F x 1 i explain in terms of the position of the points

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Unformatted text preview: s, is given by the equation s = 40t + 0.5at2, where a is a constant. If the ball reaches its maximum height when t = 25, find the value of a. Working: Answers: ………………………………………….. (Total 3 marks) 319. The equation kx2 – 3x + (k + 2) = 0 has two distinct real roots. Find the set of possible values of k. Working: Answers: ………………………………………….. (Total 3 marks) 221 320. The diagram shows the graph of the functions y1 and y2. y y2 2 0 x –2 y1 On the same axes sketch the graph of y1 . Indicate clearly where the x-intercepts and y2 asymptotes occur. Working: (Total 3 marks) 222 321. The function f is given by f : x a e (1 + sin πx ) , x ≥ 0. (a) Find f ′(x). Let xn be the value of x where the (n + l)th maximum or minimum point occurs, n ∈ . (i.e. x0 is the value of x where the first maximum or minimum occurs, x1 is the value of x where the second maximum or minimum occurs, etc). (b) Find xn in terms of n. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 3 marks) 322. Let f(x) = x cos 3x. (a) Use integration by parts to show that 1 1 ∫ f ( x) dx = 3 x sin 3x + 9 cos 3x + c. (3) 223 (b) Use your answer to part (a) to calculate the exact area enclosed by f(x) and the x-axis in each of the following cases. Give your answers in terms of π. (i) π 6 ≤x≤ 3π 6 (ii) 3π 5π ≤x≤ 6 6 (iii) 5π 7π ≤x≤ 6 6 (4) (c) Given that the above areas are the first three terms of an arithmetic sequence, find an π (2n + 1)π ≤x≤ expression for the total area enclosed by f(x) and the x-axis for , where 6 6 n∈ . Give your answers in terms of n and π. (4) (Total 11 marks) 323. The triangle ABC has vertices at the points A(–l, 2, 3), B(–l, 3, 5) and C(0, –1, 1). (a) Find the size of the angle θ between the vectors AB and AC . (4) (b) Hence, or otherwise, find the area of triangle ABC. (2) Let l1 be the line parallel to AB which passes through D(2, –1, 0) and l2 be the line parallel to AC which passes through E(–l, 1, 1). (c) (i) Find the equations of the lines l1 and l2 . (ii) Hence show that l1 and l2 do not intersect. (5) 224 (d) Find the shortest distance between l1 and l2. (5) (Total 16 marks) 324. Let f ( x) = x 3 ( x 2 – 1) 2 , – 1.4 ≤ x ≤ 1.4 (a) Sketch the graph of f(x). (An exact scale diagram is not required.) On your graph indicate the approximate position of (i) each zero; (ii) each maximum point; (iii) each minimum point. (4) (i) Find f′ (x) , clearly stating its domain. (ii) (b) Find the x-coordinates of the maximum and minimum points of f(x), for –1 < x < 1. (7) (c) Find the x-coordinate of the point of inflexion of f(x), where x > 0 , giving your answer correct to four decimal places. (2) (Total 13 marks) 325. Using mathematical induction, prove that dn nπ (cos x ) = cos x + , for all positive integer n 2 dx values of n. (Total 7 marks) 225 1 . Let X b...
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