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Unformatted text preview: quare. Q A B P R D (a) S (i) Show that PQ = 2 2 cm. (ii) C Find the area of PQRS. (3) The midpoints W, X, Y, Z of the sides of PQRS are now joined to form a third square as shown. A B Q X W P R Y Z D (b) S C (i) Write down the area of the third square, WXYZ. (ii) Show that the areas of ABCD, PQRS, and WXYZ form a geometric sequence. Find the common ratio of this sequence. (3) 315 The process of forming smaller and smaller squares (by joining the midpoints) is continued indefinitely. (i) Find the area of the 11th square. (ii) (c) Calculate the sum of the areas of all the squares. (4) (Total 10 marks) 465. The diagram below shows a sketch of the graph of the function y = sin(ex) where –1 ≤ x ≤ 2, and x is in radians. The graph cuts the y-axis at A, and the x-axis at C and D. It has a maximum point at B. y B A –1 (a) 0 1 C D2x Find the coordinates of A. (2) (b) The coordinates of C may be written as (ln k, 0). Find the exact value of k. (2) (c) (i) Write down the y-coordinate of B. (ii) Find (iii) Hence, show that at B, x = ln dy . dx π . 2 (6) 316 (d) (i) Write down the integral which represents the shaded area. (ii) Evaluate this integral. (5) (e) (i) Copy the above diagram into your answer booklet. (There is no need to copy the shading.) On your diagram, sketch the graph of y = x3. (ii) The two graphs intersect at the point P. Find the x-coordinate of P. (3) (Total 18 marks) 466. The following diagram shows the point O with coordinates (0, 0), the point A with position vector a = 12i + 5j, and the point B with position vector b = 6i + 8j. The angle between (OA) and (OB) is θ. y C B A x O Find (i) | a |; (ii) a unit vector in the direction of b; (iii) the exact value of cosθ in the form p , where, p, q ∈ q . (Total 6 marks) 317 467. In this question, s represents displacement in metres, and t represents time in seconds. (a) ds = 30 – at, where a is a dt constant. Given that s = 0 when t = 0, find an expression for s in terms of a and t. The velocity v ms–1 of a moving body may be written as v = (5) Trains approaching a station start to slow down when they pass a signal which is 200 m from the station. (b) The velocity of Train 1 t seconds after passing the signal is given by v = 30 – 5t. (i) Write down its velocity as it passes the signal. (ii) Show that it will stop before reaching the station. (5) (c) Train 2 slows down so that it stops at the station. Its velocity is given by ds v= = 30 – at, where a is a constant. dt (i) Find, in terms of a, the time taken to stop. (ii) Use your solutions to parts (a) and (c)(i) to find the value of a. (5) (Total 15 marks) 468. In a country called Tallopia, the height of adults is normally distributed with a mean of 187.5 cm and a standard deviation of 9.5 cm. (a) What percentage of adults in Tallopia have a height greater than 197 cm? (3) (b) A standard doorway in Tallopia is designed so that 99 % of adults have a space of at least 17 cm over their heads when going through a doorway. Find t...
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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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