Calculus PYQ — Page 31

The value of the integral ${\int }_{-1}^{1}{\mathrm{log}}_{e}(\sqrt{1-x}+\sqrt{1+x})dx$ is equal to:

The function $f(x)=|{x}^{2}-2x-3|\cdot {e}^{9{x}^{2}-12x+4}$ is not differentiable at exactly :

If $y\frac{dy}{dx}=x[\frac{{y}^{2}}{{x}^{2}}+\frac{\phi (\frac{{y}^{2}}{{x}^{2}})}{{\phi }^{'}(\frac{{y}^{2}}{{x}^{2}})}],x>0,\phi >0,$ and $y(1)=-1,$ then $\phi (\frac{{y}^{2}}{4})$ is equal to:

Let ${J}_{n,m}={\int }_{0}^{1/2}\frac{{x}^{n}}{{x}^{m}-1}dx,\forall n>m$ and $n,m\in N.$ Consider a matrix $A={[{a}_{ij}]}_{3\times 3}$ where ${a}_{ij}={\begin{matrix}{J}_{6+i,3}-{J}_{i+3,3} & ,i\leq j \\ 0 & ,i>j\end{matrix}$. Then $|adj{A}^{-1}|$ is :

Let $f:[0,\infty )\rightarrow [0,3]$ be a function defined by $f(x)={\begin{matrix}\mathrm{max}{\mathrm{sin}t:0\leq t\leq \pi },x\in [0,\pi ] \\ 2+\mathrm{cos}x,x>\pi \end{matrix}$. Then which of the following is true ?

A function $f$ is defined on $[-3,3]$ as $f(x)={\begin{matrix}\mathrm{min}{|x|,2-{x}^{2}},-2\leq x\leq 2 \\ [|x|],2<|x|\leq 3\end{matrix}$ where $[x]$ denotes the greatest integer $\leq x$. The number of points, where $f$ is not differentiable in $(-3,3)$ is ___ .

Let $P(x)={x}^{2}+bx+c$ be a quadratic polynomial with real coefficients such that ${\int }_{0}^{1}P(x)dx=1$ and $P(x)$ leaves remainder $5$ when it is divided by $(x-2)$ Then the value of $9(b+c)$ is equal to:

Let $F:[3,5]\rightarrow R$ be a twice differentiable function on $(3,5)$ such that $F(x)={e}^{-x}{\int }_{3}^{x}(3{t}^{2}+2t+4{F}^{'}(t))dt.$ If ${F}^{'}(4)=\frac{\alpha {e}^{\beta }-224}{{({e}^{\beta }-4)}^{2}}$, then $\alpha +\beta$ is equal to _____.

Let $f:R\rightarrow R$ be defined as $f(x)={\begin{matrix}\frac{{x}^{3}}{(1-\mathrm{cos}2x{)}^{2}}{\mathrm{log}}_{e}(\frac{1+2x{e}^{-2x}}{{(1-x{e}^{-x})}^{2}}) & ,x\neq 0 \\ \alpha & ,x=0\end{matrix}$ If $f$ is continuous at $x=0,$ then $\alpha$ is equal to:

The area (in sq. units) of the region, given by the set ${(x,y)\in R\times R\mid x\geq 0,2{x}^{2}\leq y\leq 4-2x}$ is :

Let $f:[-3,1]\rightarrow R$ be given as $f(x)={\begin{matrix}min{(x+6),{x}^{2}}, & -3\leq x\leq 0 \\ max{\sqrt{x},{x}^{2}}, & 0\leq x\leq 1\end{matrix}.$If the area bounded by $y=f(x)$ and $x$-axis is $A$ sq units, then the value of $6A$ is equal to

If the function $f(x)={\begin{matrix}\frac{1}{x}{\mathrm{log}}_{e}(\frac{1+\frac{x}{a}}{1-\frac{x}{b}}),x<0 \\ k,x=0 \\ \frac{{\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x-1}{\sqrt{{x}^{2}+1}-1},x>0\end{matrix}$ is continuous at $x=0,$ then $\frac{1}{a}+\frac{1}{b}+\frac{4}{k}$ is equal to :

$\underset{x\rightarrow 0}{\mathrm{lim}}\frac{{\mathrm{sin}}^{2}(\pi {\mathrm{cos}}^{4}x)}{{x}^{4}}$ is equal to :

$\underset{x\rightarrow 2}{\mathrm{lim}}(\sum _{n=1}^{9}\frac{x}{n(n+1){x}^{2}+2(2n+1)x+4})$ is equal to :

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{a{e}^{x}-b\mathrm{cos}x+c{e}^{-x}}{x\mathrm{sin}x}=2$, then $a+b+c$ is equal to ________.

If $\underset{x\rightarrow 0}{\mathrm{lim}}\frac{ax-({e}^{4x}-1)}{ax({e}^{4x}-1)}$ exists and is equal to $b$, then the value of $a-2b$ is ___ .

Let a function $g:[0,4]\rightarrow R$ be defined as $g(x)={\begin{matrix}\underset{0\leq t\leq x}{\mathrm{max}{{t}^{3}-6{t}^{2}+9t-3}}, & 0\leq x\leq 3 \\ 4-x, & 3<x\leq 4\end{matrix}$ then the number of points in the interval $(0,4)$ where $g(x)$ is NOT differentiable, is _________.

Let $f:R\rightarrow R$ be a function defined as $f(x)={\begin{matrix}\frac{\mathrm{sin}(a+1)x+\mathrm{sin}2x}{2x} & ,\mathrm{if}x<0 \\ b & ,\mathrm{if}x=0 \\ \frac{\sqrt{x+b{x}^{3}}-\sqrt{x}}{b{x}^{5/2}} & ,\mathrm{if}x>0\end{matrix}$ If $f$ is continuous at $x=0$, then the value of $a+b$ is equal to :

Let a function $f:R\rightarrow R$ be defined as, $f(x)={\begin{matrix}\mathrm{sin}x-{e}^{x} & \mathrm{if}x\leq 0 \\ a+[-x] & \mathrm{if}0<x<1 \\ 2x-b & \mathrm{if}x\geq 1\end{matrix}$ Where $[x]$ is the greatest integer less than or equal to $x.$ If $f$ is continuous on $R$, then $(a+b)$ is equal to:

Let $f:[-1,1]\rightarrow R$ be defined as $f(x)=a{x}^{2}+bx+c$ for all $x\in [-1,1],$ where $a,b,c\in R$ such that $f(-1)=2,{f}^{'}(-1)=1$ and for $x\in (-1,1)$ the maximum value of ${f}^{"}(x)$ is $\frac{1}{2}.$ If $f(x)\leq \alpha ,x\in [-1,1],$ then the least value of $\alpha$ is equal to

If $R$ is the least value of $a$ such that the function $f(x)={x}^{2}+ax+1$ is increasing on $[1,2]$ and $S$ is the greatest value of $a$ such that the function $f(x)={x}^{2}+ax+1$ is decreasing on $[1,2]$, then the value of $|R-S|$ is

Consider the function $f:R\rightarrow R$ defined by $f(x)={\begin{matrix}(2-\mathrm{sin}(\frac{1}{x}))|x|, & x\neq 0 \\ 0, & x=0\end{matrix}.$ Then $f$ is:

The area (in sq. units) of the part of the circle ${x}^{2}+{y}^{2}=36$, which is outside the parabola ${y}^{2}=9x$, is equal to

The value of the integral, ${\int }_{1}^{3}[{x}^{2}-2x-2]dx,$ where $[x]$ denotes the greatest integer less than or equal to $x,$ is