4 marks 70 solve the equation 3 sin2x cos2x for 0

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Unformatted text preview: ….......... (Total 4 marks) 76. The diagram shows part of the graph of y = 1 . The area of the shaded region is 2 units. x y 0 1 a x 54 Find the exact value of a. Working: Answers: ....…………………………………….......... (Total 4 marks) 77. The function f is given by f(x) = 1n ( x − 2) . Find the domain of the function. Working: Answers: ....…………………………………….......... (Total 4 marks) 55 78. A population of bacteria is growing at the rate of 2.3 % per minute. How long will it take for the size of the population to double? Give your answer to the nearest minute. Working: Answers: ....…………………………………….......... (Total 4 marks) 79. Let f(x) = √x, and g(x) = 2x. Solve the equation (f –1 o g)(x) = 0.25. Working: Answers: ....…………………………………….......... (Total 4 marks) 56 80. The diagrams show a circular sector of radius 10 cm and angle θ radians which is formed into a cone of slant height 10 cm. The vertical height h of the cone is equal to the radius r of its base. Find the angle θ radians. 10cm 10cm h r Working: Answers: ....…………………………………….......... (Total 4 marks) 57 81. The graph shows a normal curve for the random variable X, with mean µ and standard deviation σ. y 0 A 12 x It is known that p(X ≥ 12) = 0.1. (a) The shaded region A is the region under the curve where x ≥ 12. Write down the area of the shaded region A. (1) It is also known that p(X ≤ 8) = 0.1. (b) Find the value of µ, explaining your method in full. (5) (c) Show that σ = 1.56 to an accuracy of three significant figures. (5) (d) Find p(X ≤ 11). (5) (Total 16 marks) 58 82. The diagram shows the graph of the function f given by f(x) = A sin π x + B, 2 for 0 ≤ x ≤ 5, where A and B are constants, and x is measured in radians. y (1,3) (5, 3) 2 (0, 1) x 0 1 2 3 4 5 (3, –1) The graph includes the points (1, 3) and (5, 3), which are maximum points of the graph. (a) Write down the values of f(1) and f(5). (2) (b) Show that the period of f is 4. (2) The point (3, –1) is a minimum point of the graph. (c) Show that A = 2, and find the value of B. (5) (d) Show that f'(x) = π cos π x . 2 (4) 59 The line y = k – πx is a tangent line to the graph for 0 ≤ x ≤ 5. (e) Find (i) the point where this tangent meets the curve; (ii) the value of k. (6) (f) Solve the equation f(x) = 2 for 0 ≤ x ≤ 5. (5) (Total 24 marks) 83. (a) Find the equation of the tangent line to the curve y = ln x at the point (e, 1), and verify that the origin is on this line. (4) (b) Show that d (x ln x – x) = ln x. dx (2) (c) The diagram shows the region enclosed by the curve y = ln x, the tangent line in part (a), and the line y = 0. y 1 0 (e, 1) 1 2 x 3 Use the result of part (b) to show that the area of this region is 1 2 e – 1. (4) (Total 10 marks) 60 84. A curve has equation y = x(x – 4)2. (a) For this curve find (i) the x-intercepts; (ii) the coordinates of th...
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