JEE Main Mathematics — Calculus previous year questions with solutions.
Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}{x}^{5}\mathrm{sin}(\frac{1}{x})+5{x}^{2} & , & x<0 \\ 0 & , & x=0 \\ {x}^{5}\mathrm{cos}(\frac{1}{x})+\lambda {x}^{2} & , & x>0\end{matrix}$. The value of $\lambda$ for which ${f}^{"}(0)$ exists, is___.
Let $f(x)=x\cdot [\frac{x}{2}],$ for $-10<x<10,$ where $[t]$ denotes the greatest integer function. Then the number of points of discontinuity of $f(x)$ is equal to
Let $[t]$ denote the greatest integer $\leq t$ and $\underset{x\rightarrow 0}{\mathrm{lim}}x[\frac{4}{x}]=A.$ Then the function, $f(x)=[{x}^{2}]\mathrm{sin}(\pi x)$ is discontinuous, when $x$ is equal to:
If a function $f(x)$ defined by $f(x)={\begin{matrix}a{e}^{x}+b{e}^{-x}, & -1\leq x<1 \\ c{x}^{2}, & 1\leq x\leq 3 \\ a{x}^{2}+2cx, & 3<x\leq 4\end{matrix}$be continuous for some $a,b,c\in R$ and ${f}^{'}(0)+{f}^{'}(2)=e$, then the value of $a$ is
Let $f$ and $g$ be differentiable functions on $R$ such that $fog$ is the identity function. If for some $a,b\in R,{g}^{'}(a)=5$ and $g(a)=b,$ then ${f}^{'}(b)$ is equal to:
If $y(\alpha )=\sqrt{2(\frac{tan\alpha +cot\alpha }{1+ta{n}^{2}\alpha })+\frac{1}{si{n}^{2}\alpha }},\alpha \in (\frac{3\pi }{4},\pi )$, then $\frac{dy}{d\alpha }$ at $\alpha =\frac{5\pi }{6}$ is
Let ${x}^{k}+{y}^{k}={a}^{k},(a,k>0)$ and $\frac{dy}{dx}+{(\frac{y}{x})}^{\frac{1}{3}}=0,$ then $k$ is
If $x=1$ is a critical point of the function $f(x)=(3{x}^{2}+ax-2-a){e}^{x},$ then
If the surface area of a cube is increasing at a rate of $3.6c{m}^{2}/sec$, retaining its shape; then the rate of change of its volume (in $c{m}^{3}/sec$), when the length of a side of the cube is $10cm$, is:
Suppose $f(x)$ is a polynomial of degree four having critical points at $-1,0,1$. If $T={x\in R|f(x)=f(0)}$, then the sum of squares of all the elements of $T$ is :
$\underset{x\rightarrow 0}{\mathrm{lim}}{(\mathrm{tan}(\frac{\pi }{4}+x))}^{1/x}$ is equal to
Let $f(x)$, be a polynomial of degree $3$, such that $f(-1)=10, f(1)=-6, f(x)$, has a critical point at $x=-1$ and $f'(x)$, has a critical point at $x=1.$ Then $f(x)$, has local minima at $x=$
Let $f:(0,\infty )\rightarrow (0,\infty )$ be a differentiable function such that $f(1)=e$ and $\underset{t\rightarrow x}{\mathrm{lim}}\frac{{t}^{2}{f}^{2}(x)-{x}^{2}{f}^{2}(t)}{t-x}=0.$ If$f(x)=1$, then $x$ is equal to:
If ${I}_{1}={\int }_{0}^{1}{(1-{x}^{50})}^{100}dx$ and ${I}_{2}={\int }_{0}^{1}{(1-{x}^{50})}^{101}dx$ such that ${I}_{2}=\alpha {I}_{1}$ then $\alpha$ equals to :
If $\int ({e}^{2x}+2{e}^{x}-{e}^{-x}-1){e}^{({e}^{x}+{e}^{-x})}dx=g(x){e}^{({e}^{x}+{e}^{-x})}+c,$ where $c$ is a constant of integration, then $g(0)$ is
If$\int \frac{\mathrm{cos}\theta }{5+7\mathrm{sin}\theta -2{\mathrm{cos}}^{2}\theta }d\theta ={Alog}_{e}|B(\theta )|+C,$ where $C$ is a constant of integration, then $\frac{B(\theta )}{A}$ can be:
If $\int {\mathrm{sin}}^{-1}(\frac{\sqrt{x}}{1+x})dx=A(x){\mathrm{tan}}^{-1}(\sqrt{x})+B(x)+C$, where $C$ is a constant of integration, then the ordered pair $(A(x),B(x))$ can be :
The integral ${\int }_{1}^{2}{e}^{x}.{x}^{x}(2+{\mathrm{log}}_{e}x)\mathrm{dx}$ equals :
Let ${x}$ and $[x]$ denote the fractional part of $x$ and the greatest integer $\leq x$ respectively of a real number $x$. if ${\int }_{0}^{n}{x}dx,{\int }_{0}^{n}[x]dx$ and $10({n}^{2}-n),(n\in N,n>1)$ are three consecutive terms of a G.P. then $n$ 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:
The area (in sq. units) of the region ${(x,y):0\leq y\leq {x}^{2}+1,0\leq y\leq x+1,\frac{1}{2}\leq x\leq 2}$ is
The area of the region (in sq. units), enclosed by the circle ${x}^{2}+{y}^{2}=2$ which is not common to the region bounded by the parabola ${y}^{2}=x$ and the straight line $y=x$, is
If $(a+\sqrt{2}b\mathrm{cos}x)(a-\sqrt{2}b\mathrm{cos}y)={a}^{2}-{b}^{2}$, where $a>b>0,$ then$\frac{dx}{dy}$ at $(\frac{\pi }{4},\frac{\pi }{4})$ is:
A spherical iron ball of $10cm$ radius is coated with a layer of ice of uniform thickness that melts at a rate of $50{cm}^{3}/min$ . When the thickness of ice is $5cm$ , then the rate (in $cm/min$ .) at which of the thickness of ice decreases, is: