Answer in the form ln x2 x2 x2 ln 2 ln 3 y y y xm

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Unformatted text preview: ation. (ii) Find the solution to the equation g(x) = f(x) (4) (Total 10 marks) 703. Let h(x) = (x – 2)sin(x – 1) for –5 ≤ x ≤ 5. The curve of h(x) is shown below. There is a minimum point at R and a maximum point at S. The curve intersects the x-axis at the points (a, 0) (1, 0) (2, 0) and (b, 0). y 4 3 2 –5 –4 –3 –2 S 1 (a, 0) –1 (b, 0) 1 –1 R 2 3 4 5 x –2 –3 –4 –5 –6 –7 (a) Find the exact value of (i) a; (ii) b. (2) 473 The regions between the curve and the x-axis are shaded for a ≤ x ≤ 2 as shown. (b) (i) Write down an expression which represents the total area of the shaded regions. (ii) Calculate this total area. (5) (c) (i) The y-coordinate of R is –0.240. Find the y-coordinate of S. (ii) Hence or otherwise, find the range of values of k for which the equation (x – 2)sin(x – 1) = k has four distinct solutions. (4) (Total 11 marks) 704. Reaction times of human beings are normally distributed with a mean of 0.76 seconds and a standard deviation of 0.06 seconds. (a) The graph below is that of the standard normal curve. The shaded area represents the probability that the reaction time of a person chosen at random is between 0.70 and 0.79 seconds. a 0b (i) Write down the value of a and of b. (ii) Calculate the probability that the reaction time of a person chosen at random is (a) greater than 0.70 seconds; (b) between 0.70 and 0.79 seconds. (6) 474 Three percent (3 %) of the population have a reaction time less than c seconds. (b) (i) Represent this information on a diagram similar to the one above. Indicate clearly the area representing 3 %. (ii) Find c. (4) (Total 10 marks) 705. Let f(x) = (a) 1 . 1+ x2 Write down the equation of the horizontal asymptote of the graph of f. (1) (b) Find f′(x). (3) (c) The second derivative is given by f″(x) = 6x 2 − 2 . (1 + x 2 ) 3 Let A be the point on the curve of f where the gradient of the tangent is a maximum. Find the x-coordinate of A. (4) 475 (d) Let R be the region under the graph of f, between x = – 1 1 and x = , 2 2 as shaded in the diagram below y 2 1 R –1 –1 2 x 1 1 2 –1 Write down the definite integral which represents the area of R. (2) (Total 10 marks) 706. Let y = g(x) be a function of x for 1 ≤ x ≤ 7. The graph of g has an inflexion point at P, and a minimum point at M. Partial sketches of the curves of g′ and g″ are shown below. g’ ( x ) g’ ’ (x) 6 6 5 5 4 4 3 3 2 2 1 1 0 1 2 3 4 –1 5 6 7 8 x 0 1 2 3 4 –1 –2 8 x –4 –5 7 –3 –4 6 –2 –3 5 –5 –6 –6 y = g’ (x) y = g’’ (x) 476 Use the above information to answer the following. (a) Write down the x-coordinate of P, and justify your answer. (2) (b) Write down the x-coordinate of M, and justify your answer. (2) (c) Given that g (4) = 0, sketch the graph of g. On the sketch, mark the points P and M. (4) (Total 8 marks) 707. Consider f(x) = x3 – 2x2 – 5x + k. Find the value of k if(x + 2) is a factor of f(x). Working: Answer: ……………………...
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