CAT 2025 Surds and Indices Question (Hard)

Question

If 9 x 2 2 x - 3 - 4 left 3 x 2 2 x - 2 right 27 0 , then the product of all possible values of x is

Accepted Answer

Let s assume that x 2 2x-3 t 9 x 2 2x-3 -4 3 x 2 2x-2 27 0 can be written as 9 t-4 3 t 1 27 0 3 2t -12 3 t 27 0 Let s assume that 3 t y y 2-12y 27 0 y-9 y-3 0 y 3 or 9 Rightarrow t 1 or 2 Let s solve when t 1 x 2 2x-3 1 Rightarrow x 2 2x-4 0 b 2-4ac 4 16 20 Positive, so the equation has real roots.

CAT 2025

Quantitative

Surds and Indices

Hard

If ( 9 ^ { x ^ { 2 } + 2 x - 3 } - 4 \left( 3 ^ { x ^ { 2 } + 2 x - 2 } \right) + 27 = 0 ), then the product of all possible values of ( x ) is

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