t 100 1 total 70 find a the sample standard

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Unformatted text preview: lower vertices are on the x-axis and its upper vertices are on the curve y = sin x, where 0 ≤ x ≤ n. (a) Write down an expression for the area of the rectangle. 107 (b) Find the maximum area of the rectangle. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 3 marks) 157. Find the values of a > 0, such that ∫ a2 a 1 dx = 0.22. 1 + x2 Working: Answers: ....…………………………………….......... (Total 3 marks) 108 158. Let f : x (a) esin x. Find f ′ (x). There is a point of inflexion on the graph of f, for 0 < x < 1. (b) Write down, but do not solve, an equation in terms of x, that would allow you to find the value of x at this point of inflexion. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 3 marks) 159. The diagram shows the graph of y = f′ (x). y y = f’(x) x Indicate, and label clearly, on the graph (a) the points where y = f(x) has minimum points; 109 (b) the points where y = f(x) has maximum points; (c) the points where y = f(x) has points of inflexion. Working: (Total 3 marks) 160. The area of the triangle shown below is 2.21 cm2. The length of the shortest side is x cm and the other two sides are 3x cm and (x + 3) cm. x 3x x+3 (a) Using the formula for the area of the triangle, write down an expression for sin θ in terms of x. (2) (b) Using the cosine rule, write down and simplify an expression for cos θ in terms of x. (2) 110 (c) (i) Using your answers to parts (a) and (b), show that, 2 3x 2 − 2 x − 3 4.42 =1− 2 2 2x 3x 2 (1) (ii) Hence find (a) the possible values of x; (2) (b) the corresponding values of θ, in radians, using your answer to part (b) above. (3) (Total 10 marks) 161. The diagram below shows the graphs of y = –x3 + 3x2 and y = g(x), where g(x) is a polynomial of degree 3. y y = –x 3 + 3x 2 g(x) A x 0 A’ (a) If g(–2) = 0, 4. g(0) = –4, g′ (–2) = 0, and g′ (0) = (0) show that g(x) = x3 + 3x2 – (5) The graph of y = – x3 + 3x2 is reflected in the y-axis, then translated using the vector – 1 to give the graph of y = h(x). – 1 111 (b) Write h(x) in the form h(x) = ax3 + bx2 + cx + d. (5) The graph of y = x3 + 3x2 – 4 is obtained by applying a composition of two transformations to the graph of y = –x3 + 3x2. (c) State the two transformations whose composition maps the graph of y = –x3 + 3x2 onto the graph of y = x3 + 3x2 – 4 and also maps point A onto point A′. (3) (Total 14 marks) 162. Let f(x) = ln | x5 – 3x2|, –0.5 < x < 2, x ≠ a, x ≠ b ; (a, b are values of x for which f(x) is not defined). (a) (i) Sketch the graph of f(x), indicating on your sketch the number of zeros of f(x). Show also the position of any asymptotes. (2) (ii) Find all the zeros of f(x), (that is, solve f(x) = 0). (3) (b) Find the exact values of a and b. (3) (c) Find f(x), and indicate clearly wh...
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