Spun it cannot land on the lines between the sectors

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Unformatted text preview: sinθ). Working: Answer: …………………………………………........ (Total 6 marks) 720. Find the total area of the two regions enclosed by the curve y = x3 – 3x2 – 9x +27 and the line y = x + 3. Working: Answer: …………………………………………........ (Total 6 marks) 484 721. Find the range of values of m such that for all x m(x + 1) ≤ x2. Working: Answer: …………………………………………........ (Total 6 marks) 722. Find the equation of the normal to the curve x3 + y3 – 9xy = 0 at the point (2, 4). Working: Answer: …………………………………………........ (Total 6 marks) 485 723. Using the substitution 2x = sinθ, or otherwise, find ∫ 1 − 4 x 2 dx . Working: Answer: …………………………………………........ (Total 6 marks) 724. A closed cylindrical can has a volume of 500 cm3. The height of the can is h cm and the radius of the base is r cm. (a) Find an expression for the total surface area A of the can, in terms of r. (b) Given that there is a minimum value of A for r > 0, find this value of r. Working: Answers: (a) ………………………………………….. (b) ………………………………………….. (Total 6 marks) 486 725. The following diagram shows the lines x – 2y – 4 = 0, x + y = 5 and the point P(1, 1). A line is drawn from P to intersect with x – 2y – 4 = 0 at Q, and with x + y = 5 at R, so that P is the midpoint of [QR]. y 10 8 6 4 2 P(1, 1) × –10 –8 –6 –4 –2 0 2 4 6 8 10 x –2 –4 –6 –8 –10 Find the exact coordinates of Q and of R. Working: Answer: …………………………………………........ (Total 6 marks) 487 726. Consider the complex number z = cosθ + i sinθ. (a) Using De Moivre’s theorem show that zn + 1 = 2cosnθ. zn (2) 4 (b) 1 By expanding z + show that z cos4θ = 1 (cos4θ + 4cos2θ + 3). 8 (4) (c) Let g(a) = ∫ a 0 cos 4 θdθ . (i) Find g(a). (ii) Solve g(a) = 1 (5) (Total 11 marks) 727. A line l1 has equation (a) x+2 y z −9 == . 3 1 −2 Let M be a point on l1 with parameter µ. Express the coordinates of M in terms of µ. (1) (b) The line l2 is parallel to l1 and passes through P(4, 0, –3). (i) Write down an equation for l2. (ii) Express PM in terms of µ. (4) 488 (c) The vector PM is perpendicular to l1. (i) Find the value of µ. (ii) Find the distance between l1 and l2. (5) (d) The plane π1 contains l1 and l2. Find an equation for π1, giving your answer in the form Ax + By + Cz = D. (4) (e) The plane π2 has equation x – 5y – z = –11. Verify that l1 is the line of intersection of the planes π1 and π2. (2) (Total 16 marks) 728. Ian and Karl have been chosen to represent their countries in the Olympic discus throw. Assume that the distance thrown by each athlete is normally distributed. The mean distance thrown by Ian in the past year was 60.33 m with a standard deviation of 1.95 m. (a) In the past year, 80 % of Ian’s throws have been longer than x metres....
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