Let f(x) be a continuously differentiable function on the interval (0, ∞) such that f(1)=2 and t arrow x t^ 10 f(x)-x^ 10 f(t) t^9-x^9 =1 for each x>0 . Then, for all x>0, f(x) is equal to

Correct answer: B. (9)/(11 x)+(13)/(11) x^ 10 aligned & _ t arrow x t^ 10 f(x)-x^ 10 f(t) t^9-x^9 =1 & _ t arrow x 10 t^9 f(x)-x^ 10 f^ (t) 9 t^8 =1 aligned aligned & 10 x f(x)-x^2 f^ (x)=9 & x^2 f^ (x)=10 x f(x)-9 & f^ (x)=(10 f(x))/(x)-(9)/(x^2) & (d y)/(d x)-(10)/(x) y=-(9)/(x^2) & y · 1 x^ 10 =∫-(9)/(x^2) · 1 x^ 10 d x & y x^ 10 = 9 11 x^ 11 +c...(1) aligned aligned & f(1)=2 (2)/(1)=(9)/(11)+c c=(13)/(11) & f(x)= 9 11 x +(13)/(11) x^ 10 aligned Option (2) is correct.

JEE Advanced 2024Paper 1

Exam: JEE Advanced
Year: 2024
Paper: Paper 1
Subject: Mathematics
Chapter: Limits

JEE Advanced 2024 · Paper 1 · Mathematics · Limits

Let f(x) be a continuously differentiable function on the interval...

Let

f(x)

be a continuously differentiable function on the interval

(0, \infty)

such that

f(1)=2

and

\underset{t \rightarrow x}\lim \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1

for each

x>0

. Then, for all

x>0, f(x)

is equal to

Options

A $\frac{31}{11 x}-\frac{9}{11} x^{10}$
B $\frac{9}{11 x}+\frac{13}{11} x^{10}$
C $\frac{-9}{11 x}+\frac{31}{11} x^{10}$
D $\frac{13}{11 x}+\frac{9}{11} x^{10}$

Correct answer

B. $\frac{9}{11 x}+\frac{13}{11} x^{10}$

Step-by-step solution

\begin{aligned} & \lim _{t \rightarrow x} \frac{t^{10} f(x)-x^{10} f(t)}{t^9-x^9}=1 \\ & \Rightarrow \lim _{t \rightarrow x} \frac{10 t^9 f(x)-x^{10} f^{\prime}(t)}{9 t^8}=1\end{aligned}

\begin{aligned} & \Rightarrow 10 x f(x)-x^2 f^{\prime}(x)=9 \\ & \Rightarrow x^2 f^{\prime}(x)=10 x f(x)-9 \\ & \Rightarrow f^{\prime}(x)=\frac{10 f(x)}{x}-\frac{9}{x^2} \\ & \Rightarrow \frac{d y}{d x}-\frac{10}{x} y=-\frac{9}{x^2} \\ & \Rightarrow y \cdot \frac{1}{x^{10}}=\int-\frac{9}{x^2} \cdot \frac{1}{x^{10}} d x \\ & \Rightarrow \frac{y}{x^{10}}=\frac{9}{11 x^{11}}+c...(1)\end{aligned}

\begin{aligned} & \because \mathrm{f}(1)=2 \Rightarrow \frac{2}{1}=\frac{9}{11}+\mathrm{c} \Rightarrow \mathrm{c}=\frac{13}{11} \\ & \therefore \mathrm{f}(\mathrm{x})=\frac{9}{11 \mathrm{x}}+\frac{13}{11} \mathrm{x}^{10}\end{aligned}

\Rightarrow$ Option (2) is correct.

Practice Limits on Quantrex Academy →

More from Limits

Let _ x → 2 (( (x-2))(rx^2 + (p-2)x - 2p))/((x-2)^2) = 5 for some r, p ∈ ℝ . If the set… JEE Main 2026 The value of _ x → 0 ((x^2 ^2 x)/(x^2 - ^2 x)) is: JEE Main 2026 Let f(x) = _ y → 0 ((1 - (xy)) (xy))/(y^3) . Then the number of solutions of the equation… JEE Main 2026 The product of all possible values of α , for which _ x → 0 ((1 - (α x) ((α+1)x)… JEE Main 2026 If _ x → 2 (x^3 - 5x^2 + ax + b) ( x-1 - 1) _e(x-1) = m , then a + b + m is equal to : JEE Main 2026 The value of _ x arrow 0 _ e ( (e x) · (e^ 2 x) · … · (e^ 10 x)) e^ 2 -e^ 2 x is equal to JEE Main 2026 If _ x arrow 0 e^ (a-1) x +2 ~b x+(c-2) e^ -x x x- _ e (1+x) =2 , then a^ 2 +b^ 2 +c^ 2… JEE Main 2026 Let [·] denote the greatest integer function and f(x)= _ n arrow ∞ 1 n^ 3 ∑_ k=1 ^ n [ k^… JEE Main 2026 Let f: R arrow(0, ∞) be a twice differentiable function such that f(3)=18, f^ (3)=0 and… JEE Main 2026 Let α and β be the real numbers such that _ x arrow 0 (1)/(x^3)((α)/(2) ∫_0^x (1)/(1-t^2)… JEE Advanced 2025 Let x_0 be the real number such that e^ x_0 +x_0=0 . For a given real number α , define… JEE Advanced 2025 Given below are two statements : Statement I : _ x arrow 0 ( ^ -1 x+ _e (1+x)/(1-x) -2 x… JEE Main 2025 If the function f(x)=( ( x)- ( x))/( x- x) is continuous at x=0 , then f(0) is equal to… JEE Main 2025 _ x arrow 0^ + (5(x)^ (1)/(3) ) _e(1+3 x^2) ( ^ -1 3 x )^2(e^ 5(x)^ (4)/(3) -1) is equal… JEE Main 2025 If _ x arrow 1^ + ((x-1)(6+λ (x-1))+μ (1-x))/((x-1)^3)=-1 , where λ, μ ∈ ℝ , then λ+μ is… JEE Main 2025 If Lim_ x arrow 0 (( x)/(x))^ (1)/(x^2) =p , then 96 _e p is equal to ______ JEE Main 2025 Full Limits list Limits JEE Maths PYQs JEE Mathematics PYQs All JEE Advanced PYQs