Calculus PYQ — Page 56

Given $P(x)=x^4+a x^3+b x^2+c x+d$ such that $x=0$ is the only real root of $P^{\prime}(x)=0$. If $P(-1) < P(1)$, then in the interval $[-1,1]$

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

How many real solutions does the equation $x^7+14 x^5+16 x^3+30 x-560=0$ have?

Let $f(x)=\left\{\begin{array}{ll}(x-1) \sin \left(\frac{1}{x-1}\right), & \text { if } x \neq 1 \\ 0, & \text { if } x=1\end{array}\right.$. Then which one of the following is true?

Suppose the cube $x^3-p x+q$ has three distinct real roots where $p>0$ and $q>0$. Then which one of the following holds?

Let $\mathrm{I}=\int_0^1 \frac{\sin \mathrm{x}}{\sqrt{\mathrm{x}}} \mathrm{dx}$ and $\mathrm{J}=\int_0^1 \frac{\cos \mathrm{x}}{\sqrt{\mathrm{x}}} \mathrm{dx}$. Then which one of the following is true?

The solution of the differential equation $\frac{d y}{d x}=\frac{x+y}{x}$ satisfying the condition $y(1)=1$ is

The value of $\sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right)}$ is

The area of the plane region bounded by the curves $x+2 y^2=0$ and $x+3 y^2=1$ is equal to

The area enclosed between the curves $y^2=x$ and $y=|x|$ is

$\int \frac{d x}{\cos x+\sqrt{3} \sin x}$ equals

The solution for $x$ of the equation $\int_{\sqrt{2}}^x \frac{d t}{t \sqrt{t^2-1}}=\frac{\pi}{2}$ is

The function $f(x)=\tan ^{-1}(\sin x+\cos x)$ is an increasing function in

Let $F(x)=f(x)+f\left(\frac{1}{x}\right)$, where $f(x)=\int_1^x \frac{\log t}{1+t} d t$. Then $F(e)$ equals

Let $f: R \rightarrow R$ be a function defined by $f(x)=\operatorname{Min}\{x+1,|x|+1\}$. Then which of the following is true?

The function $f:R \sim\{0\} \rightarrow R$ given by $f(x)=\frac{1}{x}-\frac{2}{e^{2 x}-1}$ can be made continuous at $x=0$ by defining $f(0)$ as

The value of the integral, $\int_3^6 \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} d x$ is

If $x^m \cdot y^n=(x+y)^{m+n}$, then $\frac{d y}{d x}$ is

The value of $\int_1^a[x] f^{\prime}(x) d x, a>1$, where $[x]$ denotes the greatest integer not exceeding $x$ is

The set of points where $f(x)=\frac{x}{1+|x|}$ is differentiable is

$\int_0^\pi x f(\sin x) d x$ is equal to

If $x$ is real, the maximum value of $\frac{3 x^2+9 x+17}{3 x^2+9 x+7}$ is

The function $f(x)=\frac{x}{2}+\frac{2}{x}$ has a local minimum at

$\int_{-3 \pi / 2}^{-\pi / 2}\left[(x+\pi)^3+\cos ^2(x+3 \pi)\right] d x$ is equal to