111) f(x)=f(y) does not necessarily mean x=±y as the only possibility.
1) Let S=1+\frac{1}{2}+\frac{1}{3}+...\frac{1}{2^n-1}.
S lies between
a) 0 & n2
b) n2 & n
c) n and 2n
d) none.
2) IIT-1996.
Let f(x) be evn.
Given f(x) satisfies f(x)=f\left(\frac{x+1}{x+2} \right)
find all possible values of x.
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7 Answers
1Given ,
f ( x ) = f ( x + 1x + 2 )
Hence , x = x + 1x + 2
or , x 2 + 2 x = x + 1
or , x 2 + x - 1 = 0
So , x = - 1 ± √52
Again , as f ( x ) is an even function , hence ,
f ( - x ) = f ( x ) = f ( x + 1x + 2 )
or , - x = x + 1x + 2
or , x 2 + 2 x = - x - 1
or , x 2 + 3 x + 1 = 0
Hence , x = - 3 ± √52
So , we find out exactly 4 values of x .
11 >
S = 11 + 12 + 13 + ....... + 12 n - 1 > 12 + 12 + 12 + ...... + 12 = n2
Again , S = 11 + 12 + 13 + ....... + 12 n - 1 < 11 + 11 + 11 + ...... + 11 = n
So , I guess the answer is B .
1I agree with you , Soumik , the function should be either strictly increasing or strictly decreasing . I feel this is an old IIT - JEE question , isn ' t it ?
66@Ricky: " I feel this is an old IIT - JEE question , isn ' t it ?"
Does that change the basics?
1Of course no , sir . I did not at all wanted to give an impression that since the question is already a past IIT Qs . , so the concept is not correct . Actually , it ' s a part of my mistake , I should not have said it in the first place . Sorry , sir .
11What Gallardo did was exactly what the book did (from which the qsn is given).
So did IIT gve an incomplete qsn in 1996 ?