Preliminary Mathematics

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1 NORTH SYDNEY GIRLS HIGH SCHOOL 2011 YEARLY EXAMINATION Preliminary Mathematics General Instructions Reading Time 5 minutes Working Time 2 hours Write using black or blue pen Board-approved calculators may be used All necessary working should be shown in every question. Total Marks 84 Attempt Questions 1 7 All questions are of equal value. At the end of the examination, place your solution booklets in order and put this question paper on top. Submit ONE bundle. The bundle will be separated for marking so please ensure your name is written on each solution booklet. Student Name: Teacher: QUESTION MARK 1 /12 2 /12 3 /12 4 /12 5 /12 6 /12 7 /12 TOTAL /84

2 Total Marks 84 Attempt Questions 1 7 All questions are of equal value Begin each question in a SEPARATE writing booklet. Extra writing booklets are available. Question 1 (12 Marks) Use a SEPARATE writing booklet. (a) Evaluate cos92 9 tan130 correct to three significant figures. 2 (b) 2 3 Rationalise the denominator of Express your answer in simplest form. (c) Simplify a 1 b. 2 b 1 a (d) Solve 2x x y (e) Simplify x y. 2 (f) Solve 2 x 2x. 2 NSGHS 2011 Preliminary Mathematics Examination 2

3 Question 2 (12 Marks) Use a SEPARATE writing booklet. In the diagram below, A and C are the points 4, 2 and 2, 6 respectively. Lines AB and BC are perpendicular and intersect at B. y C (2, 6) NOT TO SCALE M. B O x A (-4, -2) (a) Find the equation of the line AB, given that it passes through the origin. 1 (b) The line BC is perpendicular to AB. Find the equation of line BC. 2 (c) Hence, or otherwise, find the coordinates of B. 2 (d) Find the coordinates of M, the midpoint of AC. 1 (e) Find the distance AM and hence show that AM = MB. 2 (f) Find the coordinates of point D that makes quadrilateral ABCD a rectangle. 1 (g) Explain why a circle can be drawn through A, B, C and D, with its centre at M. 1 (h) Write down the equation of this circle. 2 NSGHS 2011 Preliminary Mathematics Examination 3

4 Question 3 (12 Marks) Use a SEPARATE writing booklet. (a) The diagram below showss an equilateral triangle, ABC, with w AP = BQQ = CR. A P R B Q C (i) (ii) Prove that APR and BQP are congruent. Show that BPQ APR (iii) Hence prove that PQR is equilateral. 2 (b) The diagram below showss a lighthouse at Q and 40 km fromm P. The bearing of the lighthouse, at Q, from P is 115 T. R is a headland, 30 km from Q and on a bearing of 220 T from Q. (i) (ii) Find the size of PQR, correct to the nearest degree. Find the distance from f P to R, correct to 2 decimal places. p 1 2 (iii) Find the bearing of R from P.. 2 NSGHS 2011 Preliminary Mathematics Examination 4

5 Question 4 (12 Marks) Use a SEPARATE writing booklet. (a) Differentiate with respect to x: (i) 3 x 5 2 2x 1 (ii) (iii) (iv) 1 3x 2 2 x 2 2x x 2 4 x 3 2 (b) Given the function f (x) x 2 3x 3, find the values of b for which f (b) f (b). 2 (c) (i) Find dy dx where x x y. 2 x (ii) For x 0, explain why every tangent has a positive gradient. 1 Question 5 (12 Marks) Use a SEPARATE writing booklet. (a) Simplify 2 1 cot 90 sin (b) Find the exact value of sec (c) (d) Show that Solve 3 2sin 3 cos sin cos cot sin. 2 sin for (e) A function is defined as f x x 1 x. 2 (i) Simplify f x f x 2. 2 (ii) Determine whether f x is even, odd or neither. 1 NSGHS 2011 Preliminary Mathematics Examination 5

6 Question 6 (12 Marks) Use a SEPARATE writing booklet. (a) (i) Find the equation of the line, L, that passes through the point of intersection, A, 3 of the lines. l 1 :5x y 6 0 and l 2 :2x 3y 1 0 and also passes through the point B 3, 2. (ii) Find the perpendicular distance from the origin to L. 2 Leave your answer in exact form. (b) Consider the function f (x) x 2 4x 3. (i) State the domain of the function f (x). 1 (ii) Show that f (x) 2x 5x 3 4x 3. 3 (iii) Find the equation of the tangent to f (x) at the point P 3, (iv) State the values of x where the tangents to f ( x ) are horizontal. 1 NSGHS 2011 Preliminary Mathematics Examination 6

7 Question 7 (12 Marks) Use a SEPARATE writing booklet. (a) On a number plane, shade the regionn satisfying the following inequalities. 3 y x 2 1 and y 16 x 2. (b) The diagram below showss the graph of y f (x). y a O b c d e x Copy this graph into yourr answer booklet. (i) Find x,, where dy 0. dx 2 (ii) Sketch the graph y f ( x). 3 (c) The diagram below showss ABCwith AC = 7, BC = 10 andd AB = 6. ABC x and ACB y. (i) Express the size of BAC inn terms of x and y. 1 (ii) Hence, show that sin x sin y sin x y 3 End of paper NSGHS 2011 Preliminary Mathematics Examination 7

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