Assignment1

# Assignment1 - [3 marks ax 1 x 2 = 1 ax 1 x 2 = a 2(c[1...

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Student name: Student number: Assignment 1 1. In general, circles are described by equations of the form x 2 + y 2 + ax + by + c = 0 , where a , b and c are constants. Given such a description, the area of the circle is equal to π 4 ( a 2 + b 2 - 4 c ) . Let P 1 = (1 , 1), P 2 = (5 , - 3), P 3 = ( - 3 , 3) and P 4 = ( - 1 , 3). (a) [3 marks] Find the equation of the circle C passing through the points P 1 , P 2 and P 3 . (b) [2 marks] Prove that no circle passes through the points P 1 , P 2 and P 4 . (c) [1 bonus mark] Find the equation of the circle C , not equal to C , which passes through the points P 1 and P 2 and has the same area as C . 2. For each of the following systems, find all real values of a for which the system has: (i) exactly one solution; (ii) no solutions; and (iii) infinitely many solutions. (a) [2 marks] x 1 + 2 x 2 - 4 x 3 = 4 3 x 1 - x
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Unformatted text preview: [3 marks] ax 1 + x 2 = 1 ax 1 + x 2 = a 2 (c) [1 bonus mark] x 2 1 + ax 2 2 = a ax 2 1 + x 2 2 = a 4 3. Let A = ± a b c d ² and I = ± 1 1 ² . (a) [3 marks] Prove that if ad-bc 6 = 0, then A has reduced row-echelon form I . (b) [2 marks] Prove that if A has reduced row-echelon form I , then ad-bc 6 = 0. (c) [1 bonus mark] The product of two 2 × 2 matrices is deﬁned thus: ± a b c d ²± A B C D ² = ± aA + bC aB + bD cA + dC cB + dD ² . Prove that the set S = ³± a b c d ² : ad-bc 6 = 0 ´ is closed under multiplication; that is, prove that the product of any two members of the set is itself in the set....
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