Question

Let $x\left(t\right)=2\sqrt{2}\cos t\sqrt{\sin 2t}$ and $y\left(t\right)=2\sqrt{2}\sin t\sqrt{\sin 2t},t\in \left(0,\frac{\pi }{2}\right)$. Then $\frac{1+{\left(\frac{dy}{dx}\right)}^2}{\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}}$ at $t=\frac{\pi }{4}$ is equal to

JEE Main 2022 (28 Jul Shift 2)

Options

• A: $\frac{-2\sqrt{2}}{3}$
• B: $\frac{2}{3}$
• C: $\frac{1}{3}$
• D: $\frac{-2}{3}$

Question

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

JEE Main 2021 (26 Aug Shift 1)

Options

• A: $(1-x{)}^2{f}^{\text{'}}(x)+2(f\left(x\right){)}^2=0$
• B: $(1+x{)}^2{f}^{\text{'}}(x)+2(f\left(x\right){)}^2=0$
• C: $(1-x{)}^2{f}^{\text{'}}(x)-2(f\left(x\right){)}^2=0$
• D: $(1+x{)}^2{f}^{\text{'}}(x)-2(f\left(x\right){)}^2=0$

Question

If $y\left(x\right)={\cot }^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right), x\in \left(\frac{\pi }{2},\pi \right)$, then $\frac{dy}{dx}$ at $x=\frac{5\pi }{6}$ is:

JEE Main 2021 (27 Aug Shift 2)

Options

• A: $0$
• B: $-1$
• C: $\frac{-1}{2}$
• D: $\frac{1}{2}$

Question

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

JEE Main 2024 (29 Jan Shift 1)

Options

• A: $\pi$
• B: $0$
• C: $\sqrt{\pi }$
• D: $\frac{\pi }{2}$

Question

Let $f:S\to S$ where $S=\left(0,\infty \right)$ be a twice differentiable function such that $f\left(x+1\right)=xf\left(x\right)$. If $g:S\to R$ be defined as $g\left(x\right)={\log }_{\mathrm{e}}f\left(x\right)$, then the value of $\left|g^{\text{'}\text{'}}\left(5\right)-g^{\text{'}\text{'}}\left(1\right)\right|$ is equal to :

JEE Main 2021 (16 Mar Shift 2)

Options

• A: $\frac{205}{144}$
• B: $\frac{197}{144}$
• C: $\frac{187}{144}$
• D: $1$

Question

Let $f:\mathit{R}\to \mathit{R}$ be defined as $f\left(x\right)=x^3+x-5$. If $g\left(x\right)$ is a function such that $f\left(g\left(x\right)\right)=x,\forall x\in \mathit{R}$, then $g^{\text{'}}\left(63\right)$ is equal to ______

JEE Main 2022 (25 Jun Shift 1)

Options

• A: $49$
• B: $\frac{1}{49}$
• C: $\frac{43}{49}$
• D: $\frac{3}{49}$

Question

Let for a differentiable function $f:(0,\infty )\to R$, $f\left(x\right)-f\left(y\right)\ge {\log }_e\left(\frac{x}{y}\right)+x-y,\forall x,y\in \left(0,\infty \right)$. Then $\sum _{n=1}^{20}{f}^{\text{'}}\left(\frac{1}{n^2}\right)$ is equal to

JEE Main 2024 (27 Jan Shift 1)

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Question

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

JEE Main 2022 (27 Jun Shift 2)

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Question

Suppose for a differentiable function $h, h(0)=0, h(1)=1$ and $h^{\prime}(0)=h^{\prime}(1)=2$. If $\mathrm{g}(x)=h\left(\mathrm{e}^x\right) \mathrm{e}^{h(x)}$, then $g^{\prime}(0)$ is equal to:

JEE Main 2024 (06 Apr Shift 2)

Options

• A: 5
• B: 4
• C: 8
• D: 3