CAT 2020 Question Paper | Quants Slot 1

Functions as an idea is a fabulous idea if one can wrap their head around it the right way, and it has been one of the staple ideas tested by the CAT exam historically. Here's a beautiful question from Functions that tests a subtle but sophisticated concept. This question appeared in CAT 2020. Will you get that idea properly or, you'll miss it?

Question 2 : If f(5 + x) = f(5 - x) for every real x and f(x) = 0 has four distinct real roots, then the sum of the roots is

Video Explanation

Explanatory Answer

Given f (5 + x) = f (5 – x)
When x = 1, f(6) = f(4)
When x = 2, f(7) = f(3)
Imagine a graph and while joining (6,4) and (7,3) it looks symmetrical about 5
Given 4 distinct real roots. Assume 12 as one of the roots.
f(12) = 0 --> Then f(5 + 7) = 0 which is same as f(5 – 7) = f(-2) = 0
So the roots are in the form (5 + k) and (5 – k) {This is one pair}
Sum of the roots = 5 + k + 5 – k = 10
We have 4 roots, So two pairs =10 + 10 = 20

The question is "If f(5 + x) = f(5 - x) for every real x and f(x) = 0 has four distinct real roots, then the sum of the roots is"

Hence, the answer is, "20"