# Two arcs of circles the triangle abc is a right

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Unformatted text preview: y. Calculate an unbiased estimate of (a) the mean time taken to drive to school; (b) the variance of the time taken to drive to school. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks) 552. The diagram below shows the graph of y1 = f(x). y2 x 367 On the axes below, sketch the graph of y2 = f′ (x). y2 x (Total 6 marks) 553. The function f is defined by f(x) = (a) (i) Show that f′ (x) = (ii) x2 , for x > 0. 2x 2 x – x 2 ln 2 2x Obtain an expression for f″(x), simplifying your answer as far as possible. (5) (i) Find the exact value of x satisfying the equation f′(x) = 0 (ii) (b) Show that this value gives a maximum value for f(x). (4) (c) Find the x-coordinates of the two points of inflexion on the graph of f. (3) (Total 12 marks) 368 554. The variables x, y, z satisfy the simultaneous equations x + 2y + z = k 2x + y + 4z = 6 x – 4y + 5z = 9 where k is a constant. (a) (i) Show that these equations do not have a unique solution. (ii) Find the value of k for which the equations are consistent (that is, they can be solved). (6) (b) For this value of k, find the general solution of these equations. (3) (Total 9 marks) 555. (a) Prove, using mathematical induction, that for a positive integer n, (cosθ + i sinθ)n = cosnθ + i sinnθ where i2 = –1. (5) (b) The complex number z is defined by z = cosθ + i sinθ. 1 = cos(–θ) + i sin(–θ). z (i) Show that (ii) Deduce that zn + z–n = 2cos nθ. (5) (c) (i) (ii) Find the binomial expansion of (z + z–l)5. 1 (a cos5θ + b cos3θ + c cosθ), 16 where a, b, c are positive integers to be found. Hence show that cos5θ = (5) (Total 15 marks) 369 556. A business man spends X hours on the telephone during the day. The probability density function of X is given by 1 3 (8 x – x ), for 0 ≤ x ≤ 2 f(x) = 12 0, otherwise. (a) (i) Write down an integral whose value is E(X). (ii) Hence evaluate E(X). (3) (b) (i) Show that the median, m, of X satisfies the equation m4 – 16m2 + 24 = 0. (ii) Hence evaluate m. (5) (c) Evaluate the mode of X. (3) (Total 11 marks) π 557. The function f with domain 0, is defined by f(x) = cos x + 2 3 sin x. This function may also be expressed in the form R cos(x – α) where R > 0 and 0 < α < (a) π . 2 Find the exact value of R and of α. (3) (b) (i) Find the range of the function f. (ii) State, giving a reason, whether or not the inverse function of f exists. (5) (c) Find the exact value of x satisfying the equation f(x) = 2. (3) 370 (d) Using the result ∫ sec xdx = lnsec x + tan x+ C, where C is a constant, show that π 2 0 ∫ dx 1 = ln(3 + 2 3 ). f ( x) 2 (5) (Total 16 marks) 558. Give all numerical answers to this question correct to two decimal places. A radar records the speed, v kilometres per hour, of cars on a road. The speed of these cars is normally distributed. The results for 1000 cars are recorded in the following table. Speed 40...
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## This note was uploaded on 06/24/2013 for the course HISTORY exam taught by Professor Apple during the Fall '13 term at Berlin Brothersvalley Shs.

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