Solution to the problem If x=(9+4*root(5))^48=[x]+f, where [x] is defined as integral part of x and f is a fraction,then x(1–f) equals – A. 1 B. Less than 1 C. More than 1 D. Between 1 and 2 E. None of the above Sol : This was a question asked for 5 marks in XAT and sure was a difficult one. In difficult questions, always look first at the options. Now, one option is less than one and another more than one, and the third 1 so there is no chance that the answer could be E. Say, I get my answer as 1.5, then both option C and option D, would be correct so D cannot be the answer. So, we have eliminated two options without even looking at the question. Now, let us try and solve it. x=(9+4*root(5))^48, When I see an equation with large coefficients and having large even powers, most of the times, it fits into (a+b)^2. In this case a = (root(5)^2 + 2^2 + 2*2*root(5)) = (root(5)+2)^2 = 9 + 4*root(5) So, x = (a+b)^96 = (root(5) +2)^96 Now, since we see irrational coefficients in addition given in x, we define y = (root(5) - 2)^96 So x*y = (5-4)^96 = 1 Now, let us also see if we can remove the irrational coefficients.If we add x+y, all the odd powers in the binomial expansion will get cancelled. If you do not want to expand the whole thing, just try(a+b)^2 + (a-b)^2 = 2(a^2 + b^2)So, x+y is an integer.[x] + f + y is an integer. So, f +y is an integer, Now y = ( root(5)-2)^96 and since root(5)-2 is less than 1,f+y=1, or y = 1-f and hence we get our answer as 1 Now, even if you were not able to think after x*y, just take a look at the question. We have to find x(1-f), and since we found x*y, there is a high probability that y = 1-f. Just take it backwards, and you may be able to prove it.
Posted on: Tue, 19 Nov 2013 04:03:00 +0000
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