JEE Main Mathematics — Calculus previous year questions with solutions.
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:
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 $f:R\rightarrow R$ is a function defined by $f(x)=[x-1]\mathrm{cos}(\frac{2x-1}{2})\pi ,$ where $[\cdot ]$ denotes the greatest integer function, then $f$ is:
If ${I}_{m,n}={\int }_{0}^{1}{x}^{m-1}{(1-x)}^{n-1}dx$, for $m,n\geqslant 1$, and ${\int }_{0}^{1}\frac{{x}^{m-1}+{x}^{n-1}}{{(1+x)}^{m+n}}dx=\alpha {I}_{m,n},\alpha \in R$, then $\alpha$ equals ________.
If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{ax-({e}^{4x}-1)}{ax({e}^{4x}-1)}$ exists and is equal to $b$, then the value of $a-2b$ is ___ .
If $[x]$ denotes the greatest integer less than or equal to $x,$ then the value of the integral ${\int }_{-\pi /2}^{\pi /2}[[x]-\mathrm{sin}x]dx$ is equal to:
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 $f(x)=\int \frac{5{x}^{8}+7{x}^{6}}{{({x}^{2}+1+2{x}^{7})}^{2}}dx,(x\geq 0),f(0)=0$ and $f(1)=\frac{1}{K},$ then the value of $K$ is
If ${y}^{1/4}+{y}^{-1/4}=2x,$ and $({x}^{2}-1)\frac{{d}^{2}y}{d{x}^{2}}+\alpha x\frac{dy}{dx}+\beta y=0,$ then $|\alpha -\beta |$ is equal to _______.
If $y\frac{dy}{dx}=x[\frac{{y}^{2}}{{x}^{2}}+\frac{\phi (\frac{{y}^{2}}{{x}^{2}})}{{\phi }^{'}(\frac{{y}^{2}}{{x}^{2}})}],x>0,\phi >0,$ and $y(1)=-1,$ then $\phi (\frac{{y}^{2}}{4})$ 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 _______.
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 :
If a rectangle is inscribed in an equilateral triangle of side length $2\sqrt{2}$ as shown in the figure, then the square of the largest area of such a rectangle is _____. 
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 $b,{\int }_{1}^{2}f(x)dx=\frac{62}{5}?$
For real numbers $\alpha ,\beta ,\gamma$ and $\delta ,$ if $\int \frac{({x}^{2}-1)+{\mathrm{tan}}^{-1}(\frac{{x}^{2}+1}{x})}{({x}^{4}+3{x}^{2}+1){\mathrm{tan}}^{-1}(\frac{{x}^{2}+1}{x})}dx=\alpha {\mathrm{log}}_{e}({\mathrm{tan}}^{-1}(\frac{{x}^{2}+1}{x}))+\beta {\mathrm{tan}}^{-1}(\frac{\gamma ({x}^{2}-1)}{x})+\delta {\mathrm{tan}}^{-1}(\frac{{x}^{2}+1}{x})+C$ where $C$ is an arbitrary constant, then the value of $10(\alpha +\beta \gamma +\delta )$ is equal to ______ .
For $x>0$, if $f(x)={\int }_{1}^{x}\frac{{\mathrm{log}}_{e}t}{(1+t)}dt$, then $f(e)+f(\frac{1}{e})$ is equal to
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 :
Consider the function $f:R\rightarrow R$ defined by $f(x)={\begin{matrix}(2-\mathrm{sin}(\frac{1}{x}))|x|, & x\neq 0 \\ 0, & x=0\end{matrix}.$ Then $f$ is:
Consider the function $f(x)=\frac{P(x)}{\mathrm{sin}(x-2)},x\neq 2$, and $f(x)=7,x=2$where $P(x)$ is a polynomial such that ${P}^{"}(x)$ is always a constant and $P(3)=9.$ If $f(x)$ is continuous at $x=2,$ then $P(5)$ is equal to __________.
A wire of length $20m$ is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in meters) of the hexagon, so that the combined area of the square and the hexagon is minimum, is
A wire of length $36m$ is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum, and the circumference of the circle is $k$ (meter), then $(\frac{4}{\pi }+1)k$ is equal to
A function $f$ is defined on $[-3,3]$ as $f(x)={\begin{matrix}\mathrm{min}{|x|,2-{x}^{2}},-2\leq x\leq 2 \\ [|x|],2<|x|\leq 3\end{matrix}$ where $[x]$ denotes the greatest integer $\leq x$. The number of points, where $f$ is not differentiable in $(-3,3)$ is ___ .
A box open from top is made from a rectangular sheet of dimension $a\times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $x$ is equal to:
The value of ${\int }_{0}^{2\pi }\frac{x{\mathrm{sin}}^{8}x}{{\mathrm{sin}}^{8}x+{\mathrm{cos}}^{8}x}dx$ is equal to: