Calculus PYQ — Page 10

The value of ${\int }_{0}^{1}{(2{x}^{3}-3{x}^{2}-x+1)}^{\frac{1}{3}}dx$ is equal to:

Let the area of the region enclosed by the curves $y=3 x, 2 y=27-3 x$ and $y=3 x-x \sqrt{x}$ be $A$. Then $10 A$ is equal to

If the area of the region $\left\{(x, y): \frac{\mathrm{a}}{x^2} \leq y \leq \frac{1}{x}, 1 \leq x \leq 2,0 < \mathrm{a} < 1\right\}$ is $\left(\log _{\mathrm{e}} 2\right)-\frac{1}{7}$ then the value of $7 \mathrm{a}-3$ is equal to:

Let $f(x)$ be a positive function such that the area bounded by $y=f(x), y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4 a^2+a-1$. Then the differential equation, whose general solution is $y=c_1 f(x)+c_2$, where $c_1$ and $c_2$ are arbitrary constants, is

Let $r_k=\frac{\int_0^1\left(1-x^7\right)^k d x}{\int_0^1\left(1-x^7\right)^{k+1} d x}, k \in \mathbb{N}$. Then the value of $\sum_{k=1}^{10} \frac{1}{7\left(r_k-1\right)}$ is equal to________

Let $f(x)=a x^3+b x^2+c x+41$ be such that $f(1)=40, f^{\prime}(1)=2$ and $f^{\prime}(1)=4$. Then $\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2$ is equal to:

If the solution $y(x)$ of the given differential equation $\left(\mathrm{e}^y+1\right) \cos x \mathrm{~d} x+\mathrm{e}^y \sin x \mathrm{~d} y=0$ passes through the point $\left(\frac{\pi}{2}, 0\right)$, then the value of $\mathrm{e}^{y\left(\frac{\pi}{6}\right)}$ is equal to_________

The solution curve, of the differential equation $2 y \frac{\mathrm{d} y}{\mathrm{~d} x}+3=5 \frac{\mathrm{d} y}{\mathrm{~d} x}$, passing through the point $(0,1)$ is a conic, whose vertex lies on the line:

The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas $2{y}^{2}=kx$ and $k{y}^{2}=2(y-x)$ is maximum, is equal to:

If the area of the region ${(x,y):0\leq y\leq \mathrm{min}{2x,6x-{x}^{2}}}$ is $A$, then $12A$ is equal to _______.

Let the maximum and minimum values of $\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in \mathbf{R}$ be $\mathrm{M}$ and $\mathrm{m}$, respectively. Then $\mathrm{M}^2-\mathrm{m}^2$ is equal to _________

Let $f:R\rightarrow R$ be defined $f(x)=a{e}^{2x}+b{e}^{x}+cx$. If $f(0)=-1,{f}^{'}({\mathrm{log}}_{e}2)=21$ and ${\int }_{0}^{\mathrm{log}4}(f(x)-cx)dx=\frac{39}{2}$, then the value of $|a+b+c|$ equals:

If ${\int }_{0}^{1}\frac{1}{\sqrt{3+x}+\sqrt{1+x}}dx=a+b\sqrt{2}+c\sqrt{3}$, where $a,b,c$ are rational numbers, then $2a+3b-4c$ is equal to :

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{e}^{2|\mathrm{sin}x|}-2|\mathrm{sin}x|-1}{{x}^{2}}$

Let $f:(-\infty, \infty)-\{0\} \rightarrow \mathbb{R}$ be a differentiable function such that $f^{\prime}(1)=\lim _{a \rightarrow \infty} a^2 f\left(\frac{1}{a}\right)$. Then $\lim _{a \rightarrow \infty} \frac{a(a+1)}{2} \tan ^{-1}\left(\frac{1}{a}\right)+a^2-2 \log _e a$ is equal to

The derivative of sin²x is

The derivative of ln(sin x) with respect to x is:

The integral of 1/x is

If the integral from 0 to 1 of x²eˣ dx = ae - b, find a + b.

The interval in which the function $f(x)=x^x, x>0$, is strictly increasing is

If $f(x)=\left\{\begin{array}{l}x^3 \sin \left(\frac{1}{x}\right), x \neq 0 \\ 0 \quad, x=0\end{array}\right.$ then

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{3+\alpha \mathrm{sin}x+\beta \mathrm{cos}x+{\mathrm{log}}_{e}(1-x)}{3{\mathrm{tan}}^{2}x}=\frac{1}{3}$, then $2\alpha -\beta$ is equal to :

Consider the function $f(x)={\begin{matrix}\frac{a(7x-12-{x}^{2})}{b|{x}^{2}-7x+12|} & ,x<3 \\ {2}^{\frac{\mathrm{sin}(x-3)}{x-[x]}} & ,x>3 \\ b & ,x=3\end{matrix}$,where $[x]$ denotes the greatest integer less than or equal to $x$. If $S$ denotes the set of all ordered pairs $(a,b)$ such that $f(x)$is continuous at $x=3$, then the number of elements in $S$ is :

Let $g(x)$ be a linear function and $f(x)={\begin{matrix}g(x), & x\leq 0 \\ {(\frac{1+x}{2+x})}^{\frac{1}{x}}, & x>0\end{matrix}$, is continuous at $x=0$. If ${f}^{'}(1)=f(-1)$, then the value of $g(3)$ is