HKALE Pure Maths 2001 Paper01

7 a rotation which rotates any vector anticlockwise

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Unformatted text preview: 2 + " + 1] or otherwise, show that (1 + u ) n ≥ 1 + nu (1 + u ) n −1 2 and the equality holds if and only if u = 0 . (3 marks) END OF PAPER 2001-AL-P MATH 1−11 − 10 − 2001 Paper 1 Section A 1. (a) 1 2 1 −+ x−2 x x+2 2. (b) n(n + 1)(n + 2) 4. (a) bci + acj + abk 5. (a) 1 ≤ k ≤ 2n + 1 4 (b) 30 T8 = C 7 1 3 (a) 1 − 1 1 1 (b) λ= 2 7. 7 A rotation which rotates any vector anticlockwise through about the origin, followed by an enlargement with factor π 4 2. 2001 Paper 1 Section A 8. (a) Imaginary L 4 + 4i Real O (b) PQ = 2 2 − 1 The complex number representing Q is 1+ i 2 . 2001 Paper 1 Section B 9. (a) Let ∆ be the determinant of the coefficients of (S). 1λ1 ∆= λ −1 1 3 1 2 = − 2 + 3λ + λ + 3 − 1 − 2λ 2 = −2 λ (λ − 2 ) (S) has a unique solution iff iff (b) (i) ∆≠0 λ ≠ 0 and λ ≠ 2 When λ ≠ 0 and λ ≠ 2 , (S) is consistent for all values of k , kλ1 1 x= y= −1 1 −1 1 2 ∆x 3(λ + k ) − 2 k − λ + 1 − 1 − k − 2λ = = =− − 2λ (λ − 2) ∆ − 2λ ( λ − 2) 2 λ (λ − 2 ) ∆y ∆ = 1k1 λ11 3 −1 2 − 2λ (λ − 2) λk λ −1 1 3 1 −1 = 2 + 3k − λ − 3 + 1 − 2λk 2λk + λ − 3k = − 2λ (λ − 2) 2λ ( λ − 2) 1 z= ∆z 1 + 3λ + λk + 3k − 1 + λ 2 (λ + 3)(λ + k ) = = = ∆ − 2λ (λ − 2) − 2λ (λ − 2) − 2λ (λ − 2) 2001 Paper 1 Section B (ii) When λ = 0 , the augmented matrix of (S) becomes k 1 0 1 k 1 0 1 0 −1 1 1 ~ 0 −1 1 1 3 1 2 − 1 0 0 0 − 3k ∴ (S) is consistent iff k = 0 . x + z = 0 . When λ = 0 and k = 0 , − y + z = 1 S.S. of (S) = { (t , − (t + 1), − t ) : t∈R } (iii) When λ = 2 , the augmented matrix of (S) becomes k 1 2 1 k 1 2 1 ~ 2 −1 1 1 0 5 1 2k − 1 3 1 2 − 1 0 0 0 − k − 2 ∴ (S) is consistent iff k = −2 . x + 2 y + z = −2 When λ = 2 and k = −2 , . 5 y + z = −5 S.S. of (S) = (c) { (3(1 + t ), t , − 5(1 + t ) ): t∈R } The system of equations is (S) when λ = 0 and k = 0 . If some solution (t , − (t + 1), − t ) sati...
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This note was uploaded on 06/07/2012 for the course MATHS 2002 taught by Professor Lo during the Fall '02 term at Wisc Oshkosh.

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