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Midterm1Solutions

Midterm1Solutions - Faculty of Mathematics University of...

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Faculty of Mathematics University of Waterloo MATH 135 MIDTERM EXAM #1 Fall 2008 Monday 06 October 2008 19:00 – 20:15 Solutions 1. In each part of this problem, full marks will be given if the correct answer is written in the box. In part (d), if your answer is incorrect, your work will be assessed for part marks. (a) Write the converse of “If the units digit of n is 4, then n is divisible by 4”. [2] Solution “If n is divisible by 4, then the units digit of n is 4.” (b) Is the statement “ x y, x 2 + y = 1” TRUE or FALSE? [2] (The universe of discourse is Z , the integers.) Solution TRUE Reason : The statement says “For all integers x there exists an integer y such that x 2 + y = 1”. This is TRUE, because for any integer x , we can choose y = 1 - x 2 which is an integer. (c) Is the statement “ y x, x 2 + y = 1” TRUE or FALSE? [2] (The universe of discourse is Z , the integers.) Solution FALSE Reason : The statement says “For all integers y , there exists an integer x such that x 2 + y = 1”. This is FALSE, which we can see using y = 2 as a counterexample, because there is no integer x for which x 2 + 2 = 1 (or x 2 = - 1). (d) Write down the expanded and simplified form of ( x 2 + 2 y 3 ) 4 . [4] Solution ( x 2 + 2 y 3 ) 4 = 4 0 ( x 2 ) 4 + 4 1 ( x 2 ) 3 (2 y 3 ) + 4 2 ( x 2 ) 2 (2 y 3 ) 2 + 4 3 ( x 2 )(2 y 3 ) 3 + 4 4 (2 y 3 ) 4 = x 8 + 8 x 6 y 3 + 24 x 4 y 6 + 32 x 2 y 9 + 16 y 12
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MATH 135, Midterm #1 Solutions Page 2 of 6 2. Prove by induction that n r =2 1 + 1 r 2 - 1 = 2 n n + 1 for all n P with n 2. [7] Solution We prove the result by induction on n .
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