HKALE Pure Maths 2001 Paper01

A b c are the points a 0 0 0 b 0 0 0 c

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Unformatted text preview: Prove that pq ab +. pq (5 marks) 2001-AL-P MATH 1−3 −2− Go on to the next page 4. A , B , C are the points (a, 0, 0) , (0, b, 0) , (0, 0, c) respectively and O is the origin. (a) Find AB × AC . (b) Let S ∆XYZ denote the area of the triangle with vertices X , Y and Z . Prove that S ∆ABC = S ∆OAB + S ∆OBC + S ∆OCA . 2 2 2 2 (5 marks) 5. Let the kth term in the binomial expansion of (1 + x) 2 n in ascending powers of 2n x be denoted by Tk , i.e. Tk = C k −1 x k −1 . 1 , find the range of values of k such that Tk +1 ≥ Tk . 3 (a) (b) 6. If x = Find the greatest term in the expansion if x = 1 and n = 15 . 3 (5 marks) Let f( x) = ax 2 + bx + c where a, b, c are real numbers and a ≠ 0 . Show that if f [f( x)] = [f( x )] 2 for all x , then f( x) = x 2 . (4 marks) 2001-AL-P MATH 1−4 −3− 7. A 2×2 matrix M is the matrix representation of a transformation T in R2 . T transforms (1, 0) and (0, 1) to (1, 1) and (−1, 1) respectively. (a) (b) Find M . λ 0 a b Find λ > 0 such that M can be decomposed as 0 λ c d where ab =1 . cd Hence describe the geometric meaning of T . (5 marks) 8. Let L be the straight line z − (4 + 4i ) = z and C be the circle z = 1 . (a) Sketch L on an Argand diagram. (b) Let P , Q be points on L and C respectively such that PQ is equal to the shortest distance between L and C . Find PQ and the complex numbers representing P and Q . (6 marks) 2001-AL-P MATH 1−5 −4− Go on to the next page SECTION B (60 marks) Answer any FOUR questions in this section. Each question carries 15 marks. Use a separate AL(D) answer book for each question. 9. Consider the system of linear equations (S ) : z= k x + λy + y+ z= 1 λ x − 3x + y + 2z = −1 where λ , k ∈ R . (a) Show that (S) has a unique solution if and only if λ ≠ 0 and λ ≠ 2 . (2 marks) (b) For each of the following cases, determine the value(s) of k for which (S) is consistent. Solve (S) in each case. (i) λ ≠ 0 and λ ≠ 2 , (ii) λ =0 , (iii) λ=2 . (8 marks) (c) + z= 0 x −y...
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