CAT 2023 Question Paper | Quant Slot 2

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 2 : The sum of all possible values of \(x\) satisfying the equation \(2^{4 x^2}-2^{2 x^2+x+16}+2^{2 x+30}=0\), is

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Video Explanation

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Explanatory Answer

\( 2 ^ { 4 x ^ { 2 } } - 2 ^ { 2 x ^ { 2 } + x + 16 } + 2 ^ { 2 x + 30 } = 0 \)
\( \div \) by \( 2 ^ { 2 x ^ { 2 } + x + 16 } \) on both sides.
\( 2 ^ { 2 x ^ { 2 } - x - 16 } + 2 ^ { - 2 x ^ { 2 } + x + 14 } = 1 \)
\( 2 ^ { 2 x ^ { 2 } - x - 16 } + 2 ^ { - 2 x ^ { 2 } + x + 16 - 2 } = 1 \)
\( 2 ^ { 2 x ^ { 2 } - x - 16 } + 2 ^ { - \left( 2 x ^ { 2 } - x - 16 \right) - 2 } = 1 \)
let \( y = 2 ^ { 2 x ^ { 2 } - x - 16 } \)
\( y + \frac { 1 } { 4 y } = 1 \)
\( 4 y ^ { 2 } - 4 y + 1 = 0 \)
\( 4 y ^ { 2 } - 2 y - 2 y + 1 = 0 \)
\( 2 y ( 2 y - 1 ) - 1 ( 2 y - 1 ) = 0 \)
\( ( 2 y - 1 ) ^ { 2 } = 0 \)
\( \therefore 2 y = 1 \)
\( 2 \times 2 ^ { 2 x ^ { 2 } - x - 16 } = 1 = 2 ^ { 0 } \)
\( 2 ^ { 2 x ^ { 2 } - x - 15 } = 2 ^ { 0 } \)
\( 2 x ^ { 2 } - x - 15 = 0 \)
\( 2 x ^ { 2 } - 6 x + 5 x - 15 = 0 \)
\( 2 x ( x - 3 ) + 5 ( x - 3 ) = 0 \)
\( ( 2 x + 5 ) ( x - 3 ) = 0 \)
\( \therefore x = 3 \) or \( x = - 5 / 2 \)
Sum of all possible values of x \( = 3 + ( - 5 / 2 ) = \frac { 1 } { 2 } \)

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The question is " The sum of all possible values of \(x\) satisfying the equation \(2^{4 x^2}-2^{2 x^2+x+16}+2^{2 x+30}=0\), is "

Hence, the answer is '\(\frac{1}{2}\)'

Choice C is the correct answer.