Where i2 1 5 b the complex number z is defined by

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Unformatted text preview: A, B and C which are three vertices of a parallelogram ABCD. The 2 point A has position vector . 2 y 10 C B 5 A 5 (a) 10 x Write down the position vector of B and of C. (2) (b) d The position vector of point D is . Find d. 4 (3) (c) Find BD . (1) The line L passes through B and D. (d) (i) Write down a vector equation of L in the form x = y (ii) – 1 +t 7 m . n Find the value of t at point B. (3) 391 (e) Let P be the point (7, 5). By finding the value of t at P, show that P lies on the line L. (3) (f) Show that CP is perpendicular to BD . (4) (Total 16 marks) 582. The diagram below shows a circle, centre O, with a radius 12 cm. The chord AB subtends at an angle of 75° at the centre. The tangents to the circle at A and at B meet at P. A 12 cm P diagram not to scale O 75º B (a) Using the cosine rule, show that the length of AB is 12 2(1 – cos 75°) . (2) (b) Find the length of BP. (3) (c) Hence find (i) the area of triangle OBP; (ii) the area of triangle ABP. (4) (d) Find the area of sector OAB. (2) 392 (e) Find the area of the shaded region. (2) (Total 13 marks) 583. The diagrams below show the first four squares in a sequence of squares which are subdivided 1 in half. The area of the shaded square A is . 4 A A B Diagram 1 Diagram 2 A A B B C Diagram 3 (a) C Diagram 4 (i) Find the area of square B and of square C. (ii) Show that the areas of squares A, B and C are in geometric progression. (iii) Write down the common ratio of the progression. (5) (b) (i) Find the total area shaded in diagram 2. (ii) Find the total area shaded in the 8th diagram of this sequence. Give your answer correct to six significant figures. (4) 393 (c) The dividing and shading process illustrated is continued indefinitely. Find the total area shaded. (2) (Total 11 marks) 584. Consider the function f(x) = 1 + e–2x. (i) Find f′(x). (ii) (a) Explain briefly how this shows that f(x) is a decreasing function for all values of x (i.e. that f(x) always decreases in value as x increases). (2) Let P be the point on the graph of f where x = – (b) 1 . 2 Find an expression in terms of e for (i) the y-coordinate of P; (ii) the gradient of the tangent to the curve at P. (2) (c) Find the equation of the tangent to the curve at P, giving your answer in the form y = ax + b. (3) (d) (i) Sketch the curve of f for –1 ≤ x ≤ 2. (ii) Draw the tangent at x = – (iii) Shade the area enclosed by the curve, the tangent and the y-axis. (iv) Find this area. 1 . 2 (7) (Total 14 marks) 394 585. Note: Radians are used throughout this question. A mass is suspended from the ceiling on a spring. It is pulled down to point P and then released. It oscillates up and down. diagram not to scale P Its distance, s cm, from the ceiling, is modelled by the function s = 48 + 10cos2πt where t is the time in seconds from release. (a) (i) What is the distance of the point P from the ceiling? (ii) How long is it until the mass is next at P? (5) (b) ds . dt (i) Find (ii) Where is the mass when the velocity...
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