June 2007 Question Paper

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1. Find the exact solutions to the equations(a) ln x + ln 3 = ln 6,(b) e x + 3e –x = 4.(2)(4)2. f(x) =2x+ 3x + 2–9 + 2x, x >221.2x+ 3x− 24x− 6(a) Show that f (x) = .2x−1(b) Hence, or otherwise, find f ′(x) in its simplest form.(7)(3)3. A curve C has equation y = x 2 e x .(a) Findd y , using the product rule for differentiation.dx(3)(b) Hence find the coordinates of the turning points of C.(3)2d y(c) Find2 .dx(2)(d) Determine the nature of each turning point of the curve C.(2)N26109A 2
4. f(x) = – x 3 + 3x 2 – 1.(a) Show that the equation f(x) = 0 can be rewritten as(b) Starting with x 1 = 0.6, use the iterationx =x n + 1 =⎛⎜⎝13 − x⎛ 1⎜⎝ 3⎞⎟ .⎠− x n⎞⎟⎠(2)to calculate the values of x 2 , x 3 and x 4 , giving all your answers to 4 decimal places.(c) Show that x = 0.653 is a root of f(x) = 0 correct to 3 decimal places.(2)(3)5. The functions f and g are defined byf : a ln (2x – 1), x ∈ R, x > 21,g : a2, x ∈ R, x ≠ 3.x − 3(a) Find the exact value of fg(4).(b) Find the inverse function f –1 (x), stating its domain.(c) Sketch the graph of y = |g(x)|. Indicate clearly the equation of the vertical asymptote and thecoordinates of the point at which the graph crosses the y-axis.(3)(d) Find the exact values of x for which2x − 3= 3.(2)(4)(3)N26109A 3
Page 1: Paper Reference(s)6665/01Edexcel GC
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June 2007 Question Paper

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