3U Test 6 (60 marks) 1. (3 marks) Solve: |4x − 1| > 2(x(1 − x) 7. (6 marks) Sketch the functions 2. (2 marks) Find the Range of π(π₯) without using calculus cos π₯ + sin π₯ π(π₯) = cos π₯ + sin π₯ + 2 3. (4 marks) Simplify and find the exact value: a) cos42π‘ππ!" (−3)9 " b) π‘ππ!" π₯ + π‘ππ!" # 4. (4 marks) Sketch $ a) π¦ = % − πππ !" >1 − b) π¦ = πππ !" (cos π₯) a) y = f !" (|π₯|) b) π¦ = π '(#!") c) π¦ = ln(π(π₯)) 8. (6 marks) Given &# ? % 1 π₯ Find the stationary points and π(π₯) = π₯ + a) determine their nature. 5. (2 marks) All the letters of the word πΈππΆπππ΅π π΄ππΆπΈ are arranged in a line. b) Sketch π¦ = π(π₯) Find the total number of arrangements c) What is the largest domain which contain all of the vowels in containing π₯ = ±3 for which π(π₯) alphabetical order but separated by at least has an inverse function? one consonant. 6. (3 marks) A cylinder, open at both ends, is inscribed in a sphere of constant radius R. d) Hence, sketch π¦ = π !" (π₯) e) Find the equation of the inverse function. The radius of the cylinder is r and the height is h. Find the maximum surface area of the cylinder in terms of R. 9. (3 marks) The cost of running a ship at a *! constant speed of v km/hr is 160 + "+++ dollars per hour. If the ship’s maximum speed is 16km/hr in a 1000km journey, find what the minimum cost would be. 10. (5 marks) In the diagram, ABCD is a unit square. ∠π΄πΈπΊ = ∠πΉπ»πΆ = πΌ. π΄πΈ = π and πΉπΆ = π 13. (3 marks) 3 numbers are chosen at random from the numbers 1 to 100. Calculate the probability of the three numbers is divisible by 3, 6 and 12 in any order. 14. (8 marks) A convex quadrilateral πππ π has side lengths π, π, π, π and an area of √ππππ units2. The quadrilateral has an inscribed circle as shown in the diagram below. Let π be the length of the diagonal ππ and a) Prove that π + π + ππ = 1 ∠πππ = πΌ and ∠ππ π = π½. b) Find the max area of EBFD (in exact value) 11. (3 marks) A circle centre O with radius r inscribed in βπ΄π΅πΆ. Prove that π π΄ π΅ πΆ = cos πππ ππ πππ ππ π 2 2 2 a) Prove that (π − π)% = (π − π)% 12. (8 marks) Given π₯ π(π₯) = ± (4 − π₯ % 2 , a) Find π (π₯) b) Hence, show that b) Find stationary points and determine their nature. c) Hence, by using the area of the c) Find the points which have vertical tangents. d) Find the gradient at π₯ = 0 e) Sketch the graph. (show all important points) ππ(1 − cos πΌ) = ππ(1 − cos π½) quadrilateral πππ π, show that the quadrilateral πππ π is a cyclic quadrilateral