JEE Main Mathematics — Calculus previous year questions with solutions.
The area (in square units) bounded by the curves $y=\sqrt{x},2y-x+3=0$, $X$-axis and lying in the first quadrant is
The cost of running a bus from $A$ to $B$, is $₹\left(a v+\frac{b}{v}\right)$, where $v \mathrm{~km} / \mathrm{h}$ is the average speed of the bus. When the bus travels at $30 \mathrm{~km} / \mathrm{h}$, the cost comes out to be $₹ 75$ while at $40 \mathrm{~km} / \mathrm{h}$, it is $₹ 65$. Then the most economical speed (in $\mathrm{km} / \mathrm{h}$ ) of the bus is :
The value of $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{(1-\mathrm{cos}2x)(3+\mathrm{cos}x)}{x\mathrm{tan}4x}$ is equal to
If $\int \frac{x^2-x+1}{x^2+1} e^{\cot ^{-1} x} d x=A(x) e^{\cot ^{-1} x}+C$, then $A(x)$ is equal to :
If $\int f(x)dx=\psi (x)$, then $\int {x}^{5}f({x}^{3})dx$, is equal to
Let $f(x)=\frac{x^2-x}{x^2+2 x} x \neq 0,-2$. Then $\frac{d}{d x}\left[f^{-1}(x)\right]$ (wherever it is defined) is equal to:
The area bounded by the curve $y=\ln (x)$ and the lines $y=0, y=\ln (3)$ and $x=0$ is equal to:
Let $f(x)=-1+|x-2|$, and $g(x)=1-|x|$; then the set of all points where $f_{o g}$ is discontinuous is :
The area of the region (in sq. units), in the first quadrant bounded by the parabola $y=9 x^2$ and the lines $x=0, y=1$ and $y=4$, is :
The maximum area of a right angled triangle with hypotenuse $h$ is :
Statement-1: The equation $x \log x=2-x$ is satisfied by at least one value of $x$ lying between 1 and 2. Statement-2: The function $f(x)=x \log x$ is an increasing function in $[1,2]$ and $g(x)=2-x$ is a decreasing function in $[1,2]$ and the graphs represented by these functions intersect at a point in $[1,2]$
If $f(x)=\sin (\sin x)$ and $f^{\prime \prime}(x)+\tan x f^{\prime}(x)+g(x)$ $=0$, then $g(x)$ is :
The equation of the curve passing through the origin and satisfying the differential equation $\left(1+x^2\right) \frac{d y}{d x}+2 x y=4 x^2$ is
The area enclosed by the curves $y=x^2, y=x^3$, $x=0$ and $x=p$, where $p>1$, is $1 / 6$. The $\mathrm{p}$ equals
If $f(x)=\int\left(\frac{x^2+\sin ^2 x}{1+x^2}\right) \sec ^2 x d x$ and $f(0)=0$, then $f(1)$ equals
If $\int_e^x t f(t) d t=\sin x-x \cos x-\frac{x^2}{2}$, for all $x \in R-\{0\}$, then the value of $f\left(\frac{\pi}{6}\right)$ is
If a metallic circular plate of radius $50 \mathrm{~cm}$ is heated so that its radius increases at the rate of $1 \mathrm{~mm}$ per hour, then the rate at which, the area of the plate increases (in $\mathrm{cm}^2 /$ hour) is
$\lim _{x \rightarrow 0}\left(\frac{x-\sin x}{x}\right) \sin \left(\frac{1}{x}\right)$
The area bounded by the parabola $y^2=4 x$ and the line $2 x-3 y+4=0$, in square unit, is
If $f^{\prime}(x)=\sin (\log x)$ and $y=f\left(\frac{2 x+3}{3-2 x}\right)$, then $\frac{d y}{d x}$ equals
The area bounded between the parabolas $x^2=\frac{y}{4}$ and $x^2=9 y$, and the straight line $y=2$ is
The population $\mathrm{p}(\mathrm{t})$ at time $t$ of a certain mouse species satisfies the differential equation $\frac{\mathrm{dp}(\mathrm{t})}{\mathrm{dt}}=0.5 \mathrm{~p}(\mathrm{t})$ $-450$. If $p(0)=850$, then the time at which the population becomes zero is
Statement 1: The degrees of the differential equations $\frac{d y}{d x}+y^2=x$ and $\frac{d^2 y}{d x^2}+y=\sin x$ are equal. Statement 2: The degree of a differential equation, when it is a polynomial equation in derivatives, is the highest positive integral power of the highest order derivative involved in the differential equation, otherwise degree is not defined.
The integral of $\frac{x^2-x}{x^3-x^2+x-1}$ w.r.t. $x$ is