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CAT Previous Year Paper | CAT Number theory Questions | Question 18

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Question 18 : If the sum of squares of two numbers is 97, then which one of the following cannot be their product?

  1. 64
  2. -32
  3. 16
  4. 48

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Explanatory Answer

Method of solving this CAT Question from Number theory

Given that the sum of squares of two numbers is 97 i.e. a2 + b2 = 97
From the given options we have to find which one cannot be their product i.e. ab
A. 64 ⟹ 2ab = 128
B. −32 ⟹ 2ab = -64
C. 16 ⟹ 2ab = 32
D. 48 ⟹ 2ab = 96
2ab is found because we know that
a2 + b2 + 2ab ≥ 0
a2 + b2 - 2ab ≥ 0
By this we can know that 97 + 128 works but 97 - 128 doesn’t works so we can understand option A cannot be the product and the rest can be.
a2 + b2 ≥ |2ab|
a2 + b2 ≥ 2ab
a2 + b2 ≥ -2ab
⟹ \\frac{a^2 + b^2}{2}) ≥ |ab|
So here 2ab should lie between +97 and -97 or ab should be less than \\frac{97}{2}) or greater than −\\frac{97}{2}), so except option A all the other options works so option A 64 cannot be the product

The question is "If the sum of squares of two numbers is 97, then which one of the following cannot be their product?"

Hence, the answer is 64

Choice A is the correct answer.


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