Q JEE MAIN 2024

If $y=\frac{(\sqrt{x}+1)\left(x^2-\sqrt{x}\right)}{x \sqrt{x}+x+\sqrt{x}}+\frac{1}{15}\left(3 \cos ^2 x-5\right) \cos ^3 x$, then $96 y^{\prime}\left(\frac{\pi}{6}\right)$ is equal to :

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Q Jee Main 2024

Let $\mathrm{y}=\log _{\mathrm{e}}\left(\frac{1-\mathrm{x}^2}{1+\mathrm{x}^2}\right),-1

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Q JEE MAIN 2026

Let $f(x)=x^3+x^2 f^{\prime}(1)+2 x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in \mathbf{R}$. Then the value of $f^{\prime}(5)$ is:

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Q JEE MAIN 2019

$$ \text { The derivative of } \tan ^{-1}\left(\frac{\sin x-\cos x}{\sin x+\cos x}\right) \text {, with respect to } \frac{x}{2}...

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Q JEE MAIN 2019

If $x=3$ tan $t$ and $y=3$ sect, then the value of $\frac{d^2 y}{d x^2}$ at $t=\frac{\pi}{4}$, is

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Q JEE MAIN 2019

If $x=3 \tan t$ and $y=3$ sect, then the value of $\frac{d^2 y}{d x^2}$ at $t=\frac{\pi}{4}$, is

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Q JEE MAIN 2020

The derivative of $\tan ^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to $\tan ^{-1}\left(\frac{2 x \sqrt{1-x^2}}{1-2 x^2}\right)$ at $x=\frac{1}{2}$ is :

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Q JEE MAIN 2019

If $x \log _e\left(\log _e x\right)-x^2+y^2=4(y>0)$, then $\frac{d y}{d y}$ at $x=e$ is equal to:

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Q JEE MAIN 2019

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Q JEE Main 2019

Let $f: R \rightarrow R$ be a function such that $f(x)=x^3+x^2 f^{\prime}(1)+x f^{\prime \prime}(2)+f^{\prime \prime \prime}(3), x \in R$. Then...

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Q JEE MAIN 2020

If $(a+\sqrt{2} b \cos x)(a-\sqrt{2 b} \cos y)=a^2-b^2$, where $a >\mathrm{b}>0$, then $\frac{\mathrm{dx}}{\mathrm{dy}} \mathrm{at}\left(\frac{\pi}{4}, \frac{\pi}{4}\right)$ is :

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Q JEE MAIN 2020

Let $f$ and $g$ be differentiabel functions on $R$ such that fog is the identity function. If for some $...

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Q JEE MAIN 2020

If $y(\alpha)=\sqrt{2\left(\frac{\tan \alpha+\cot \alpha}{1+\tan ^2 \alpha}\right)+\frac{1}{\sin ^2 \alpha}}, \alpha \in\left(\frac{3 \pi}{4}, \pi\right)$ then $\frac{\mathrm{dy}}{\mathrm{d} \alpha}$ at $\alpha=\frac{5 \pi}{6}$ is

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Q JEE MAIN 2020

Let $x^k+y^k=a^k,(a, k>0)$ and $\frac{d y}{d x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0$, then k is :

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Q JEE Main 2021

If $y=y(x)$ is an implicit function of $x$ such that $\log _e(x+y)=4 x y$, then $\frac{d^2 y}{d x^2}$ at $x=0$...

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Q JEE MAIN 2021

If $y^{1 / 4}+y^{-1 / 4}=2 x$, and $\left(x^2-1\right) \frac{d^2 y}{d x^2}+\alpha x \frac{d y}{d t}+\beta y=0$, then $|\alpha-\beta|$ is...

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Q JEE Main 2021

Let $f(x)=\cos \left(2 \tan ^{-1} \sin \left(\cos ^{-1} \sqrt{\frac{1-x}{x}}\right)\right) .0

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Q JEE MAIN 2019

If $f(1)=1, f^{\prime}(1)=3$, then the derivative of $f(f(f(x)))+(f(x))^2$ at $x=1$ is :

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Q JEE MAIN 2021

Consider the function $$ \begin{array}{ll} f(x)=\frac{P(x)}{\sin (x-2)}, & x \neq 2 \\ =7 & x=2 \end{array} $$ Where $\mathrm{P}(\mathrm{x})$ is...

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Q JEE MAIN 2022

If $y(x)=\left(x^{x^x}\right), x>0$ then $\frac{d^2 x}{d y^2}+20$ at $x=1$ is equal to :

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Q JEE MAIN 2022

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfy $f(x+y)=2 x f(y)+4 y f(x), \forall x, y \in \mathbb{R}$. If $f(2)=3$, then 14...

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Q JEE MAIN 2022

If $\cos ^{-1}\left(\frac{y}{2}\right)=\log _e\left(\frac{x}{5}\right)^5,|y|

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Q JEE MAIN 2022

Let $f(x)$ be a polynomial function such that $f(x)+f^{\prime}(x)+f^{\prime \prime}(x)=x^5+64$. Then, the value of $\lim _{x \rightarrow 1} \frac{f(x)}{x-1}$

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Q JEE MAIN 2023

For the differentiable function $f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$, let $3 f(x)+2 f\left(\frac{1}{x}\right)=\frac{1}{x}-10$, then $\left|f(3)+f^{\prime}\left(\frac{1}{4}\right)\right|$ is equal to

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Q JEE-Main 2023

If $2 x^y+3 y^x=20$, then $\frac{d y}{d x}$ at $(2,2)$ is equal to

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Q JEE MAIN

If a function f satisfies f(m+n)=f(m)+f(n) for all m,n∈N and f(1)=1, then the largest natural number λ such that $...

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Q JEE MAIN

Let $f(x)=a x^3+b x^2+e x+41$ be such that $f(1)=40, f^{\prime}(1)=2$ and $f^{\prime \prime}(1)=4$. Then $a^2+b^2+c^2$ is equal to :

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Q JEE MAIN 2024

Let $f(\mathrm{x})=\mathrm{x}^5+2 \mathrm{e}^{\mathrm{x} / 4}$ for all $\mathrm{x} \in \mathrm{R}$. Consider a function $g(x)$ such that (gof) $(x)=x$ for all...

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Q JEE-Main 2024

If $f(x)=\left\{\begin{array}{cc}x^3 \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x=0\end{array}\right.$, then

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Q JEE MAIN 2024

$f(x)=\left|\begin{array}{ccc}3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2\end{array}\right|$ for all $x \in \mathbb{R}$, then $...

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Q JEE MAIN 2024

If $...

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Q JEE MAIN

Suppose $f(x)=\frac{\left(2^x+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^2-x+1\right)}}{\left(7 x^2+3 x+1\right)^3}$ Then the value of f^' (0) is equal to

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Q JEE MAIN 2024

Let for a differentiable function $f:(0, \infty) \rightarrow R$, $f(x)-f(y) \geq \log _{\varepsilon}\left(\frac{x}{y}\right)+x-y, \forall x, y \in(0, \infty).$ Then $...

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Q JEE MAIN 2025

If $y(x) = \left| {\begin{array}{*{20}{c}} {\sin x}&{\cos x}&{\sin x + \cos x + 1}\\ {27}&{28}&{27}\\ 1&1&1 \end{array}} \right|$, $...

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