JEE Main Mathematics — Calculus previous year questions with solutions.
If $\alpha ,\beta$ are the distinct roots of ${x}^{2}+bx+c=0,$ then $\underset{x\rightarrow \beta }{\mathrm{lim}}\frac{{e}^{2({x}^{2}+bx+c)}-1-2({x}^{2}+bx+c)}{(x-\beta {)}^{2}}$ is equal to
If the function $f(x)=\frac{\mathrm{cos}(\mathrm{sin}x)-\mathrm{cos}x}{{x}^{4}}$ is continuous at each point in its domain and $f(0)=\frac{1}{k},$ then $k$ is _________.
Let $f(x)$ be a polynomial of degree $6$ in $x,$ in which the coefficient of ${x}^{6}$ is unity and it has extrema at $x=-1$ and $x=1$. If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{f(x)}{{x}^{3}}=1,$ then $5\cdot f(2)$ is equal to
Let $f(x)={\int }_{0}^{x}{e}^{t}f(t)dt+{e}^{x}$ be a differentiable function for all $x\in R$. Then $f(x)$ equals :
Let $g(t)={\int }_{-\pi /2}^{\pi /2}(\mathrm{cos}\frac{\pi }{4}t+f(x))dx,$ where $f(x)={\mathrm{log}}_{e}(x+\sqrt{{x}^{2}+1}),x\in R.$ Then which one of the following is correct?
If $f(x)={\begin{matrix}\frac{1}{|x|} & ; & |x|\geq 1 \\ a{x}^{2}+b & ; & |x|<1\end{matrix}$ is differentiable at every point of the domain, then the values of $a$ and $b$ are respectively:
Let $y=y(x)$ be the solution of the differential equation $\frac{dy}{dx}=(y+1)((y+1){e}^{{x}^{2}/2}-x),0<x<2.1$, with $y(2)=0$. Then the value of $\frac{dy}{dx}$ at $x=1$ is equal to
$\int \frac{2{e}^{x}+3{e}^{-x}}{4{e}^{x}+7{e}^{-x}}dx=\frac{1}{14}(ux+v{\mathrm{log}}_{e}(4{e}^{x}+7{e}^{-x}))+C$, where $C$ is a constant of integration, then $u+v$ is equal to
If $\int \frac{\mathrm{cos}x-\mathrm{sin}x}{\sqrt{8-\mathrm{sin}2x}}dx=a{\mathrm{sin}}^{-1}(\frac{\mathrm{sin}x+\mathrm{cos}x}{b})+c,$ where $c$ is a constant of integration, then the ordered pair $(a,b)$ is equal to:
Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}-55x, & \mathrm{if}x<-5 \\ 2{x}^{3}-3{x}^{2}-120x, & \mathrm{if}-5\leq x\leq 4 \\ 2{x}^{3}-3{x}^{2}-36x-336, & \mathrm{if}x>4\end{matrix}$ Let $A={x\in R:f$ is increasing}. Then $A$ is equal to:
If $y=y(x)$ is the solution of the equation ${e}^{\mathrm{sin}y}\mathrm{cos}y\frac{dy}{dx}+{e}^{\mathrm{sin}y}\mathrm{cos}x=\mathrm{cos}x,y(0)=0$; then $1+y(\frac{\pi }{6})+\frac{\sqrt{3}}{2}y(\frac{\pi }{3})+\frac{1}{\sqrt{2}}y(\frac{\pi }{4})$ is equal to _______.
Let $f$ be any function defined on $R$ and let it satisfy the condition: $|f(x)-f(y)|\leq |{(x-y)}^{2}|,\forall (x,y)\in R$. If $f(0)=1,$ then :
The value of the integral ${\int }_{-1}^{1}\mathrm{log}(x+\sqrt{{x}^{2}+1})dx$ is:
Let $y=y(x)$ satisfies the equation $\frac{dy}{dx}-|A|=0,$ for all $x>0,$ where $A=[\begin{matrix}y & \mathrm{sin}x & 1 \\ 0 & -1 & 1 \\ 2 & 0 & \frac{1}{x}\end{matrix}]$. If $y(\pi )=\pi +2,$ then the value of $y(\frac{\pi }{2})$ is:
If the curve $y=y(x)$ is the solution of the differential equation $2({x}^{2}+{x}^{5/4})dy-y(x+{x}^{1/4})dx=2{x}^{9/4}dx,x>0$ which passes through the point $(1,1-\frac{4}{3}{\mathrm{log}}_{e}2),$ then the value of $y(16)$ is equal to
Let ${I}_{n}={\int }_{1}^{e}{x}^{19}(\mathrm{log}|x|{)}^{n}dx,$ where $n\in N$. If $(20){I}_{10}=\alpha {I}_{9}+\beta {I}_{8},$ for natural numbers $\alpha$ and $\beta$, then $\alpha -\beta$ equal to _______.
If a curve $y=f(x)$ passes through the point $(1,2)$ and satisfies $x\frac{dy}{dx}+y=b{x}^{4},$ then for what value of $b,{\int }_{1}^{2}f(x)dx=\frac{62}{5}?$
Let $y=y(x)$ be the solution of the differential equation $dy={e}^{\alpha x+y}dx;\alpha \in N.$ If $y({\mathrm{log}}_{e}2)={\mathrm{log}}_{e}2$ and $y(0)={\mathrm{log}}_{e}(\frac{1}{2}),$ then the value of $\alpha$ is equal to ___.
Let $g(x)={\int }_{0}^{x}f(t)dt$, where $f$ is continuous function in $[0,3]$ such that $\frac{1}{3}\leq f(t)\leq 1$ for all $t\in [0,1]$ and $0\leq f(t)\leq \frac{1}{2}$ for all $t\in (1,3]$. The largest possible interval in which $g(3)$ lies is :
If $[x]$ is the greatest integer $\leq x,$ then ${\pi }^{2}{\int }_{0}^{2}(\mathrm{sin}\frac{\pi x}{2}){(x-[x])}^{[x]}dx$ is equal to :
The local maximum value of the function, $f(x)={(\frac{2}{x})}^{{x}^{2}},x>0,$ is
Let $f$ be a twice differentiable function defined on $R$ such that $f(0)=1,{f}^{'}(0)=2$ and ${f}^{'}(x)\neq 0$ for all $x\in R.$ If $|\begin{matrix}f(x) & {f}^{'}(x) \\ {f}^{'}(x) & {f}^{''}(x)\end{matrix}|=0,$ for all $x\in R,$ then the value of $f(1)$ lies in the interval
If $y(x)={\mathrm{cot}}^{-1}(\frac{\sqrt{1+\mathrm{sin}x}+\sqrt{1-\mathrm{sin}x}}{\sqrt{1+\mathrm{sin}x}-\sqrt{1-\mathrm{sin}x}}),x\in (\frac{\pi }{2},\pi )$, then $\frac{dy}{dx}$ at $x=\frac{5\pi }{6}$ is:
The area of the region bounded by the parabola $(y-2{)}^{2}=(x-1)$, the tangent to it at the point whose ordinate is $3$ and the $x$-axis, is: