Calculus PYQ — Page 55

$\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^2 x\right)}{x^2}$ equals

Consider the function $f(x)=|x-2|+|x-5|, x \in R$. Statement $1$: $f^{\prime}(4)=0$ Statement $2$: $f$ is continuous in $[2,5]$, differentiable in $(2,5)$ and $f(2)=f(5)$.

Consider a rectangle whose length is increasing at the uniform rate of $2 \mathrm{~m} / \mathrm{sec}$, breadth is decreasing at the uniform rate of $3 \mathrm{~m} / \mathrm{sec}$ and the area is decreasing at the uniform rate of $5 \mathrm{~m}^2 / \mathrm{sec}$. If after some time the breadth of the rectangle is $2 \mathrm{~m}$ then the length of the rectangle is

A spherical balloon is filled with $4500 ~\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 ~\pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases $49$ minutes after the leakage began is

The value of $\int_0^1 \frac{8 \log (1+x)}{1+x^2} d x$ is

The value of $p$ and $q$ for which the function $f(x)=\left\{\begin{array}{cl}\frac{\sin (p+1) x+\sin x}{x} & x < 0 \\ q & , x=0 \\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3 / 2}} & , x>0\end{array}\right.$ is continuous for all $\mathrm{x}$ in $\mathrm{R}$, is

The shortest distance between line $y-x=1$ and curve $x=y^2$ is

The area of the region enclosed by the curves $y=x, x=e, y=\frac{1}{x}$ and the positive $x$-axis is

$$ \lim _{x \rightarrow 2}\left(\frac{\sqrt{1-\cos \{2(x-2)\}}}{x-2}\right)

Let I be the purchase value of an equipment and $\mathrm{V}(\mathrm{t})$ be the value after it has been used for t years. The value $\mathrm{V}(\mathrm{t})$ depreciates at a rate given by differential equation $\frac{\mathrm{dV}(\mathrm{t})}{\mathrm{dt}}=-\mathrm{k}(\mathrm{T}-\mathrm{t})$, where $\mathrm{k}>0$ is a constant and $\mathrm{T}$ is the total life in years of the equipment. Then the scrap value $\mathrm{V}(\mathrm{T})$ of the equipment is

If $\frac{d y}{d x}=y+3>0$ and $y(0)=2$, then $y(\ln 2)$ is equal to

For $x \in\left(0, \frac{5 \pi}{2}\right)$, define $f(x)=\int_0^x \sqrt{t} \sin t d t$. Then $f$ has

$\frac{d^2 x}{d y^2}$ equals

The area bounded by the curves $y=\cos x$ and $y=\sin x$ between the ordinates $x=0$ and $x=\frac{3 \pi}{2}$ is

Solution of the differential equation $\cos x d y=y(\sin x-y) d x, 0 < x < \frac{\pi}{2}$ is

Let $f: R \rightarrow R$ be defined by $f(x)=\left\{\begin{array}{ll}k-2 x, & \text { if } x \leq-1 \\ 2 x+3, & \text { if } x>-1\end{array}\right.$. If $f$ has a local minimum at $x=-1$, then a possible value of $\mathrm{k}$ is

Let $f: R \rightarrow R$ be a positive increasing function with $\lim _{x \rightarrow \infty} \frac{f(3 x)}{f(x)}=1$. Then $\lim _{x \rightarrow \infty} \frac{f(2 x)}{f(x)}=$

Let $p(x)$ be a function defined on $R$ such that $p^{\prime}(x)=p^{\prime}(1-x)$, for all $x \in[0,1], p(0)=1$ and $p(1)=41$. Then $\int_0^1 p(x) d x$ equals

Let $f:(-1,1) \rightarrow R$ be a differentiable function with $f(0)=-1$ and $f^{\prime}(0)=1$. Let $g(x)=[f(2 f(x)+2)]^2$. Then $g^{\prime}(0)=$

Let $f: R \rightarrow R$ be a continuous function defined by $f(x)=\frac{1}{e^x+2 e^{-x}}$. Statement-1: $f(c)=\frac{1}{3}$, for some $c \in R$. Statement-2: $0 < f(x) \leq \frac{1}{2 \sqrt{2}}$, for all $x \in R$

The shortest distance between the line $\mathrm{y}-\mathrm{x}=1$ and the curve $\mathrm{x}=\mathrm{y}^2$ is

The area of the region bounded by the parabola $(y-2)^2=x-1$, the tangent to the parabola at the point $(2,3)$ and the $x$-axis is

Let $y$ be an implicit function of $x$ defined by $x^{2 x}-2 x^x \cot y-1=0$. Then $y^{\prime}(1)$ equals

Let $f(x)=x|x|$ and $g(x)=\sin x$. Statement-1 : gof is differentiable at $x=0$ and its derivative is continuous at that point. Statement-2 : gof is twice differentiable at $x=0$.