JEE Main Mathematics — Calculus previous year questions with solutions.
If $\underset{x\rightarrow \infty }{\mathrm{lim}}(\sqrt{{x}^{2}-x+1}-ax)=b$, then the ordered pair $(a,b)$ is:
Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}2\mathrm{sin}(-\frac{\pi x}{2}), & \mathrm{if}x<-1 \\ |a{x}^{2}+x+b|, & \mathrm{if}-1\leq x\leq 1 \\ \mathrm{sin}(\pi x), & \mathrm{if}x>1\end{matrix}$ If $f(x)$ is continuous on $R$, then $a+b$ equals :
If the integral ${\int }_{0}^{10}\frac{[\mathrm{sin}2\pi x]}{{e}^{x-[x]}}dx=\alpha {e}^{-1}+\beta {e}^{-\frac{1}{2}}+\gamma ,$ where $\alpha ,\beta ,\gamma$ are integers and $[x]$ denotes the greatest integer less than or equal to $x,$ then the value of $\alpha +\beta +\gamma$ is equal to:
$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{\int }_{0}^{{x}^{2}}(\mathrm{sin}\sqrt{t})dt}{{x}^{3}}$ is equal to:
If $\int \frac{dx}{{({x}^{2}+x+1)}^{2}}=a{\mathrm{tan}}^{-1}(\frac{2x+1}{\sqrt{3}})+b(\frac{2x+1}{{x}^{2}+x+1})+C,x>0$ where $C$ is the constant of integration, then the value of $9(\sqrt{3}a+b)$ is equal to _________.
If ${\int }_{0}^{100\pi }\frac{{\mathrm{sin}}^{2}x}{{e}^{(\frac{x}{\pi }-[\frac{x}{\pi }])}}dx=\frac{\alpha {\pi }^{3}}{1+4{\pi }^{2}},\alpha \in R$ where $[x]$ is the greatest integer less than or equal to $x,$ then the value of $\alpha$ is:
Consider the integral $I={\int }_{0}^{10}\frac{[x]{e}^{[x]}}{{e}^{x-1}}dx$ where $[x]$ denotes the greatest integer less than or equal to $x$. Then the value of $I$ is equal to :
The sum of all the local minimum values of the twice differentiable function $f:R\rightarrow R$ defined by $f(x)={x}^{3}-3{x}^{2}-\frac{3{f}^{"}(2)}{2}x+{f}^{"}(1)$ is:
If $\alpha =\underset{x\rightarrow \pi /4}{\mathrm{lim}}\frac{{\mathrm{tan}}^{3}x-\mathrm{tan}x}{\mathrm{cos}(x+\frac{\pi }{4})}$ and $\beta =\underset{x\rightarrow 0}{\mathrm{lim}}{(\mathrm{cos}x)}^{\mathrm{cot}x}$ are the roots of the equation, $a{x}^{2}+bx-4=0,$ then the ordered pair $(a,b)$ is :
The area of the region bounded by $y-x=2$ and ${x}^{2}=y$ is equal to :-
If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{\mathrm{sin}}^{-1}x-{\mathrm{tan}}^{-1}x}{3{x}^{3}}$ is equal to $L,$ then the value of $(6L+1)$ is
If ${\int }_{0}^{\pi }({\mathrm{sin}}^{3}x){e}^{-{\mathrm{sin}}^{2}x}dx=\alpha -\frac{\beta }{e}{\int }_{0}^{1}\sqrt{t}{e}^{t}dt,$ then $\alpha +\beta$ is equal to
If ${I}_{n}={\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}{\mathrm{cot}}^{n}xdx$, then
Let $[t]$ denote the greatest integer $\leq t.$ The number of points where the function $f(x)=[x]|{x}^{2}-1|+\mathrm{sin}(\frac{\pi }{[x]+3})-[x+1],x\in (-2,2)$ is not continuous is _____ .
Let $a,b\in R,b\neq 0$. Defined a function, $f(x)={\begin{matrix}a\mathrm{sin}\frac{\pi }{2}(x-1),\mathrm{for}x\leq 0 \\ \frac{\mathrm{tan}2x-\mathrm{sin}2x}{b{x}^{3}},\mathrm{for}x>0\end{matrix}$ If $f$ is continuous at$x=0$, then $10-ab$ is equal to
Let $f:[0,\infty )\rightarrow [0,\infty )$ be defined as $f(x)={\int }_{0}^{x}[y]dy$ where $[x]$ is the greatest integer less than or equal to $x$. Which of the following is true?
Let $f:(-\frac{\pi }{4},\frac{\pi }{4})\rightarrow R$ be defined as, $f(x)={\begin{matrix}{(1+|\mathrm{sin}x|)}^{\frac{3a}{|\mathrm{sin}x|}} & ,-\frac{\pi }{4}<x<0 \\ b & ,x=0 \\ {e}^{\mathrm{cot}4x/\mathrm{cot}2x} & ,0<x<\frac{\pi }{4}\end{matrix}$ If $f$ is continuous at $x=0$ then the value of $6a+{b}^{2}$ is equal to:
Let $y=y(x)$ be the solution of the differential equation $\mathrm{cos}x(3\mathrm{sin}x+\mathrm{cos}x+3)dy=(1+y\mathrm{sin}x(3\mathrm{sin}x+\mathrm{cos}x+3))dx,0\leq x\leq \frac{\pi }{2},y(0)=0.$ Then, $y(\frac{\pi }{3})$ is equal to:
If $y=y(x)$ is an implicit function of $x$ such that ${\mathrm{log}}_{e}(x+y)=4xy$, then $\frac{{d}^{2}y}{d{x}^{2}}$ at $x=0$ is equal to
If $f(x)=\mathrm{sin}({\mathrm{cos}}^{-1}(\frac{1-{2}^{2x}}{1+{2}^{2x}}))$ and its first derivative with respect to $x$ is $-\frac{b}{a}{\mathrm{log}}_{e}2$ when $x=1,$ where $a$ and $b$ are integers, then the minimum value of $|{a}^{2}-{b}^{2}|$ is _______.
Let $f(x)=\mathrm{cos}(2{\mathrm{tan}}^{-1}\mathrm{sin}({\mathrm{cot}}^{-1}\sqrt{\frac{1-x}{x}})),0<x<1$. Then:
The number of distinct real roots of the equation $3{x}^{4}+4{x}^{3}-12{x}^{2}+4=0$ is _________.
Let $P(x)$ be a real polynomial of degree $3$ which vanishes at $x=-3$. Let $P(x)$ have local minima at $x=1$, local maxima at $x=-1$ and ${\int }_{-1}^{1}P(x)dx=18$, then the sum of all the coefficients of the polynomial $P(x)$ is equal to ___ .
Let $a$ be an integer such that all the real roots of the polynomial $2{x}^{5}+5{x}^{4}+10{x}^{3}+10{x}^{2}+10x+10$ lie in the interval $(a,a+1)$. Then, $|a|$ is equal to ______.