Calculus PYQ — Page 38

If $I=\int _{1}^{2}\frac{dx}{\sqrt{2{x}^{3}-9{x}^{2}+12x+4}}$, then

Let $y=y(x)$ be a function of $x$ satisfying $y\sqrt{1-{x}^{2}}=k-x\sqrt{1-{y}^{2}}$ where $k$ is a constant and $y(\frac{1}{2})=-\frac{1}{4}$.Then $\frac{dy}{dx}$ at $x=\frac{1}{2}$ , is equal to

Area (in sq. units) of the region outside $\frac{|x|}{2}+\frac{|y|}{3}=1$ and inside the ellipse $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}=1$ is

If y = x³ + x², then dy/dx at x = 1 is:

The area bounded by y = x², x-axis, and x = 2 is:

Let $f$ be any function continuous on $[a,b]$ and twice differentiable on $(a,b)$ . If all $x\in (a,b),{f}^{'}(x)>0$ and ${f}^{''}(x)<0$ , then for any $c\in (a,b),\frac{f(c)-f(a)}{f(b)-f(c)}$

Let $S$, be the set of all functions $f:[0,1]\rightarrow R$, which are continuous on $[0,1]$, and differentiable on $(0,1)$. Then for every $f$ in $S$, there exists $c\in (0,1)$, depending on $f$, such that.

The area (in sq. units) of the region enclosed by the curves $y={x}^{2}-1$ and $y=1-{x}^{2}$ is equal to:

Let $f(x)=\int \frac{\sqrt{x}}{{(1+x)}^{2}}dx(x\geq 0)$. Then $f(3)-f(1)$ is equal to :

Consider a region $R={(x,y)\in {R}^{2}:{x}^{2}\leq y\leq 2x}$. If a line $y=\alpha$ divides the area of region $R$ into two equal parts, then which of the following is true ?

The position of a moving car at time $t$ is given by $f(t)=a{t}^{2}+bt+c,t>0$, where $a,b\text{and}c$ are real numbers greater than $1$. Then the average speed of the car over the time interval $[{t}_{1},{t}_{2}]$ is attained at the point:

If the function $f$ defined on $(-\frac{1}{3},1/3)$ by $f(x)={\begin{matrix}\frac{1}{x}{log}_{e}(\frac{1+3x}{1-2x}), & \text{when }x\neq 0 \\ k & ,\text{when }x=0\end{matrix}$, is continuous, then $k$ is equal to.

If $f(x)={\begin{matrix}\frac{sin(a+2)x+sinx}{x} & ;x<0 \\ b & ;x=0 \\ \frac{{(x+3{x}^{2})}^{1/3}-{x}^{1/3}}{{x}^{1/3}} & ;x>0\end{matrix}$ is continuous at $x=0$ , then $a+2b$ is equal to:

If $\alpha$ is the positive root of the equation, $p(x)={x}^{2}-x-2=0$, then $\underset{x\rightarrow {\alpha }^{+}}{\mathrm{lim}}\frac{\sqrt{1-\mathrm{cos}p(x)}}{x+\alpha -4}$is equal to

If $\underset{x\rightarrow 1}{\mathrm{lim}}\frac{x+{x}^{2}+{x}^{3}+...+{x}^{n}-n}{x-1}=820,(n\in N)$ then the value of $n$ is equal to....

$\underset{x\rightarrow a}{\mathrm{lim}}\frac{{(a+2x)}^{\frac{1}{3}}-{(3x)}^{\frac{1}{3}}}{{(3a+x)}^{\frac{1}{3}}-{(4x)}^{\frac{1}{3}}}(a\neq 0)$ is equal to:

$\underset{x\rightarrow 0}{\mathrm{lim}}{(\frac{3{x}^{2}+2}{7{x}^{2}+2})}^{\frac{1}{{x}^{2}}}$ is equal to

If for all real triplets $(a,b,c),f(x)=a+bx+c{x}^{2};$ then $\int _{0}^{1}f(x)dx$ is equal to:

The solution curve of the differential equation, $(1+{e}^{-x})(1+{y}^{2})\frac{dy}{dx}={y}^{2}$ which passes through the point $(0,1)$, is

If the function $f(x)={\begin{matrix}{k}_{1}(x-\pi {)}^{2}-1, & x\leq \pi \\ {k}_{2}\mathrm{cos}x, & x>\pi \end{matrix}$ is twice differentiable, then the ordered pair $({k}_{1},{k}_{2})$ is equal to:

If for $x\geq 0,y=y(x)$ is the solution of the differential equation, $(x+1)dy=({(x+1)}^{2}+y-3)dx,y(2)=0$ then $y(3)$ is equal to ________

Let $y=y(x)$ be the solution curve of the differential equation, $({y}^{2}-x)\frac{dy}{dx}=1$ , satisfying $y(0)=1$ . This curve intersects the $X-$axis at a point whose abscissa is

The integral ${\int }_{0}^{2}||x-1|-x|dx$ is equal to

The value of $\alpha$ for which $4\alpha \int _{-1}^{2}{e}^{-\alpha |x|}dx=5$ , is