Property 5

Q.

How do you evaluate $\sin \left(\frac{3\mathrm{\pi }}{2}\right)$?

Q.

If $\underset{x\to 0}{lim}\frac{{\sin }^{-1}x-{\tan }^{-1}x}{3x^3}$ is equal to $L$, then the value of $(6L+1)$ is:

1. $\frac{1}{2}$

2. $2$

3. $\frac{1}{6}$

4. $6$

Q.

$\sin \left(\frac{\pi }{10}\right)\sin \left(\frac{3\pi }{10}\right)=$

1. $\frac{1}{2}$

2. $-\frac{1}{2}$

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

4. $1$

Q. The value of the integral $\underset{-1}{\overset{1}{\int }}\log (x+\sqrt{x^2+1})dx$ is

1. $0$
2. $-1$
3. $2$
4. $1$

Q.

If $x=\sin t\cos 2t$ and $y=\cos t\sin 2t$, then at $t=\frac{\mathrm{\pi }}{4}$, the value of $\frac{dy}{dx}$ is equal to

1. $-\frac{1}{2}$

2. $\frac{1}{2}$

3. $-2$

4. $2$

Q.

${\int }_0^{\frac{\mathrm{\pi }}{2}}\log \sin xdx=$

1. $-\mathrm{\pi }\log 2$

2. $\mathrm{\pi }\log 2$

3. $-\frac{\mathrm{\pi }}{2}\log 2$

4. $\frac{\mathrm{\pi }}{2}\log 2$

Q.

How do you find the exact value of $\sin \left(\frac{\mathrm{\pi }}{3}\right)$ ?

Q. If $f\left(x\right)={\int }_0^x(1+t^3{)}^{-1/2}dt$ and g(x) is the inverse of f, then the value of $\frac{g^{\prime \prime }\left(x\right)}{g^2\left(x\right)}$ is

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

Q.

If in a $∆\mathrm{ABC}$, $a=6\mathrm{cm},b=8\mathrm{cm},c=10\mathrm{cm}$, then $\sin 2A$ is

1. $\frac{6}{25}$

2. $\frac{8}{25}$

3. $\frac{10}{25}$

4. $\frac{24}{25}$

Q.

If $x=A\cos 4t+B\sin 4t$, then $\frac{d^{2}x}{d{t}^2}$ is equal to:

1. $-16x$

2. $16x$

3. $x$

4. $-x$

Q.

Find the integral of $\frac{x\log x}{{(1+x^2)}^2}$ under the limits $0$ to infinity.

Q. Evaluate the limit:
$\underset{x\to \frac{\pi }{3}}{\text{lim}}\frac{\sqrt{3}-\tan x}{\pi -3x}$

Q.

The value of ${\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{1}{1+{\tan }^3\left(x\right)}dx$ is

1. $\mathrm{\pi }$

2. $0$

3. $2\mathrm{\pi }$

4. $\frac{\mathrm{\pi }}{4}$

Q.

If $e^x+e^y=e^{(x+y)}$, then $\frac{dy}{dx}at(2,2)$ is

Q. Let $f\left(x\right)$ be a twice differentiable function for all real values of $x$ and satisfies $f\left(1\right)=1,f\left(2\right)=4,f\left(3\right)=9$. Then which of the following is definitely true?

1. $f^{\prime \prime }\left(x\right)=2$ $\forall x\in (1,3)$
2. $f^{\prime \prime }\left(x\right)=f^{\prime }\left(x\right)=5$ for some $x\in (2,3)$
3. $f^{\prime \prime }\left(x\right)=3\forall x\in (2,3)$
4. $f^{\prime \prime }\left(x\right)=2$ for some $x\in (1,3)$

Q. $\underset{x\to \frac{\pi }{2}}{lim}\frac{sinx-(sinx{)}^{sinx}}{1-sinx+lnsinx}$ is equal to

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

Q.

The value of ${\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{1}{1+\cot \left(x\right)}dx$is

1. $\mathrm{\pi }$

2. $\frac{\mathrm{\pi }}{2}$

3. $\frac{\mathrm{\pi }}{3}$

4. $\frac{\mathrm{\pi }}{4}$

Q. Prove that ${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx,$
hence evaluate ${\int }_0^{\pi }\frac{x\sin x}{1+{\cos }^{2}x}dx.$

Q. If the sum of maximum and minimum values of $E={\left({\sin }^{-1}x\right)}^2+2\pi {\cos }^{-1}x+{\pi }^2$ is $\frac{a{\pi }^2}{b},$where $a$ and $b$ are co-prime, then the value of $(a-b)$ is

Q.

The eccentricity of the conic $\left[\frac{{\left(\mathrm{x}+2\right)}^2}{7}\right]+{\left(\mathrm{y}-1\right)}^2=14$ is

1. $\sqrt{\frac{7}{8}}$

2. $\sqrt{\frac{6}{17}}$

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

4. $\sqrt{\frac{6}{11}}$

5. $\sqrt{\frac{6}{7}}$

Q.

Evaluate the definite integrals.
${\int }_0^2\frac{6x+3}{x^2+4}dx.$

Q.

If $f\left(x\right)={\int }_a^x{t}^3{e}^{t}dt,then\frac{d}{dx}f\left(x\right)=$ [MP PET 1989]

1. 
   

2. 
   

3. None of these

4. 
   

Q. The value of $\underset{0}{\overset{\pi /2}{\int }}\left(\cos \right(2018\pi {\sin }^{2}x)+\cos (1009\pi \sin x\left)\right)dx$ is

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

Q. Let $I_1=\underset{0}{\overset{\pi /3}{\int }}(2\sec x{)}^{100}dx,$ and $I_2=\underset{0}{\overset{\ln (2+\sqrt{3})}{\int }}(e^x+e^{-x}{)}^{99}dx$. Then the value of $\frac{I_1}{I_2}$ is

1. $2$
2. $\frac{1}{2}$
3. $1$
4. $2^{99}$

Q.

If $f\left(x\right)={\int }_a^x{t}^3{e}^{t}dt,then\frac{d}{dx}f\left(x\right)=$ [MP PET 1989]

1. $e^x(x^3+3x^2)$
2. $x^3{e}^x$

3. $a^3{e}^x$

4. $Noneofthese$

Q. If given $g\left(x\right)={\int }_2^{x^2}\frac{1}{ln(1+t^2)}dtthenfind{g}^{\prime }\left(\sqrt{2}\right)$ .

1. $Noneofthese$
2. $\frac{2.\sqrt{2}}{ln\left(2\right)}$
3. $\frac{\sqrt{2}}{ln\left(3\right)}$
4. $\frac{2.\sqrt{2}}{ln\left(3\right)}$

Q. $\int \frac{dx}{(x+1)\sqrt{2x-3}}$ equals to (where $C$ is integration constant)

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

Q. Let $f\left(x\right)=x^3-x^2+1$ be a continuous function in $R$.
Statement $1:f\left(x\right)$ has atleast one real solution in $(-2,2).$
Statement $2:f^{-1}\left(x\right)$ has atleast one real solution in $(-2,2).$
Then

1. Statement $1$ is true, Statement $2$ is true and Statement $2$ is the correct explanation of Statement $1.$
2. Statement $1$ is true, Statement $2$ is true and Statement $2$ is not the correct explanation of Statement $1.$
3. Statement $1$ is true and Statement $2$ is false.
4. Statement $1$ is false and Statement $2$ is true.

Q. $\underset{-\pi }{\overset{\pi }{\int }}|\pi -|x\left|\right|dx$ is equal to

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

Q. Differentiate the function w.r.t. $x$.
$x^{\sin x}+(\sin x{)}^{\cos x}$