairs only. (a) Sifiso paints one lab stool every x minutes. Write down an expression, in terms of x, for the number of lab stools he paints in an hour. 60 x []

24 (b) lpheous takes minutes longer to paint a chair than Sifiso takes to paint a stool. Write down an expression, in terms of x, for the number of chairs lpheous paints in an hour. 60 x (c) Sifiso and lpheous paint a total of items in one hour. Form an equation in x and show that it reduces to x 98x 0 = x x 60(x + ) + 60x = x(x + ) 60x x = x + x 0x + 0 = x + x x + x 0x 0 = 0 x 98x 0 = 0 [] (d) (i) Solve the equation x 98x 0 = 0. x = 0 or x =.09 [4] (ii) Hence find the number of chairs lpheous painted. 5 chairs [] 7 0 BCD is a parallelogram. D = and B =. 4 5 C B D Find (a) C, 9 C [] 9 4

25 (b) the magnitude of C,.78 (c) angle DC, 4.5º [5] (d) the area of triangle BC..5 [] v Speed (m/s) Time (secs) The diagram shows the speed-time graph of a moving object. (a) Express, in terms of v, the acceleration of the object in the first ten seconds. v 0 [] (b) Given that the total distance in the first 50 seconds is 675m, find the value of v. v = 5 [] (c) Calculate the average speed in the first 0 seconds..5m/s [] 5

26 the whole of this question on a sheet of graph paper. In a youth rally, ribbons and pins were needed for the participants. The leader decided to buy at least 0 ribbons and at least 40 pins. ribbon costs 60c and pins cost 40c each. He wanted to spend no more than E48. (a) Using x for the number of ribbons and y for the number of pins, form three inequalities to represent this information. x 0 ; y 40 60x 40 y 4800 (the total cost in cents) x y 40 (simplify the inequality) [] (b) Using a scale of cm to 0 units on both axes, draw a graph for the inequalities in (a), by shading the unwanted region y (c) Use your graph to find the maximum number of pins he could buy. x 90 pins [] [5] 6