Let f: R-\ 0\ arrow R be a function such that f(x)-6 f((1)/(x))=(35)/(3 x)-(5)/(2) . If the _ x arrow 0 ((1)/(α x)+f(x))=β ; α, β ∈ R , then α+2 β is equal to

Correct answer: C. 4 aligned & f(x)-6 f((1)/(x))=(35)/(3 x)-(5)/(2)...(1) & 6(f((1)/(x))-6 f(x)=(35 x)/(3)-(5)/(2)) & 6 f((1)/(x))-36 f(x)=(210 x)/(3)-(30)/(2)...(2) & (1)+(2) & -35 f(x)=(35)/(3)[(1)/(x)+6 x]-(5)/(2)(1+6) & -f(x)=(1)/(3)((1)/(x)+6 x)-(1)/(2) & f(x)=-(1)/(3 x)-2 x+(1)/(2) & _ x arrow 0 [(1)/(α x)-(1)/(3 x)-2 x+(1)/(2)]=β & α=3 & β+2 β=3+2 × (1)/(2)=4 aligned

JEE Main 202524 JanuaryMorning Shift

Exam: JEE Main
Year: 2025
Date: 24 January
Shift: Morning Shift
Subject: Mathematics
Chapter: Functions

JEE Main 2025 · 24 January · Morning Shift · Mathematics · Functions

Let f: R-\ 0\ arrow R be a function such that f(x)-6...

Let

f: R-\{0\} \rightarrow R

be a function such that

f(x)-6 f\left(\frac{1}{x}\right)=\frac{35}{3 x}-\frac{5}{2}

. If the

\lim _{x \rightarrow 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ; \alpha, \beta \in R

, then

\alpha+2 \beta

is equal to

Options

A $5$
B $3$
C $4$
D $6$

Correct answer

C. $4$

Step-by-step solution

\begin{aligned} & f(x)-6 f\left(\frac{1}{x}\right)=\frac{35}{3 x}-\frac{5}{2}...(1) \\ & 6\left(f\left(\frac{1}{x}\right)-6 f(x)=\frac{35 x}{3}-\frac{5}{2}\right) \\ & 6 f\left(\frac{1}{x}\right)-36 f(x)=\frac{210 x}{3}-\frac{30}{2}...(2) \\ & (1)+(2) \\ & -35 f(x)=\frac{35}{3}\left[\frac{1}{x}+6 x\right]-\frac{5}{2}(1+6) \\ & -f(x)=\frac{1}{3}\left(\frac{1}{x}+6 x\right)-\frac{1}{2} \\ & f(x)=-\frac{1}{3 x}-2 x+\frac{1}{2} \\ & \lim _{x \rightarrow 0}\left[\frac{1}{\alpha x}-\frac{1}{3 x}-2 x+\frac{1}{2}\right]=\beta \\ & \Rightarrow \alpha=3 \\ & \beta+2 \beta=3+2 \times \frac{1}{2}=4 \end{aligned}

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