JEE Main 2025 Jan 29 Shift 2, Mathematics Q4: Introduction Of Functions

JEE Main2025Jan 29, Shift 2Mathematics

If the domain of the function $lo g_{5} (18 x - x^{2} - 77)$ is $(α, β)$ and the domain of the function $lo g_{(x - 1)} (\frac{2 x ^{2} + 3 x - 2}{x ^{2} - 3 x - 4})$ is $(γ, δ)$ , then $α^{2} + β^{2} + γ^{2}$ is equal to :

A
$195$
B
$174$
C
$186$
D
$179$

Solution

Function 1: Need $18 x - x^{2} - 77 > 0$ , i.e. $x^{2} - 18 x + 77 < 0 \Rightarrow (x - 7) (x - 11) < 0$ , so $x \in (7, 11)$ . Hence $α = 7$ , $β = 11$ .

Function 2: The base $x - 1$ must be positive and not equal to $1$ , so $x > 1$ , $x \neq = 2$ . The argument needs $\frac{( 2 x - 1 ) ( x + 2 )}{( x - 4 ) ( x + 1 )} > 0$ .

Critical points: $- 2, - 1, 1/2, 4$ . Sign chart on $(1, \infty)$ : between $1$ and $4$ the fraction is negative (numerator positive, denominator negative); for $x > 4$ it is positive. The exclusion $x \neq = 2$ falls inside the negative region, so it does not add to the domain.

Combining the base and argument constraints leaves $x \in (4, \infty)$ , giving $γ = 4$ .

$∴ α^{2} + β^{2} + γ^{2} = 49 + 121 + 16 = 186$ .

Concept behind this question

Introduction Of FunctionsNotes, formulas and examples →

Practice more Mathematics

Concept-wise practice with instant solutions on Fundamenthol.

Start practicing →

JEE Main 2025 Jan 29 Shift 2, Mathematics Q4: Introduction Of Functions

Concept behind this question

More previous year questions on Introduction Of Functions

Practice more Mathematics