JEE Main 2016 (10)

Algebra Level 3

If f ( x ) = ( 9 log ⁡ 2 ( 3 − 2 x ) − 1 ) 1 3 f(x)=(\dfrac{9}{\log_{2}(3-2x)}-1)^{\frac{1}{3}} , then the value of a a which satisfies f − 1 ( 2 a − 4 ) = 0.5 f^{-1}(2a-4)=0.5 , is given by

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Shivam Jadhav by Shivam Jadhav , 22, India

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1 solution

Chew-Seong Cheong
Mar 21, 2016

For f ( x ) = ( 9 log ⁡ 2 ( 3 − 2 x ) − 1 ) 1 3 f(x) = \left(\dfrac{9}{\log_2(3-2x)} -1 \right)^\frac{1}{3} and f − 1 ( 2 a − 4 ) = 1 2 f^{-1}(\color{#3D99F6}{2a-4}) = \color{#D61F06}{\dfrac{1}{2}} , we have:

2 a − 4 = ( 9 log ⁡ 2 ( 3 − 2 × 1 2 ) − 1 ) 1 3 = ( 9 log ⁡ 2 2 − 1 ) 1 3 = ( 9 1 − 1 ) 1 3 = 8 1 3 = 2 ⇒ 2 a = 6 a = 3 \begin{aligned} \color{#3D99F6}{2a-4} & = \left(\frac{9}{\log_2 \left(3-2\times \color{#D61F06}{\frac{1}{2}}\right)} -1 \right)^\frac{1}{3} \\ & = \left(\frac{9}{\log_2 2} -1 \right)^\frac{1}{3} \\ & = \left(\frac{9}{1} -1 \right)^\frac{1}{3} = 8^\frac{1}{3} = 2 \\ \Rightarrow 2a & = 6 \\ a & = \boxed{3} \end{aligned}

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Not a level 4 question surely!

Ayushman Chahar - 4 years, 2 months ago

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