# M17 LE3 0909 - Mathematics 17 First Semester AY 2009-2010...

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Mathematics 17 First Semester AY 2009-2010 Third Long Exam 5 September 2009 General Instructions. Use only blue or black ballpen. Show all necessary solutions and box all final answers. I. MODIFIED TRUE OR FALSE. Write TRUE if the statement is true. Otherwise, provide a brief justification why the statement is false. 2 pts each 1. For any positive real number a , a x > a y implies that x > y . 2. If a polynomial function f has real coefficients and odd degree, then f has to have at least one real root. 3. By Descartes’ Rule of Signs, the polynomial x 4 - x 2 + 2 x - 5 has exactly one negative real root. 4. If f ( x ) = x 2 - 1 x - 1 , then f - 1 exists and ( f - 1 f )( x ) = x for all real number x . 5. The sequence log 20 , log 40 , log 80 is an arithmetic progression with a common difference of 2. II. Do as indicated. 1. Given f ( x ) = 6 x 4 - x 3 + 5 x 2 - x - 1. (a) List all possible rational zeroes of f . 2 pts (b) Given that i is a zero of f , find all its other zeroes. (Do not use long division.) 4 pts 2. Determine the inverse and range of the function h ( x ) = x 3 + 1 x 3 - 1 . 5 pts 3. Find the exact value of (log
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