Calculus PYQ — Page 52

If for a continuous function $\mathrm{f}(\mathrm{x})$, $\int_{-\pi}^t(f(x)+x d x)=\pi^2-t^2$, for all $t \geq-\pi$, then $\mathrm{f}\left(-\frac{\pi}{3}\right)$ is equal to:

If [ ] denotes the greatest integer function, then the integral $\int_0^\pi[\cos x d x$ is equal to:

If $\frac{dy}{dx}+y\mathrm{tan}x=sin2x$ and $y(0)=1$, then $y(\pi )$ is equal to

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \mathrm{log}|x|+\beta {x}^{2}+x$, then

The value of $\lim _{x \rightarrow 0} \frac{1}{x}\left[\tan ^{-1}\left(\frac{x+1}{2 x+1}\right)-\frac{\pi}{4}\right]$ 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

The value of $\int_{-\pi / 2}^{\pi / 2} \frac{\sin ^2 x}{1+2^x} d x$ is :

The maximum area of a right angled triangle with hypotenuse $h$ is :

The integral $\int_{7 \pi / 4}^{7 \pi / 3} \sqrt{\tan ^2 x} d x$ is equal to :

The integral $\int \frac{x d x}{2-x^2+\sqrt{2-x^2}}$ equals :

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 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 area under the curve $y=|\cos x-\sin x|$, $0 \leq x \leq \frac{\pi}{2}$, and above $x$-axis 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 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 area bounded by the curve $y=\ln (x)$ and the lines $y=0, y=\ln (3)$ and $x=0$ is equal to:

Statement-1: The function $x^2\left(e^x+e^{-x}\right)$ is increasing for all $x>0$. Statement-2: The functions $x^2 e^x$ and $x^2 e^{-x}$ are increasing for all $x>0$ and the sum of two increasing functions in any interval $(a, b)$ is an increasing function in $(a, b)$.

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 $[1,2]$

Statement - I : The value of the integral $\int _{\pi /6}^{\pi /3}\frac{dx}{1+\sqrt{\mathrm{tan}x}}$ is equal to $\frac{\pi }{6}.$ Statement - II : $\int _{a}^{b}f(x)dx=\int _{a}^{b}f(a+b-x)dx.$

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:

Let $f:[-2,3] \rightarrow[0, \infty)$ be a continuous function such that $f(1-x)=f(x)$ for all $x \in[-2,3]$. If $\mathrm{R}_1$ is the numerical value of the area of the region bounded by $y=f(x), x=-2, x=3$ and the axis of $x$ and $R_2=\int_{-2}^3 x f(x) d x$, then :

Let $f$ be a composite function of $x$ defined by $f(u)=\frac{1}{u^2+u-2}, u(x)=\frac{1}{x-1}$. Then the number of points $x$ where $f$ is discontinuous is :

Let $f(x)=-1+|x-2|$, and $g(x)=1-|x|$; then the set of all points where $f_{o g}$ is discontinuous is :

Let $f(1)=-2$ and $f^{\prime}(x) \geq 4.2$ for $1 \leq x \leq 6$. The possible value of $f(6)$ lies in the interval :