CAT 2023 Question Paper | Quant Slot 1

CAT Questions

CAT 2023 Quant was dominated by Algebra followed by Arithmetic. 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 5 : The number of integer solutions of equation \( 2|x|\left(x^2+1\right)=5 x^2 \) is

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Video Explanation

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Explanatory Answer

\( 2 | x | \left( x ^ { 2 } + 1 \right) = 5 x ^ { 2 } \)
Let |x| = k
\( 2 k \left( k ^ { 2 } + 1 \right) = 5 k ^ { 2 } \)
Either k = 0; or 2(k² + 1) = 5k
2k² – 5k + 2 = 0
2k² – 4k – k + 2 = 0
2k(k – 2) –1(k – 2) = 0
(2k – 1)(k – 2) = 0
k = 0.5 or k = 2
Therefore, k which is |x|, can take the values 0, 0.5 or 2
So, x can take the values 0, -0.5, 0.5, -2, 2
Since we are looking for integral solutions, x can only take the values 0, 0.5 or 2
Therefore, there are only 3 integral solutions to \( 2 | x | \left( x ^ { 2 } + 1 \right) = 5 x ^ { 2 } \).

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The question is " The number of integer solutions of equation \( 2|x|\left(x^2+1\right)=5 x^2 \) is "

Hence, the answer is '3'