Property 2

Q.

Let $\left[t\right]$ denote the greatest integer less than or equal to $t$. Then the value of ${\int }_1^2|2x-[3x\left]\right|dx$is.

Q.

If $a^2{x}^4+b^2{y}^4=c^6$, then maximum value of $xy$ is

1. $\frac{c^2}{\sqrt{ab}}$

2. $\frac{c^3}{ab}$

3. $\frac{c^3}{\sqrt{2ab}}$

4. $\frac{c^3}{2ab}$

Q.

$ifx=\sqrt{a^{si{n}^{-1}t}},y=\sqrt{a^{co{s}^{-1}t}},thenshowthat\frac{dy}{dx}=-\frac{y}{x}.$

Q. If $y=\cos x^{\circ },$ then $\frac{dy}{dx}=$

1. $-\sin x^{\circ }$
2. $1$
3. $-\frac{\pi }{180}\sin x^{\circ }$
4. $\frac{180}{\pi }\sin x^{\circ }$

Q. If $\int f\left(x\right)\sin x\cos xdx=\frac{1}{2(b^2-a^2)}\log f\left(x\right)+c,$ where $c$ is the constant of integration, then $f\left(x\right)=$

1. $\frac{2}{(b^2-a^2)\sin 2x}$
2. $\frac{2}{ab\sin 2x}$
3. $\frac{2}{(b^2-a^2)\cos 2x}$
4. $\frac{2}{ab\cos 2x}$

Q. $\int \frac{2^{x+1}-5^{x-1}}{{10}^x}dx$ is equal to
(where $C$ is constant of integration)

1. $\frac{2{\left(\frac{1}{5}\right)}^x}{\ln \left(\frac{1}{5}\right)}-\frac{1}{5}\frac{{\left(\frac{1}{2}\right)}^x}{\ln \left(\frac{1}{2}\right)}+C$
2. $\frac{3{\left(\frac{1}{5}\right)}^x}{\ln \left(\frac{1}{5}\right)}-\frac{1}{5}\frac{{\left(\frac{1}{2}\right)}^x}{\ln \left(\frac{1}{2}\right)}+C$
3. $\frac{4{\left(\frac{1}{5}\right)}^x}{\ln \left(\frac{1}{5}\right)}-\frac{1}{5}\frac{{\left(\frac{1}{2}\right)}^x}{\ln \left(\frac{1}{2}\right)}+C$
4. $\frac{5{\left(\frac{1}{5}\right)}^x}{\ln \left(\frac{1}{5}\right)}-\frac{1}{5}\frac{{\left(\frac{1}{2}\right)}^x}{\ln \left(\frac{1}{2}\right)}+C$

Q.

$ta{n}^{-1}\left(\frac{4}{3}\right)value$

Q. Function $f\left(x\right)=\{\begin{array}{cc}({\log }_{2}2x{)}^{{\log }_{x}8};& x\ne 1\\ (k-1{)}^3;& x=1\end{array}$

is continuous at $x=1$, then $k=$____

1. $e+1$
2. $e^{\frac{1}{3}}$
3. $e^3$
4. $e-1$

Q.

Integrate $\frac{x-\sin x}{1-\cos x}$.

Q. Find the principal value of ${\tan }^{-1}(-1).$

Q. Let $f:R\to R$ be a continuous function which satisfies $f\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt$. Then the value of $f\left({\log }_{e}5\right)$ is

1. $0$ (Zero)
2. $2$
3. $5$
4. $3$

Q. The angle between the curves $y=x^2$ and $x=y^2$ at (1, 1) is

1. ${90}^{\circ }$
   
2. $ta{n}^{-1}\left(\frac{4}{3}\right)$
   
3. $ta{n}^{-1}\left(1\right)$
4. $ta{n}^{-1}\left(\frac{3}{4}\right)$

Q. If the function $f$ defined on $(\frac{\pi }{6},\frac{\pi }{3})$ by $f\left(x\right)=\{\begin{array}{cc}\frac{\sqrt{2}\cos x-1}{\cot x-1},& x\ne \frac{\pi }{4}\\ k,& x=\frac{\pi }{4}\end{array}$, is continuous, then $k$ is equal to:

1. 2
2. $\frac{1}{2}$
3. 1
4. $\frac{1}{\sqrt{2}}$

Q.

If $\tan h\left(x\right)=\frac{3}{4}$, then the value of $x$ is

1. $\sqrt{7}$

2. $-\sqrt{7}$

3. $\log \left(\sqrt{7}\right)$

4. $-\log \left(\sqrt{7}\right)$

Q. $\int \frac{1}{{\tan }^{2}x+{\sec }^{2}x}dx$ is equal to
(where $C$ is constant of integration)

1. $\frac{1}{\sqrt{2}}{\tan }^{-1}\left(\sqrt{2}\tan x\right)-{\tan }^{-1}\left(\tan x\right)+C$
2. $\frac{1}{\sqrt{2}}{\tan }^{-1}\left(\sqrt{2}\tan x\right)-{\tan }^{-1}\left(\sqrt{2}\tan x\right)+C$
3. $\sqrt{2}{\tan }^{-1}\left(\sqrt{2}\tan x\right)-{\tan }^{-1}\left(\tan x\right)+C$
4. $\sqrt{2}{\tan }^{-1}\left(\sqrt{2}\tan x\right)-{\tan }^{-1}\left(\sqrt{2}\tan x\right)+C$

Q. Find the integral:
$\int (4e^{3x}+1)dx$

Q. If $\int \frac{dx}{\sin x\cdot \cos x({\tan }^{9}x+1)}=\frac{1}{k}\ln \left|\frac{(\sin x{)}^9}{(\sin x{)}^9+(\cos x{)}^9}\right|+C$, then the value of $k^2+1$ is:
(where $C$ is integration constant)

Q.

Evaluate $\int e^3\log x{(x^4+1)}^{-1}dx$

1. $e^{3\log x}+c$

2. $\left(\frac{1}{4}\right)\log \left|x^4+1\right|+c$

3. $\log (x^4+1)+c$

4. $\frac{1}{2}\log (x^4+1)+c$

5. $\left(\frac{x^4}{x^4+1}\right)+c$

Q. $\int \cos \sqrt{x}dx.$

Q.

If $\int \frac{dx}{(1+\sqrt{x})\left(\sqrt{x-x^2}\right)}=\frac{A\sqrt{x}}{\sqrt{1-x}}+\frac{B}{\sqrt{1-x}}+C$ where $C$ is real constant, then $A+B=$

1. $3$

2. $0$

3. $1$

4. $2$

Q. If $x=\log t$ and $y=t^2-1,$ then $y^{\prime \prime }\left(1\right)$ at $t=1$ is

1. $4$
2. $2$
3. $3$
4. None of these

Q. The value of the integral $\underset{4}{\overset{10}{\int }}\frac{\left[x^2\right]}{[x^2-28x+196]+\left[x^2\right]}dx,$ where $\left[x\right]$ denotes the greatest integer less than or equal to $x$, is

Q. if a function f(x) is discontinuous in the interval (a, b) then ${\int }_a^{b}f\left(x\right)dx$ never exists.

1. True
2. False

Q.

The number of distinct real roots of $\left|\begin{array}{ccc}sinx& cosx& cosx\\ cosx& sinx& cosx\\ cosx& cosx& sinx\end{array}\right|=0$ in the interval $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ is

(a) 0
(b) 2
(c) 1
(d) 3

Q. If $I_n=\int {\cot }^{n}xdx,$ then $I_0+I_1+2(I_2+I_3)+I_4+I_5=$
(where $C$ is integration constant)

1. $-[\cot x+\frac{{\cot }^{2}x}{2}+\frac{{\cot }^{3}x}{3}+\frac{{\cot }^{4}x}{4}]+C$
2. $-[\cot x+\frac{{\cot }^{3}x}{3}]+C$
3. $-[\frac{{\cot }^{4}x}{4}+\frac{{\cot }^{2}x}{2}]+C$
4. $-[\cot x+\frac{{\cot }^{3}x}{3}+\frac{{\cot }^{5}x}{5}]+C$

Q.

Evaluate the integrals using substitution.
${\int }_0^2\frac{1}{x+4-x^2}dx.$

Q. Let $f$ be a twice differentiable function such that $f^{\prime \prime }\left(x\right)+f^{\prime }\left(x\right)=2e^{-x},f\left(0\right)=0$, and $f^{\prime }\left(0\right)=-2.$
Then the area(in sq. units) of region enclosed by $y=f\left(x\right)$ and $x-$axis is

Q.

Solve:

${\int }_0^{\frac{\pi }{2}}\frac{\sin xdx}{9+{\cos }^{2}x}$

Q.

Evaluate the integrals using substitution.
${\int }_0^{2}x\sqrt{x+2}dx.$

Q. $A$ point of discontinuity of $f\left(x\right)=tanx$ is

1. $x=\frac{\pi }{3}$
2. $x=0$
3. $x=\frac{\pi }{2}$
4. $x=\frac{\pi }{4}$