JEE Main Mathematics — Calculus previous year questions with solutions.
If $\frac{dy}{dx}+\frac{{2}^{x-y}({2}^{y}-1)}{{2}^{x}-1}=0,x,y>0,y(1)=1$, then $y(2)$ is equal to
$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{\mathrm{cos}(\mathrm{sin}x)-\mathrm{cos}x}{{x}^{4}}$ is equal to
Let the function $f(x)={\begin{matrix}\frac{{\mathrm{log}}_{e}(1+5x)-{\mathrm{log}}_{e}(1+\alpha x)}{x} & \mathrm{if}x\neq 0 \\ 10 & \mathrm{if}x=0\end{matrix}$ be continuous at $x=0$. Then $\alpha$ is equal to
If the solution curve $y=y(x)$ of the differential equation ${y}^{2}dx+({x}^{2}-xy+{y}^{2})dy=0$, which passes through the point $(1,1)$ and intersects the line $y=\sqrt{3}x$ at the point $(\alpha ,\sqrt{3}\alpha )$, then value of ${\mathrm{log}}_{e}(\sqrt{3}\alpha )$ is equal to
If $\int \frac{1}{x}\sqrt{\frac{1-x}{1+x}}dx=g(x)+c,g(1)=0$, then $g(\frac{1}{2})$ is equal to
$\underset{x\rightarrow \frac{1}{\sqrt{2}}}{\mathrm{lim}}\frac{\mathrm{sin}({\mathrm{cos}}^{-1}x)-x}{1-\mathrm{tan}({\mathrm{cos}}^{-1}x)}$ is equal to
Let $[t]$ denote the greatest integer $\leq t$ and ${t}$ denote the fractional part of $t$. Then integral value of $\alpha$ for which the left hand limit of the function $f(x)=[1+x]+\frac{{\alpha }^{2[x]+{x}}+[x]-1}{2[x]+{x}}$ at $x=0$ is equal to $\alpha -\frac{4}{3}$ is _____
If $\underset{x\rightarrow 1}{\mathrm{lim}}(\frac{\mathrm{sin}(3{x}^{2}-4x+1)-{x}^{2}+1}{2{x}^{3}-7{x}^{2}+ax+b})=-2$, then the value of $(a-b)$ is equal to
Let $\underset{0\leqslant x\leqslant 2}{Max}{\frac{9-{x}^{2}}{5-x}}=\alpha$ and $\underset{0\leqslant x\leqslant 2}{\mathrm{Min}}{\frac{9-{x}^{2}}{5-x}}=\beta$. If ${\int }_{\beta -\frac{8}{3}}^{2\alpha -1}Max{\frac{9-{x}^{2}}{5-x},x}dx={\alpha }_{1}+{\alpha }_{2}{\mathrm{log}}_{e}(\frac{8}{15})$, then ${\alpha }_{1}+{\alpha }_{2}$ is equal to ______
Let $a$ be an integer such that $\underset{x\rightarrow 7}{\mathrm{lim}}\frac{18-[1-x]}{[x-3a]}$ exists, where $[t]$ is greatest integer $\leq t$. Then $a$ is equal to
If $y=y(x),x\in (0,\frac{\pi }{2})$ be the solution curve of the differential equation $({\mathrm{sin}}^{2}2x)\frac{dy}{dx}+(8{\mathrm{sin}}^{2}2x+2\mathrm{sin}4x)y=$ $2{e}^{-4x}(2\mathrm{sin}2x+\mathrm{cos}2x),$ with $y(\frac{\pi }{4})={e}^{-\pi }$, then $y(\frac{\pi }{6})$ is equal to
The slope of the tangent to a curve $C:y=y(x)$ at any point $[x,y)$ on it is $\frac{2{e}^{2x}-6{e}^{-x}+9}{2+9{e}^{-2x}}$. If $C$ passes through the points $(0,\frac{1}{2}+\frac{\pi }{2\sqrt{2}})$ and $(\alpha ,\frac{1}{2}{e}^{2\alpha })$ then ${e}^{\alpha }$ is equal to
The area of the region enclosed between the parabolas ${y}^{2}=2x-1$ and ${y}^{2}=4x-3$ is.
If $f(\alpha )={\int }_{1}^{\alpha }\frac{{\mathrm{log}}_{10}t}{1+t}dt,\alpha >0$, then $f({e}^{3})+f({e}^{-3})$ is equal to
If the absolute maximum value of the function $f(x)=({x}^{2}-2x+7){e}^{(4{x}^{3}-12{x}^{2}-180x+31)}$in the interval $[-3,0]$ is $f(\alpha )$, then
Let the solution curve $y=y(x)$ of the differential equation $(1+{e}^{2x})(\frac{dy}{dx}+y)=1$ pass through the point $(0,\frac{\pi }{2})$. Then, $\underset{x\rightarrow \infty }{\mathrm{lim}}{e}^{x}y(x)$ is equal to
The number of points, where the function $f:R\rightarrow R,f(x)=|x-1|\mathrm{cos}|x-2|\mathrm{sin}|x-1|+(x-3)|{x}^{2}-5x+4|$, is NOT differentiable, is
The function $f:R\rightarrow R$ defined by $f(x)=\underset{n\rightarrow \infty }{\mathrm{lim}}\frac{\mathrm{cos}(2\pi x)-{x}^{2n}\mathrm{sin}(x-1)}{1+{x}^{2n+1}-{x}^{2n}}$ is continuous for all $x$ in
Let $f:R\rightarrow R$ be a function defined by : $f(x)={\begin{matrix}\underset{t\leq x}{\mathrm{max}}{{t}^{3}-3t}; & x\leq 2 \\ {x}^{2}+2x-6; & 2<x<3 \\ [x-3]+9; & 3\leq x\leq 5 \\ 2x+1; & x>5\end{matrix}$ Where $[t]$ is the greatest integer less than or equal to $t$. Let $m$ be the number of points where $f$ is not differentiable and $I={\int }_{-2}^{2}f(x)dx$. Then the ordered pair $(m,I)$ is equal to
If the function $f(x)={\begin{matrix}\frac{{\mathrm{log}}_{e}(1-x+{x}^{2})+{\mathrm{log}}_{e}(1+x+{x}^{2})}{secx-\mathrm{cos}x}, & x\in (\frac{-\pi }{2},\frac{\pi }{2})-{0} \\ k & ,x=0\end{matrix}$ is continuous at $x=0$, then $k$ is equal to:
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined as $f(x)=[\begin{matrix}[{e}^{x}], & x<0 \\ a{e}^{x}+[x-1], & 0\leq x<1 \\ b+[\mathrm{sin}(\pi x)], & 1\leq x<2 \\ [{e}^{-x}]-c, & x\geq 2\end{matrix}$ where $a,b,c\in \mathbb{R}$ and $[t]$ denotes greatest integer less than or equal to $t$. Then, which of the following statements is true?
If $y(x)={({x}^{x})}^{x},x>0$ then $\frac{{d}^{2}x}{d{y}^{2}}+20$ at $x=1$ is equal to
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 $14\cdot \frac{{f}^{'}(4)}{{f}^{'}(2)}$ is equal to _____.
If $y={\mathrm{tan}}^{-1}(\mathrm{sec}{x}^{3}-\mathrm{tan}{x}^{3}),\frac{\pi }{2}<{x}^{3}<\frac{3\pi }{2}$, then