Prove that dn n cos x cos x for all

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Unformatted text preview: ….......... (Total 4 marks) 234 340. The diagram below shows a sector AOB of a circle of radius 15 cm and centre O. The angle θ at the centre of the circle is 2 radians. diagram not to scale B A O (a) Calculate the area of the sector AOB. (b) Calculate the area of the shaded region. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks) 235 341. Solve the equation log9 81 + log9 1 + log9 3 = log9 x. 9 Working: Answers: ………………………………………….. (Total 4 marks) 342. The following Venn diagram shows the universal set U and the sets A and B. U B A (a) Shade the area in the diagram which represents the set B ∩ A'. n(U) = 100, n(A) = 30, n(B) = 50, n(A ∪ B) = 65. (b) Find n(B ∩ A'). 236 (c) An element is selected at random from U. What is the probability that this element is in B ∩ A′? Working: Answers: (c) ………………………………………….. (Total 4 marks) 343. Consider the function f(x) = k sin x + 3x, where k is a constant. (a) Find f ′(x). (b) When x = π 3 , the gradient of the curve of f(x) is 8. Find the value of k. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks) 237 344. The diagram below shows the graph of y = x sin x , for 0 ≤ x < m, and 0 ≤ y < n, where x is in 3 radians and m and n are integers. y n n–1 0 m–1 m x Find the value of (a) m; (b) n. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks) 238 345. Given that f(x) = 2e3x, find the inverse function f –1(x). Working: Answers: ………………………………………….. (Total 4 marks) 4 4 4 346. Consider the binomial expansion (1 + x) 4 =1 + x + x 2 x 3 + x 4 . 1 2 3 (a) 4 4 4 By substituting x = 1 into both sides, or otherwise, evaluate + + . 1 2 3 239 (b) 9 9 9 9 9 9 9 9 Evaluate + + + + + + + . 1 2 3 4 5 6 7 8 Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks) 2x x + 1 347. The vectors x – 3 and 5 are perpendicular for two values of x. (a) Write down the quadratic equation which the two values of x must satisfy. (b) Find the two values of x. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 4 marks) 240 348. The diagrams below show two triangles both satisfying the conditions ˆ AB = 20 cm, AC = 17 cm, ABC = 50°. diagrams not to scale Triangle 1 Triangle 2 A A B C (a) C ˆ Calculate the size of ACB in Triangle 2. (b) B Calculate the area of Triangle 1. Working: Answers: (a) ………………………………………….. (b) .......................................................
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