JEE Main Mathematics — Calculus previous year questions with solutions.
For $0<a<1$, the value of the integral ${\int }_{0}^{\pi }\frac{dx}{1-2a\mathrm{cos}x+{a}^{2}}$ is :
Consider the function $f(x)={\begin{matrix}\frac{a(7x-12-{x}^{2})}{b|{x}^{2}-7x+12|} & ,x<3 \\ {2}^{\frac{\mathrm{sin}(x-3)}{x-[x]}} & ,x>3 \\ b & ,x=3\end{matrix}$,where $[x]$ denotes the greatest integer less than or equal to $x$. If $S$ denotes the set of all ordered pairs $(a,b)$ such that $f(x)$is continuous at $x=3$, then the number of elements in $S$ is :
If the function $f(x)={\begin{matrix}\frac{1}{|x|}, & |x|\geq 2 \\ a{x}^{2}+2b, & |x|<2\end{matrix}$ is differentiable on $R$, then $48(a+b)$ is equal to _______.
$\lim _{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x}$ is equal to
The value of $\lim _{x \rightarrow 0} 2\left(\frac{1-\cos x \sqrt{\cos 2 x} \sqrt[3]{\cos 3 x} \ldots \ldots \sqrt[10]{\cos 10 x}}{x^2}\right)$ is
If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{a{x}^{2}{e}^{x}-b{\mathrm{log}}_{e}(1+x)+cx{e}^{-x}}{{x}^{2}\mathrm{sin}x}=1$, then $16({a}^{2}+{b}^{2}+{c}^{2})$ is equal to ______.
Let $f(x)=\left\{\begin{array}{lr}-2, & -2 \leq x \leq 0 \\ x-2, & 0 < x \leq 2\end{array}\right.$ and $h(x)=f(|x|)+|f(x)|$. Then $\int_{-2}^2 h(x) \mathrm{d} x$ is equal to :
Let $x=x(t)$ and $y=y(t)$ be solutions of the differential equations $\frac{\mathrm{dx}}{\mathrm{dt}}+\mathrm{ax}=0$ and $\frac{\mathrm{dy}}{\mathrm{dt}}+\mathrm{by}=0$ respectively, $a,b\in R$. Given that $x(0)=2;y(0)=1$ and $3y(1)=2x(1)$, the value of $t$, for which $x(t)=y(t)$, is :
If $y(\theta)=\frac{2 \cos \theta+\cos 2 \theta}{\cos 3 \theta+4 \cos 2 \theta+5 \cos \theta+2}$, then at $\theta=\frac{\pi}{2}, y^{\prime \prime}+y^{\prime}+y$ is equal to :
The value of $\int_{-\pi}^\pi \frac{2 y(1+\sin y)}{1+\cos ^2 y} d y$ is :
Let $y=y(x)$ be the solution of the differential equation $\left(x^2+4\right)^2 d y+\left(2 x^3 y+8 x y-2\right) d x=0$. If $y(0)=0$, then $y(2)$ is equal to
For $\mathrm{a}, \mathrm{b}>0$, let $f(x)=\left\{\begin{array}{cc}\frac{\tan ((\mathrm{a}+1) x)+\mathrm{b} \tan x}{x}, & x < 0 \\ 3, & x=0 \\ \frac{\sqrt{\mathrm{a} x+\mathrm{b}^2 x^2}-\sqrt{\mathrm{a} x}}{\mathrm{~b} \sqrt{\mathrm{a}} x \sqrt{x}}, & x>0\end{array}\right.$ be a continous function at $x=0$. Then $\frac{\mathrm{b}}{\mathrm{a}}$ is equal to :
If the solution $y=y(x)$ of the differential equation $\left(x^4+2 x^3+3 x^2+2 x+2\right) \mathrm{d} y-\left(2 x^2+2 x+3\right) \mathrm{d} x=0$ satisfies $y(-1)=-\frac{\pi}{4}$, then $y(0)$ is equal to :
If $\log _e y=3 \sin ^{-1} x$, then $\left(1-x^2\right) y^{\prime \prime}-x y^{\prime}$ at $x=\frac{1}{2}$ is equal to
$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{e}^{2|\mathrm{sin}x|}-2|\mathrm{sin}x|-1}{{x}^{2}}$
Let $f:[-\frac{\pi }{2},\frac{\pi }{2}]\rightarrow R$ be a differentiable function such that $f(0)=\frac{1}{2}$, If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{x{\int }_{0}^{x}f(t)dt}{{e}^{{x}^{2}}-1}=\alpha$, then $8{\alpha }^{2}$ is equal to :
Let $\mathrm{A}$ be the region enclosed by the parabola $y^2=2 x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $\mathrm{A}$ is________
Let $f:(-\infty, \infty)-\{0\} \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(1)=\lim _{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)$. Then $\lim _{a \rightarrow \infty} \frac{a(a+1)}{2} \tan ^{-1}\left(\frac{1}{a}\right)+a^2-2 \log _e a$ is equal to
If $\lim _{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{\mathrm{m} \sqrt{5}}{\mathrm{n}(2 \mathrm{n})^{2 / 3}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $8 \mathrm{~m}+12 \mathrm{n}$ is equal to______
Let $\int_0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} d t=\int_0^x y(t) d t, 0 \leq x \leq 3, y \geq 0, y(0)=0$. Then at $x=2, y^{\prime \prime}+y+1$ is equal to
Let $f(x)=|2{x}^{2}+5|x|-3|,x\in R$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m+n$ is equal to:
Let $\mathrm{a}>0$ be a root of the equation $2 x^2+x-2=0$. If $\lim _{x \rightarrow \frac{1}{\mathrm{a}}} \frac{16\left(1-\cos \left(2+x-2 x^2\right)\right)}{(1-\mathrm{a} x)^2}=\alpha+\beta \sqrt{17}$, where $\alpha, \beta \in Z$, then $\alpha+\beta$ is equal to_______
For the function $f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],$ consider the following two statements : (I) $\mathrm{f}$ is increasing in $\left(0, \frac{\pi}{2}\right)$. (II) $f^{\prime}$ is decreasing in $\left(0, \frac{\pi}{2}\right)$. Between the above two statements,
Let $a$ be the sum of all coefficients in the expansion of $(1–2x+2{x}^{2}{)}^{2023}(3-4{x}^{2}+2{x}^{3}{)}^{2024}$ and $b=\underset{x\rightarrow 0}{\mathrm{lim}}(\frac{{\int }_{0}^{x}\frac{\mathrm{log}(1+t)}{{t}^{2024}+1}dt}{{x}^{2}})$. If the equations $c{x}^{2}+dx+e=0$ and $2b{x}^{2}+ax+4=0$ have a common root, where $c,d,e\in R$, then $d:c:e$ equals