JEE Main Mathematics — Calculus previous year questions with solutions.
If $\int {e}^{secx}(secx\mathrm{tan}xf(x)+(secx\mathrm{tan}x+se{c}^{2}x))dx={e}^{secx}f(x)+C,$ then a possible choice of $f(x)$ is:
If $f(x)={\begin{matrix}\frac{sin(p+1)x+sinx}{x} & , & x<0 \\ q & , & x=0 \\ \frac{\sqrt{x+{x}^{2}}-\sqrt{x}}{{x}^{3/2}} & , & x>0\end{matrix}$ is continuous at $x=0$ , then the ordered pair $(p, q)$ is equal to:
For each $x\in R$, let $[x]$ be the greatest integer less than or equal to $x$. Then $\underset{x\rightarrow {0}^{-}}{lim}\frac{x([x]+|x|)\mathrm{sin}[x]}{|x|}$ is equal to
The value of $\underset{y\rightarrow 0}{lim}\frac{\sqrt{1+\sqrt{1+{y}^{4}}}-\sqrt{2}}{{y}^{4}}$
The value of $\int _{0}^{\pi /2}\frac{{sin}^{3}x}{sinx+cosx}dx$ is:
Let $f:R\rightarrow R$ be differentiable at $c\in R$ and $f(c)=0$. If $g(x)=|f(x)|$, then at $x=c, g$ is:
Let $S$ be the set of all points in $(-\pi ,\pi )$ at which the function, $f(x)=\mathrm{min}{\mathrm{sin}x,\mathrm{cos}x}$ is not differentiable. Then $S$ is a subset of which of the following?
If $x \log _{e}\left(\log _{e} x\right)-x^{2}+y^{2}=4(y>0),$ then $\frac{d y}{d x}$ at $x=e$ is equal to :
Let, $f:R\rightarrow R$ be a function such that $f(x)={x}^{3}+{x}^{2}f'(1)+xf''(2)+f'''(3),\forall x\in R.$ Then $f(2)$ equals
Let $f(x)=\frac{x}{\sqrt{a^{2}+x^{2}}}-\frac{d-x}{\sqrt{b^{2}+(d-x)^{2}}}, x \in \mathbb{R}$ wherea, b and d are non-zero real constants. Then :
The maximum volume $(in cu.m)$ of the right circular cone having slant height $3 m$ is:
Let $\sum _{k=1}^{10}f(a+k)=16({2}^{10}-1),$ where the function $f$ satisfies $f(x+y)=f(x)f(y)$ for all natural numbers $x, y$ and $f(1)=2.$ Then the natural number $'a'$ is:
If $\int _{0}^{\frac{\pi }{2}}\frac{cotx}{cotx+\mathrm{cosec}x}dx=m(\pi +n),$ then $mn$ is equal to
Let $f:R\rightarrow R$ be a continuous and differentiable function such that $f(2)=6$ and ${f}^{'}(2)=\frac{1}{48}$. If ${\int }_{6}^{f(x)}4{t}^{3}dt=(x-2)g(x),$ then $\underset{x\rightarrow 2}{lim}g(x)$ is equal to
The value of the integral ${\int }_{0}^{1}x{cot}^{-1}(1-{x}^{2}+{x}^{4})dx$ is
If $\int _{0}^{x}f(t)dt={x}^{2}+\int _{x}^{1}{t}^{2}f(t)dt,$ then ${f}^{'}(\frac{1}{2})$ is
The value of ${\int }_{0}^{\pi }{|\mathrm{cos}x|}^{3}dx$ is
Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x)-f(y)|\leq 2{|x-y|}^{3/2},$ for all $x,y\in R\text{.}$ If $f(0)=1$ then $\int _{0}^{1}{f}^{2}(x)dx$ is equal to
Let $f(x)=\left\{\begin{array}{cl}-1, & -2 \leq x < 0 \\ x^{2}-1, & 0 \leq x \leq 2\end{array}\right.$ and $g(x)=|\eta(x)|+f(x \mid) .$ Then, in the interval $(-2,2), g$ is:
If the area (in sq. units) bounded by the parabola ${y}^{2}=4\lambda x$ and the line $y=\lambda x, \lambda >0,$ is $\frac{1}{9}$, then $\lambda$ is equal to
The area (in sq. units) of the region $A={(x,y):\frac{{y}^{2}}{2}\leq x\leq y+4}$ is:
The area (in sq. units) in the first quadrant bounded by the parabola, $y=x^{2}+1$, the tangent to it at the point (2,5) and the coordinate axes is :
The area (in sq. units) bounded by the parabola $y={x}^{2}-1,$ the tangent at the point $(2,3)$ to it and the $y$-axis is
If the function $f$ given by $f(x)={x}^{3}-3(a-2){x}^{2}+3ax+7$, for some $a\in R$ is increasing in $(0, 1]$ and decreasing in $[1, 5)$, then a root of the equation, $\frac{f(x)-14}{{(x-1)}^{2}}=0, (x\neq 1)$ is :