ARC 1999S - SUGGESTED SOLUTIONS for 3 UNIT MATHS - 1999...

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Unformatted text preview: SUGGESTED SOLUTIONS for 3 UNIT MATHS - 1999 Question One (a) 1 ) ( + = = x e y x f 1 + = y e x x y ln 1 = + 1 ln = x y the inverse function ) ( 1 x f )] ( [ 1 x f f 1 1 ln + = x e x e x = = ln x x e x e x f f x = + = + = = + 1 1 1 ln ) 1 ( 1 ] ln[ )] ( [ 1 1 (b) 3 2 2 1 + x x 2 2 ) 2 )( 3 ( 2 ) 2 ( ) 3 ( + + x x x x ) 2 )( 3 ( 2 ) 2 ( ) 3 ( 2 2 + + x x x x )] 2 ( 2 3 )[ 2 )( 3 ( + + x x x x ] 4 2 3 )[ 2 )( 3 ( + x x x x ) 7 )( 2 )( 3 ( + x x x .(1) Consider ) 7 )( 2 )( 3 ( = + x x x Test x = -10, inequality (1) holds true Test x = -5, inequality (1) does not hold true Test x = 0, inequality (1) holds true Test x = 4, inequality (1) does not hold true Solution is 7 x or 3 2 < < x ) 3 ( ), 2 ( x x From the graph of ) 7 )( 2 )( 3 ( + = x x x y-7 -2 3 0 y x -7 -2 3 Solution is 7 x or 3 2 < < x (c) ( ) ( ) ( ) n n r r 2 2 3 ... 2 2 3 2 2 3 2 3 ) 2 ( 2 3 2 1 1 + + + + = = 2 ; 2 3 = = r a 1 ) 1 ( = r r a S n n ( ) 2 1 2 3 766 1 2 1 2 = n 2 1533 ) 1 2 ( 2 3 = n 9 512 2 511 1 2 1533 ) 1 2 ( 3 = = = = n n n n (d) There are 5 vowels and 3 consonants. For words with 2 vowels and 1 consonant the number of selections is 30 1 3 2 5 = C C . For words with 1 vowel and 2 consonants the number of selections is 15 2 3 1 5 = C C . The total number of selections = 45 But each selection may be arranged 3! ways. The total number of words = 45 3! = 270 Question Two (a) RHS 2 tan 2 cos 2 sin sin 2 1 cos sin 2 sin sin 2 1 cos 2 sin 2 cos 2 2sinA- cosecA 2cosA LHS 2 2 sin 1 = = = = = = = A A A A A A A A A A A A (b) Let = x 1 tan x = tan From the triangle 2 1 sin x x + = + = 2 1 1 sin x x 2 1 1 1 sin tan x x x + = (c) sin cos cos sin ) sin( = Put = and = sin sin...
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This note was uploaded on 11/01/2011 for the course ECON 101 taught by Professor Adam during the Two '11 term at University of Technology, Sydney.

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ARC 1999S - SUGGESTED SOLUTIONS for 3 UNIT MATHS - 1999...

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