Let the domains of the functions f(x)= _4 _3 _7(8- _2(x^2+4 x+5)) and g(x)= ^ -1 ((7 x+10)/(x-2)) be (α, β) and [γ, ] , respectively. Then α^2+β^2+γ^2+ ^2 is equal to :-

Correct answer: A. 15 aligned & _3( _7(8- _2(x^2+4 x+5))) > 0 & _2(x^2+4 x+5) < 1 & x^2+4 x+3 < 0 & x ∈(-3,-1) & -1 ≤ 7 x+10 x-2 ≤ 1 & x ∈[-2,-1] & α=-3, β=-1, γ=-2, =-1 & α^2+β^2+γ^2+ ^2=15 & option (1) aligned

JEE Main 20254 AprilEvening Shift

Exam: JEE Main
Year: 2025
Date: 4 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Functions

JEE Main 2025 · 4 April · Evening Shift · Mathematics · Functions

Let the domains of the functions f(x)= _4 _3 _7(8- _2(x^2+4 x+5))...

Let the domains of the functions

\mathrm{f}(\mathrm{x})=\log _4 \log _3 \log _7\left(8-\log _2\left(\mathrm{x}^2+4 \mathrm{x}+5\right)\right)

and

g(x)=\sin ^{-1}\left(\frac{7 x+10}{x-2}\right)

(\alpha, \beta)

and

[\gamma, \delta]

, respectively. Then

\alpha^2+\beta^2+\gamma^2+\delta^2

is equal to :-

Options

A $15$
B $13$
C $16$
D $14$

Correct answer

A. $15$

Step-by-step solution

\begin{aligned} & \log _3\left(\log _7\left(8-\log _2\left(\mathrm{x}^2+4 \mathrm{x}+5\right)\right)\right) > 0 \\ & \log _2\left(\mathrm{x}^2+4 \mathrm{x}+5\right) < 1 \\ & \mathrm{x}^2+4 \mathrm{x}+3 < 0 \\ & \Rightarrow \mathrm{x} \in(-3,-1) \\ & -1 \leq \frac{7 \mathrm{x}+10}{\mathrm{x}-2} \leq 1 \\ & \Rightarrow \mathrm{x} \in[-2,-1] \\ & \alpha=-3, \beta=-1, \gamma=-2, \delta=-1 \\ & \alpha^2+\beta^2+\gamma^2+\delta^2=15 \\ & \text{ option }(1)\end{aligned}

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