HKALE Pure Maths 2001 Paper01

# 5 marks 2001 al p math 16 5 10 a n a n 1 2bn 1 a1

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Unformatted text preview: + z= If some solution (x, y, z) of 1 3 x + y + 2z = −1 satisfies ( x − p ) 2 + y 2 + z 2 = 1 , find the range of values of p . (5 marks) 2001-AL-P MATH 1−6 −5− 10. a n = a n −1 + 2bn −1 a1 = 1 and , n = 2, 3, 4, … . Let b1 = 1 bn = a n −1 + bn−1 (a) Show that for any positive integer n , (i) a n , b n > 0 and a n 2 − 2bn 2 = (−1) n ; (ii) (1 + 2 ) n = a n + bn 2 . (4 marks) (b) For n = 1, 2, 3, … , define u n = an . bn un + 2 . un +1 (i) Show that u n +1 = (ii) Show that u 2 n −1 < 2 and u 2 n > 2 . (iii) Show that u n + 2 = 3u n + 4 . 2u n + 3 { u1 , u 3 , u 5 , " } is strictly { u 2 , u 4 , u 6 , " } is strictly Hence show that the sequence increasing and the sequence decreasing. (iv) Show that the sequences { u2 , u4 , u6 , " } { u1 , u 3 , u 5 , " } and converge to the same limit. Find this limit. (11 marks) 2001-AL-P MATH 1−7 −6− Go on to the next page 11. (a) Show that θ 1 + cosθ + i sin θ = i cot . 1 − cosθ − i sin θ 2 (3 marks) (b) Let n be a positive integer. Show that all the roots of the equation ( z − 1) n + ( z + 1) n = 0 ..........(*) can be written as iα k , where α k ∈ R , k = 0, 1, ! , n − 1 . (4 marks) (c) If iα k ( k = 0, 1, ! , n − 1 ) are the roots of (*) in (b), using the relations between the roots and coefficients, show that ∑ n −1 k =0 α k 2 = n(n − 1) . (5 marks) (d) Let P0 , P1 , ! , Pn −1 be the n points in an Argand plane representing the roots of (*) in (b), and O be the origin. Q is the point representing r (cos β + i sin β ) where r ≥ 0 and β ∈ R . If d k is the distance between Pk and Q , show that ∑ n −1 k =0 dk 2 is independent of β . (3 marks) 2001-AL-P MATH 1−8 −7− 12. The position vectors of four points A , B , C and D are a = 7i + 8 j + 3k , b = 2i − 7 j + 13k , c = 17i − 6 j + 3k and d = r (2i − 4 j − 5k ) respectively, where r is a non-zero real number. (a) Show that a , b and c are linearly independent. (3 mark...
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