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Unformatted text preview: ng: Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks) 431. A rectangle is drawn so that its lower vertices are on the x-axis and its upper vertices are on the 2 curve y = e – x . The area of this rectangle is denoted by A. (a) Write down an expression for A in terms of x. (b) Find the maximum value of A. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks) 294 432. The diagram below shows the graph of y1 = f(x), 0 ″ x ″ 4. y 0 1 2 3 4 x x On the axes below, sketch the graph of y2 = ∫ f (t )dt , marking clearly the points of inflexion. 0 y 0 1 2 3 4 x (Total 6 marks) 433. The points A, B, C, D have the following coordinates A : (1, 3, 1) B : (1, 2, 4) C : (2, 3, 6) D : (5, – 2, 1). (a) (i) Evaluate the vector product AB × AC , giving your answer in terms of the unit vectors i, j, k. (ii) Find the area of the triangle ABC. (6) 295 The plane containing the points A, B, C is denoted by Π and the line passing through D perpendicular to Π is denoted by L. The point of intersection of L and Π is denoted by P. (i) Find the cartesian equation of Π. (ii) (b) Find the cartesian equation of L. (5) (c) Determine the coordinates of P. (3) (d) Find the perpendicular distance of D from Π. (2) (Total 16 marks) 434. The function y = f(x) satisfies the differential equation 2x 2 (a) (i) dy = x2 + y2 dx ( x > 0) Using the substitution y = vx, show that 2x dv = ( v – 1) 2 dx (ii) Hence show that the solution of the original differential equation is 2x y=x– , where c is an arbitrary constant. (ln x + c ) (iii) Find the value of c given that y = 2 when x = 1. (7) 296 (b) The graph of y = f(x) is shown below. The graph crosses the x-axis at A. y 0 1 5 A (i) Write down the equation of the vertical asymptote, (ii) Find the exact value of the x-coordinate of the point A. (iii) x Find the area of the shaded region. (5) (Total 12 marks) 435. (a) Find the determinant of the matrix 1 1 2 1 2 1 2 1 5 (1) (b) Find the value of λ for which the following system of equations can be solved. 1 1 2 x 3 1 2 1 y = 4 2 1 5 z λ (3) (c) For this value of λ, find the general solution to the system of equations. (3) (Total 7 marks) 297 436. (a) Prove using mathematical induction that n 2 2 n – 1 2 1 , for all positive integer values of n. = 0 1 0 1 (5) (b) Determine whether or not this result is true for n = –1. (2) (Total 7 marks) 437. Two children, Alan and Belle, each throw two fair cubical dice simultaneously. The score for each child is the sum of the two numbers shown on their respective dice. (a) (i) Calculate the probability that Alan obtains a score of 9. (ii) Calculate the probability that Alan and Belle both obtain a score of 9. (2) (b) (i) Calculate the probability that Alan and Belle obtain the same score, (ii) Deduce the probability that Alan’s score exceeds...
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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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