CAT 2022 Question Paper | Quant Slot 2

CAT 2022 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.

Question 10 : Suppose for all integers \(x\), there are two functions \(f\) and \(g\) such that \(f(x)+f(x-1)-1=0\) and \(g(x)=x^2\). If \(f\left(x^2-x\right)=5\), then the value of the sum \(f(g(5))+g(f(5))\) is

Video Explanation

Explanatory Answer

Let us look closely at “f(x) + f(x - 1) - 1 = 0”.
f(x) + f(x - 1) - 1 = 0
f(x) = 1 - f(x - 1)
This means if we know the value of f(n), we can find the value of f(n + 1), using the value of f(n + 1) we can find the value of f(n + 2)... So on and so forth, we can find the value of f(x) for any x if we find the value of f(x) for some x.
f(x² - x) = 5
f(x(x - 1)) = 5
This means if any integer k can be expressed as a product of two consecutive integers, f(k) = 5
2 = 2 (2 - 1)
Therefore, f(2) = 5.
f(3) = 1 - f(2) = 1 - 5 = -4
f(4) = 1 - f(3) = 1 - (-4) = 5
f(5) = 1 - f(4) = 1- 5 = -4
This keeps on repeating…
Therefore,
f(odd) = -4
f(even) = 5
f(g(5)) + g(f(5))
= f(25) + g(f(5))
f(odd) = -4
f(g(5)) + g(f(5)) = f(25) + g(f(5)) = (-4) + (-4)² = -4 + 16 = 12

The question is " Suppose for all integers \(x\), there are two functions \(f\) and \(g\) such that \(f(x)+f(x-1)-1=0\) and \(g(x)=x^2\). If \(f\left(x^2-x\right)=5\), then the value of the sum \(f(g(5))+g(f(5))\) is "

Hence, the answer is '12'