JEE Main Mathematics — Calculus previous year questions with solutions.
The parabola $y^2=4 x$ divides the area of the circle $x^2+y^2=5$ in two parts. The area of the smaller part is equal to:
Let $f(x)=3 \sqrt{x-2}+\sqrt{4-x}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha^2+2 \beta^2$ is equal to
Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=\frac{(\mathrm{tan}x)+y}{\mathrm{sin}x(\mathrm{sec}x-\mathrm{sin}x\mathrm{tan}x)},x\in (0,\frac{\pi }{2})$ satisfying the condition $y(\frac{\pi }{4})=2$. Then, $y(\frac{\pi }{3})$ is
Let the maximum and minimum values of $\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in \mathbf{R}$ be $\mathrm{M}$ and $\mathrm{m}$, respectively. Then $\mathrm{M}^2-\mathrm{m}^2$ is equal to _________
If $x=x(t)$ is the solution of the differential equation $(t+1)dx=(2x+{(t+1)}^{4})dt,x(0)=2$, then $x(1)$ equals ________
The area enclosed by the curves $xy+4y=16$ and $x+y=6$ is equal to:
Let $y=y(x)$ be the solution of the differential equation $\left(2 x \log _e x\right) \frac{d y}{d x}+2 y=\frac{3}{x} \log _e x, x>0$ and $y\left(e^{-1}\right)=0$. Then, $y(e)$ is equal to
The area (in sq. units) of the region described by $\left\{(x, y): y^2 \leq 2 x\right.$, and $\left.y \geq 4 x-1\right\}$ is
A function $y=f(x)$ satisfies $f(x)\mathrm{sin}2x+\mathrm{sin}x-(1+{\mathrm{cos}}^{2}x){f}^{'}(x)=0$ with condition $f(0)=0$. Then $f(\frac{\pi }{2})$ is equal to
The integral $\int_0^{\pi / 4} \frac{136 \sin x}{3 \sin x+5 \cos x} d x$ is equal to :
If the area of the region ${(x,y):0\leq y\leq \mathrm{min}{2x,6x-{x}^{2}}}$ is $A$, then $12A$ is equal to _______.
Let $\int_\alpha^{\log _e 4} \frac{\mathrm{d} x}{\sqrt{\mathrm{e}^x-1}}=\frac{\pi}{6}$. Then $\mathrm{e}^\alpha$ and $\mathrm{e}^{-\alpha}$ are the roots of the equation :
Let $y={\mathrm{log}}_{e}(\frac{1-{x}^{2}}{1+{x}^{2}}),-1<x<1$. Then at $x=\frac{1}{2}$, the value of $225({y}^{'}-{y}^{"})$ is equal to
Let $f(x)=\int_0^x\left(t+\sin \left(1-e^{\prime}\right)\right) d t, x \in \mathbb{R}$. Then, $\lim _{x \rightarrow 0} \frac{f(x)}{x^3}$ is equal to
If ${\int }_{-\pi /2}^{\pi /2}\frac{8\sqrt{2}\mathrm{cos}xdx}{(1+{e}^{\mathrm{sin}x})(1+{\mathrm{sin}}^{4}x)}=\alpha \pi +\beta {\mathrm{log}}_{e}(3+2\sqrt{2})$, where $\alpha ,\beta$ are integers, then ${\alpha }^{2}+{\beta }^{2}$ equals __________
The derivative of sin(x) with respect to x is:
Let $f$ be a differentiable function in the interval $(0, \infty)$ such that $f(1)=1$ and $\lim _{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1$ for each $x>0$. Then $2 f(2)+3 f(3)$ is equal to _______
$\underset{x\rightarrow \frac{\pi }{2}}{\mathrm{lim}}(\frac{1}{{(x-\frac{\pi }{2})}^{2}}{\int }_{{x}^{3}}^{{(\frac{\pi }{2})}^{3}}\mathrm{cos}(\frac{1}{{t}^{3}})dt)$ is equal to
Let $f:R-{0}\rightarrow R$ be a function satisfying $f(\frac{x}{y})=\frac{f(x)}{f(y)}$ for all $x,y,f(y)\neq 0$. If ${f}^{'}(1)=2024$, then
Consider the function $f:(0,\infty )\rightarrow R$ defined by $f(x)={e}^{-|{\mathrm{log}}_{e}x|}$. If $m$ and $n$ be respectively the number of points at which $f$ is not continuous and $f$ is not differentiable, then $m+n$ is
If $\alpha=\lim _{x \rightarrow 0^{+}}\left(\frac{\mathrm{e}^{\sqrt{\tan x}}-\mathrm{e}^{\sqrt{x}}}{\sqrt{\tan x}-\sqrt{x}}\right)$ and $\beta=\lim _{x \rightarrow 0}(1+\sin x)^{\frac{1}{2} \cot x}$ are the roots of the quadratic equation $a x^2+b x-\sqrt{\mathrm{e}}=0$, then $12 \log _{\mathrm{e}}(\mathrm{a}+\mathrm{b})$ is equal to__________
Three points $O(0,0),P(a,{a}^{2}),Q(-b,{b}^{2}),a>0,b>0,$ are on the parabola $y={x}^{2}$. Let ${S}_{1}$ be the area of the region bounded by the line $PQ$ and the parabola, and ${S}_{2}$ be the area of the triangle $OPQ$. If the minimum value of $\frac{{S}_{1}}{{S}_{2}}$ is $\frac{m}{n},\mathrm{gcd}(m,n)=1,$ then $m+n$ is equal to:
If the value of the integral${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}(\frac{{x}^{2}\mathrm{cos}x}{1+{\pi }^{x}}+\frac{1+{\mathrm{sin}}^{2}x}{1+{e}^{{(\mathrm{sin}x)}^{2023}}})dx=\frac{\pi }{4}(\pi +a)-2,$ then the value of $a$ is
The solution curve, of the differential equation $2 y \frac{\mathrm{d} y}{\mathrm{~d} x}+3=5 \frac{\mathrm{d} y}{\mathrm{~d} x}$, passing through the point $(0,1)$ is a conic, whose vertex lies on the line: