CAT 2019 Question Paper | Quants Slot 1

CAT Questions

Often in CAT Question Paper, we come across questions which appear easy but are actually curveballs. This is one such question. It appears very easy but is likely to consume a lot of time while the accuracy levels may be low. Questions like this separate the ones who have done their CAT online preparation vis-a-vis the folks who have just appeared after seeing the CAT syllabus. Please solve this question carefully.

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Question 9 : For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equals [TITA]

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Video Explanation

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Explanatory Answer

If n is even, f(n) = n (n + 1)
So, f (2) = 2(2+1) = 2 (3) = 6
If n is odd, f(n) = n + 3
f (1) = 1 + 3 = 4
It is given that, 8 x f(m+1) - f(m) =2
So, m can either be even or odd
Case-1: If m were even and m+1 odd
So, 8 x f(m+1) - f(m) =2
8(m + 4) - m (m + 1) = 2
8m + 32 - m² - m = 2
m² - 7m - 30 = 0
(m-10) (m+3) = 0
m = 10 or -3
m = 10, since m is positive
Case-2: If m were even and m+1 odd
8 x f(m+1) - f(m) =2
8 (m +1) (m + 2) - (m + 3) = 2
Now, let us substitute m = 1 which is the minimum possible value
8 (1 + 1) (1 + 2) - (1 + 3) 3
Case 2 does not work

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The question is "For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equals [TITA]"

Hence, the answer is 10