Winter2008 with solutions

Winter2008 with solutions - MATH 135 Midterm Solutions 1....

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MATH 135 Winter 2008 Midterm Solutions 1. (a) Write down the converse of “If x is a perfect square, then x 0”. (b) Write down the contrapositive of “If b > 8, then f ( b ) < 0 or f ( b ) = 99”. (c) Determine (with brief justification) the second smallest positive integer s such that 70 x + 42 y = s has a solution x, y Z . (d) Could mathematical induction be used to prove that | cos( x ) | ≤ 1 for all x R ? Briefly explain why or why not. (e) With the universe of discourse as the rationals, state (but do not prove) the following using quantifiers. “There is no smallest positive rational number”. Solution: (a) The converse is “If x 0, then x is a perfect square”. (b) The contrapositive is “If f ( b ) 0 and f ( b ) 6 = 99, then b 8”. (c) By inspection, gcd(70 , 42) = 14. (Alternately, the gcd could be determined using the Extended Euclidean Algorithm). By the Linear Diophantine Equation Theorem 2.31 Part 1, 70 x + 42 y = s has a solution if and only if gcd(70 , 42) | s . The smallest positive integer s for which 14 | s is 14 and the second smallest positive integer s is 14 · 2 = 28. (d) Mathematical induction cannot be used to prove that | cos( x ) | ≤ 1 for all x R . It can be used to prove that a statement P ( x ) is true for all x P , but not for all x R . (e) Original Solution: x y, 0 < y < x . A student rightly pointed out that this implies that implies that all rational numbers x are positive, and the universe of discourse was the rational numbers, not the positive rationals. Corrected Solution: x, x > 0 → ∃ y, 0 < y < x . 2. Prove that if 8 ab - 3 a + ab 3 is even, then a is even or b is odd. Solution: Suppose that 8 ab - 3 a + ab 3 is even and a is odd. We must show that b is odd. 8 ab is even, and since 3 and a are both odd, 3 a is odd. Hence 8 ab - 3 a is odd. Since 8 ab - 3 a + ab 3 is even and 8 ab - 3 a is odd, then ab 3 must be odd and b 3 must be odd. Therefore b is odd. 3. A sequence { y n } is given by y 1 = 4, y 2 = 16, For each n P , n 3 , y n = 2 y n - 1 + 3 y n - 2 .
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Winter2008 with solutions - MATH 135 Midterm Solutions 1....

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