An eulerian trail for the graph g starting with

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Unformatted text preview: he height of a standard doorway in Tallopia. Give your answer to the nearest cm. (4) (Total 7 marks) 318 469. Let g(x) = x4 – 2x3 + x2 – 2. (a) Solve g(x) = 0. (2) Let f(x) = 2x 3 + 1 . A part of the graph of f(x) is shown below. g ( x) y C A (b) 0 B x The graph has vertical asymptotes with equations x = a and x = b where a < b. Write down the values of (i) a; (ii) b. (2) (c) The graph has a horizontal asymptote with equation y = l. Explain why the value of f(x) approaches 1 as x becomes very large. (2) (d) The graph intersects the x-axis at the points A and B. Write down the exact value of the x-coordinate at (i) A; (ii) B. (2) 319 (e) The curve intersects the y-axis at C. Use the graph to explain why the values of f′(x) and f′′(x) are zero at C. (2) (Total 10 marks) 470. The diagram below shows the shaded region R enclosed by the graph of y = 2x 1 + x 2 , the x-axis, and the vertical line x = k. y y = 2x 1+x 2 R k (a) Find x dy . dx (3) (b) Using the substitution u = 1 + x2 or otherwise, show that ∫ 3 2 x 1 + x 2 dx = 2 ( 1 + x2 ) 2 + c. 3 (3) (c) Given that the area of R equals 1, find the value of k. (3) (Total 9 marks) 320 1 471. Consider the function h(x) = x 5 . (i) Find the equation of the tangent to the graph of h at the point where x = a, (a ≠ 0). Write the equation in the form y = mx + c. (ii) Show that this tangent intersects the x-axis at the point (–4a, 0). (Total 5 marks) 472. When the polynomial x4 + ax + 3 is divided by (x – 1), the remainder is 8. Find the value of a. Working: Answer: ………………………………………….. (Total 6 marks) 321 473. The graph of the function f(x) = 2x3 – 3x2 + x + 1 is translated to its image, g(x), by the vector 1 . Write g(x) in the form g(x) = ax3 + bx2 +cx + d. – 1 Working: Answer: ………………………………………….. (Total 6 marks) 8 1 474. Find the coefficient of x3 in the binomial expansion of 1 – x . 2 Working: Answer: ………………………………………….. (Total 6 marks) 322 475. Find the equations of all the asymptotes of the graph of y = x 2 – 5x – 4 . x 2 – 5x + 4 Working: Answers: ………………………………………….. ………………………………………….. (Total 6 marks) 476. An integer is chosen at random from the first one thousand positive integers. Find the probability that the integer chosen is (a) a multiple of 4; (b) a multiple of both 4 and 6. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks) 323 ∑ ln (2 ), giving the answer in the form a ln 2, where a ∈ 50 477. Find r . r =1 Working: Answer: ………………………………………….. (Total 6 marks) 478. The functions f(x) and g(x) are given by f(x) = (f ° g)(x) is defined for x ∈ x – 2 and g(x) = x2 + x. The function , except for the interval ] a, b [. (a) Calculate the value of a and of b. (b) Find the r...
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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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