Vector n is perpendicular to ab 4 iii by

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Unformatted text preview: .. (Total 3 marks) 377. (a) Solve the following system of linear equations x + 3y – 2z = –6 2x + y + 3z = 7 3x – y + z = 6. (3) (b) Find the vector v = (i + 3j – 2k) × (2i + j + 3k). (3) (c) If a = i + 3j – 2k, b = 2i + j + 3k and u = ma + nb where m, n are scalars, and u ≠ 0 , show that v is perpendicular to u for all m and n. (3) (d) The line l lies in the plane 3x – y + z = 6 , passes through the point (1, –1, 2) and is perpendicular to v. Find the equation of l. (4) (Total 13 marks) 260 378. Let y = sin(kx) – kx cos(kx), where k is a constant. Show that dy 2 = k x sin(kx). dx (Total 3 marks) 379. A particle is moving along a straight line so that t seconds after passing through a fixed point O on the line, its velocity v (t) m s–1 is given by π v (t ) = t sin t . 3 (a) Find the values of t for which v(t) = 0, given that 0 ≤ t ≤ 6. (3) (b) (i) Write down a mathematical expression for the total distance travelled by the particle in the first six seconds after passing through O. (ii) Find this distance. (4) (Total 7 marks) 380. The line segment [AB] has length l, gradient m, (0 < m < 1), and passes through the point (0, 1). It meets the x-axis at A and the line y = x at B, as shown in the diagram. y y=x B (0,1) A (a) O x Find the coordinates of A and B in terms of m. (4) 261 (b) Show that l 2 = m2 +1 . m 2 (1 – m) 2 (3) (c) Sketch the graph of y = x 2 +1 x 2 (1 – x) 2 , for x ≠ 0, x ≠ 1 , indicating any asymptotes and the coordinates of any maximum or minimum points. (4) (d) Find the value of m for which l is a minimum, and find this minimum value of l. (2) (Total 13 marks) 381. Consider the sequence {an} = {1, 1, 2, 3, 5, 8, 13, ...} where a1 = a2 = 1 and an+1 = an + an–1 for all integers n ≥ 2. 1 1 Given the matrix Q = 1 0 , use the principle of mathematical induction a n +1 a n to prove that Qn = a n a n – 1 for all integers n ≥ 2. (Total 7 marks) 382. (a) Express z5 – 1 as a product of two factors, one of which is linear. (2) (b) Find the zeros of z5 – 1, giving your answers in the form r(cos θ + i sin θ) where r > 0 and –π < 6 ≤ π. (3) (c) Express z4 + z3 + z2 + z + 1 as a product of two real quadratic factors. (5) (Total 10 marks) 262 383. Two women, Ann and Bridget, play a game in which they take it in turns to throw an unbiased six-sided die. The first woman to throw a '6' wins the game. Ann is the first to throw. (a) Find the probability that (i) Bridget wins on her first throw; (ii) Ann wins on her second throw; (iii) Ann wins on her nth throw. (6) (b) 1 25 Let p be the probability that Ann wins the game. Show that p = + p. 6 36 (4) (c) Find the probability that Bridget wins the game. (2) (d) Suppose that the game is played six times. Find the probability that Ann wins more games than Bridget. (5) (Total 17 marks) 384. Roger uses public transport to go to school each morning. The time he waits each morning for the transport is normally distributed with a mean of 15 minute...
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