Mathematics Department

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1 Mathematics Department Grade 10 Placement Test Name This assessment will help the maths department to make a provisional placement. The final placement will be determined after a suitable period of in-class assessments by the classroom teacher. Instructions: Answer all the questions in the spaces provided. Show all your working. Calculators allowed Unless otherwise stated in the question, all numerical answers must be given exactly or to three significant figures. Formulas - quadratic formula: x =!!±!!!!!"!! sin α =!""!#$%&!!"#$%&'(% cos α =!"#!$%&'!!"#$%&'(% tan α =!""!#$%&!"#!$%&' sin rule:!"#!! =!"#!! =!"#!! cos rule: a! = b! + c! 2bc cos A or cos A =!!!!!!!!!!"

2 1) Factorise each of the following quadratic expressions: a) 7x! 42x b) 25 9x! c) x! + 9x ) Solve: a) x 3 x + 7 = 0 b) x! + 5x 36 = 0 c) 3x! 5x 2 = 0 3) Solve the simultaneous equations: 4x 2y = 16 3x + y = 4 x =, y =

3 4) a) Express as a single fractions:!!!!!!!! b) Hence or otherwise, solve!!!!!!!! =!! 5) A snowball is rolled downhill, picking up snow as it goes. Ten such balls were rolled different distances and then weighed. The scatter graph shows the data collected. A line of best fit has been drawn. a) Work out the equation of the line of best fit. b) Use the equation to estimate the weight of a snowball that has been rolled 6.5 metres.

4 6) Simplify the following, leaving your answers with positive indices: a) x! x! b)!!!!! c)!!!! d) x!!!!" e) 2x!!! f)!!!!!! g) x! x!!! x!

5 7) Solve the following equations: a)!"!!! = x b) x + 3 x 4 = x 2 x + 5

6 8) The four diagrams below show the graphs of four different straight lines, all drawn to the same scale. Each diagram is numbered and c is a positive constant. y y c Number 1 Number 3 c 0 x 0 x y c y c Number 2 Number 4 0 x 0 x In the table below, write the number of the diagram whose straight line corresponds to the equation in the table. Equation Diagram number y = c y = x + c y = 3 x + c y = 1 3 x + c

7 9) The diagram shows the graph of y = x! 2x 8. The graph crosses the x- axis at the point A, and has a vertex at B. y A O x B a) Factorise x! 2x 8. b) Write down the coordinates of each of these points (i) A; (ii) B.

8 10) f x = x! 4x + 3 and g x = 2x 1. a) Solve f x = 0. b) Find g!! (x). c) Solve f x = g(x), giving your answers, simplified, in surd form. d) Find the value of g f 2. e) Find f g(x). Simplify your answer.

9 11) The diagram shows a point P, 12.3 m from the base of a building of height h m. The angle measured to the top of the building from point P is 63. h m P a) Calculate the height h of the building to the nearest metre. Consider the formula h = 4.9t 2, where h is the height of the building and t is the time in seconds to fall to the ground from the top of the building. b) Calculate, to the nearest tenth of a second, how long it would take for a stone to fall from the top of a building 30 m high. 12) A man walks a certain distance due North and then the same distance plus a further 7km due East. The final direct distance he walked from the starting point is 17km. Find the distance he walked due North.

10 13) The diagram shows the position of four cities in Africa, Windhoek (W), Johannesburg (J), Harare (H) and Lusaka (L). WL = 1400 km and WH = 1600 km. Angle LWH = 13 o, angle HWJ = 36 o and angle WJH = 95 o. a) Calculate the distance LH. b) Calculate the distance WJ. c) Calculate the area of the quadrilateral WJHL. d) The bearing of Lusaka from Windhoek is 060 o. Calculate the bearing of (i) Harare to Windhoek, (ii) Windhoek from Johannesburg. e) On the map the distance between Windhoek and Harare is 8 cm. Calculate the scale of the map in the form 1:n.

11 14) Workmen dig a trench on the level ground. The cross-section of the trench in a trapezium ABCD with parallel sides of length 1.1m and 1.4m and a vertical height of 0.7m. a) The trench is 500m long. Calculate the volume of soil removed. b) One cubic metre of soil has mass of 4.8 tonnes. Calculate the mass of soil removed, giving your answer in tonnes and in scientific notation. c) Change your answer to part b) to grams giving your answer in scientific notation. (1 tonne = 1000 kg) d) The workmen put in a cylindrical pipe, radius 0.2m and length 500m, along the bottom of the trench, as shown in the diagram. Calculate the volume of the cylindrical pipe. e) The trench is then refilled with soil. Calculate the volume of soil put back as a percentage of the amount removed. Round your answer to the nearest %.

12 15) Give your answer to this question as fractions. a) The probability that it rains today is!.! If it rains today, the probability that it rains tomorrow is!.! If it does not rain today, the probability that it will rain tomorrow is!.! The tree diagram below shows this information. (i) Write down the value of s, t and u. s =, t =, u = (ii) Calculate the probability that it will rain on both days. (iii) Calculate the probability that it will not rain tomorrow. b) Each time Christina throws a ball at a target, the probability that she hits is!!. She throws the ball three times. Find the probability that she hits the target (i) three times, (ii) at least once.

13 c) Each time Edwardo throws the ball at the target, the probability that he hits is!!. He throws the ball until he hits the target. Find the probability that he first hit the target with his (i) 4 th throw, (ii) n th throw.

14 16) OPMQ is a parallelogram and O is the origin. OP = p and OQ = q L is on PQ so that PL : LQ = 2 : 1. Find the following vectors in terms of p and q. Write your answers in their simplest form. a) PQ b) PL c) ML d) OL N is a point on OQ so that NL is parallel to LM. e) Find the ratio ON:NQ.

15 17) a) The quiz scores of a class of n students are shown in the table. Quiz score Frequency (number of students) 9 3 a 5 The mean score is 7.2. Find (i) a, (ii) n, (iii) the median score.

16 b) 200 students take a mathematics test. The cumulative frequency diagram shows the results. Write down (i) the median mark, (ii) the lower quartile, (iii) the upper quartile, (iv) the inter-quartile range, (v) the lowest possible mark scored by the top 40 students, (vi) the number of students scoring more than 25 marks.

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0580/04 058/04 www.xtremepapers.com Paper 4 (Extended) October/November 006 Additional