JEE Main Mathematics — Calculus previous year questions with solutions.
If a curve passes through the point $(1,-2)$ and has slope of the tangent at any point $(x, y)$ on it as $\frac{{x}^{2}-2y}{x}$, then the curve also passes through the point
If $cosx\frac{dy}{dx}-ysinx=6x$, $(0<x<\frac{\pi }{2})$ and $y(\frac{\pi }{3})=0,$ then $y(\frac{\pi }{6})$ is equal to
If $y(x)$ is the solution of the differential equation $\frac{d y}{d x}+\left(\frac{2 x+1}{x}\right) y=e^{-2 x}, x>0,$ where $y(1)=\frac{1}{2} e^{-2},$ then:
If $\frac{dy}{dx}+\frac{3}{{\mathrm{cos}}^{2}x}y=\frac{1}{{\mathrm{cos}}^{2}x}$ , $x\in (-\frac{\pi }{3},\frac{\pi }{3}),$ and $y(\frac{\pi }{4})=\frac{4}{3},$ then $y(-\frac{\pi }{4})$ equals
The derivative of ${tan}^{-1}(\frac{sinx-cosx}{sinx+cosx})$ with respect to $\frac{x}{2},$ where $x\in (0,\frac{\pi }{2})$, is
If $\int {x}^{5}{e}^{-4{x}^{3}}dx=\frac{1}{48}{e}^{-4{x}^{3}}f(x)+C$, where $C$ is a constant of integration, then $f(x)$ is equal to
The area (in sq. units) of the region bounded by the parabola, $y={x}^{2}+2$ and the lines, $y=x+1, x=0$ and $x=3$, is
Let $[\mathrm{x}]$ denote the greatest integer less than or equal to $\mathrm{X}$. Then : $\lim _{x \rightarrow 0} \frac{\tan \left(\pi \sin ^{2} x\right)+(|\mathrm{x}|-\sin (x[x]))^{2}}{x^{2}}$
The integral $\int_{\pi / 6}^{\pi / 4} \frac{\mathrm{d} x}{\sin 2 x\left(\tan ^{5} x+\cot ^{5} x\right)}$ equals:
A $2$m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate $25cm/sec$ , then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is $1$ m above the ground is:
If $f(1)=1,{f}^{'}(1)=3$, then the derivative of $f(f(f(x)))+{(f(x))}^{2}$ at $x=1$ is:
Consider the differential equation, ${y}^{2}dx+(x-\frac{1}{y})dy=0$. If value of $y$ is $1$ when $x=1$, then the value of $x$ for which $y=2,$is
The general solution of the differential equation $({y}^{2}-{x}^{3})dx-xydy=0,(x\neq 0)$ is (where c is a constant of integration)
The height of a right circular cylinder of maximum volume inscribed in a sphere of radius $3$ is:
Let $f(x)=5-|x-2|$ and $g(x)=|x+1|,$ $x \in R.$ If $f(x)$ attains maximum value at $\alpha$ and $g(x)$ attains minimum value at $\beta ,$ then $\underset{x\rightarrow -\alpha \beta }{lim}\frac{(x-1)({x}^{2}-5x+6)}{{x}^{2}-6x+8}$ is equal to
If $f(x)$ is a non-zero polynomial of degree four, having local extreme points at $x= –1, 0, 1;$ then the set $S={x\in R :f(x)=f(0)}$ contains exactly
Let $\alpha \in (0,\frac{\pi }{2})$, be constant.If the integral $\int \frac{tanx+tan\alpha }{tanx-\mathrm{tan}\alpha }dx=A(x)cos2\alpha +B(x)sin2\alpha +C$, where C is a constant of integration, then the functions $A(x)$ and $B(x)$ are respectively
If the area (in sq. units) of the region ${(x,y):{y}^{2}\leq 4x,x+y\leq 1,x\geq 0,y\geq 0}$ is $a\sqrt{2}+b,$ then $a-b$ is equal to
A value of $\alpha$ such that $\int _{\alpha }^{\alpha +1}\frac{dx}{(x+\alpha )(x+\alpha +1)}=lo{g}_{e}(\frac{9}{8})$ is
Themaximum value of the finction $f(x)=3 x^{3}-18 x^{2}+27 x-40$ on the set $\mathrm{S}=\left\{x \in R: x^{2}+30 \leq 11 x\right\}$ is :
The integral $\int \frac{2{x}^{3}-1}{{x}^{4}+x}dx$, is equal to
The area (in sq. units) of the region bounded by the curves $y={2}^{x}$ and $y=|x+1|$, in the first quadrant is
Let $f$ be a differentiable function such that $f(1)=2$ and ${f}^{'}(x)=f(x)$ for all $x\in R$. If $h(x)=f(f(x))$, then ${h}^{'}(1)$ is equal to :
If $\int \frac{dx}{{x}^{3}{(1+{x}^{6})}^{\frac{2}{3}} }=xf(x){(1+{x}^{6})}^{\frac{1}{3}}+C$, where $C$ is a constant of integration, then the function $f(x)$ is equal to