Property 1

Q.

How do you differentiate $y=\log x^2$?

Q.

The value of $4+\frac{1}{5+{\displaystyle \frac{1}{4+{\displaystyle \frac{1}{5+{\displaystyle \frac{1}{4+.........\infty }}}}}}}$ is

1. $2+\frac{4}{\sqrt{5}}\sqrt{30}$

2. $4+\frac{4}{\sqrt{5}}\sqrt{30}$

3. $2+\frac{2}{5}\sqrt{30}$

4. $5+\frac{2}{5}\sqrt{30}$

Q.

The negation of the compound proposition$p\vee (~p\vee q)$ is

1. $(p\wedge ~q)\wedge ~p$

2. $(p\wedge ~q)\vee ~p$

3. $(p\vee ~q)\vee ~p$

4. None of these

Q.

If $A\left(\alpha \right)=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{array}\right]$, Then the matrix $A^2\left(\alpha \right)=$

1. $A\left(2\alpha \right)$

2. $A\left(\alpha \right)$

3. $A\left(3\alpha \right)$

4. $A\left(4\alpha \right)$

Q.

If $x=a\left\{\cos \theta +\log \tan \left(\frac{\theta }{2}\right)\right\}$ and $y=a\sin \theta$, then $\frac{dy}{dx}$ is equal to:

1. $\cot \theta$

2. $\tan \theta$

3. $\sin \theta$

4. $\cos \theta$

Q. Let $f$ be a differentiable function from $R$ to $R$ such that $|f\left(x\right)-f\left(y\right)|\le 2|x-y{|}^{3/2}$, for all $x,y\in R$. If $f\left(0\right)=1$, then $\underset{0}{\overset{1}{\int }}{f}^2\left(x\right)dx$ is equal to :

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

Q. Let $A=\left(\begin{array}{ccc}[x+1]& [x+2]& [x+3]\\ \left[x\right]& [x+3]& [x+3]\\ \left[x\right]& [x+2]& [x+4]\end{array}\right),$ where $\left[t\right]$ denotes the greatest integer less than or equal to $t.$ If $det\left(A\right)=192,$ then the set of values of $x$ is the interval

1. $[60,61)$
2. $[65,66)$
3. $[62,63)$
4. $[68,69)$

Q. $If{\int }_0^k\frac{dx}{2+8x^2}=\frac{\pi }{16},thenk=$

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

Q.

The minimum value of ${27}^{\cos \left(x\right)}·{81}^{\sin \left(2x\right)}$ is

1. $-5$

2. $\frac{1}{5}$

3. $\frac{1}{243}$

4. $\frac{1}{27}$

Q. If $\left[x\right]$ denotes the greatest integer less than or equal to $x,$ then the value of the integral $\underset{-\pi /2}{\overset{\pi /2}{\int }}\left[\right[x]-\sin x]dx$ is equal to:

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

Q.

Integrate the following functions.
$\int \frac{5x-2}{1+2x+3x^2}dx.$

Q.

Let $f$ be a twice differentiable function on $\left(1,6\right)$. If $f\left(2\right)=8$, $f\left(2\right)=5$, $f\left(x\right)\ge 1$ and $f"\left(x\right)\ge 4$, for all $x\in \left(1,6\right)$, then:

1. $f\left(5\right)+f\left(5\right)\ge 28$

2. $f\left(5\right)+f"\left(5\right)\le 20$

3. $f\left(5\right)\le 10$

4. $f\left(5\right)+f\left(5\right)\le 26$

Q.

If$\sqrt{1-x^6}+\sqrt{1-y^6}=a^3(x^3-y^3)$, then$\frac{dy}{dx}=$

1. $\begin{array}{rcl}& & \frac{x^2\sqrt{1-x^6}}{y^2\sqrt{1-y^6}}\end{array}$

2. $\begin{array}{rcl}\frac{dy}{dx}& =& \frac{y^2\sqrt{1-y^6}}{x^2\sqrt{1-x^6}}\end{array}$

3. $\begin{array}{rcl}\frac{dy}{dx}& =& \frac{x^2\sqrt{1-y^6}}{y^2\sqrt{1-x^6}}\end{array}$

4. None of these

Q.

Oil is leaking from a tanker at the rate of $R\left(t\right)=2000e^{(-0.2t)}\mathrm{gallons}\mathrm{per}\mathrm{hour}$, where $t$ is measured in hours. How much oil leaks out of the tanker from the time $t=0$ to $t=10$?

Q. If $\frac{dy}{dx}=(e^y-x{)}^{-1},$ where $y\left(0\right)=0,$ then $y$ is expressed explicitly as

1. $\frac{1}{2}\ln (1+x^2)$
2. $\ln (1+x^2)$
3. $\ln (x+\sqrt{1+x^2})$
4. $\ln (x+\sqrt{1-x^2})$

Q. ${\int }_0^{\frac{\pi }{2}}\frac{dx}{a^{2}co{s}^{2}x+b^{2}si{n}^{2}x}$ where (a, b >0) is equal to-

1. $\frac{\pi }{2ab}$
2. $\frac{\pi }{2a}$
3. $\frac{\pi }{2b}$
4. $\frac{2}{\pi ab}$

Q. $I_n={\int }_0^{\frac{\pi }{4}}ta{n}^{n}xdx,thenli{m}_{n\to \infty }n[I_n+I_{n+2}]equals$

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

Q.

$x^{{\log }_{5}x}>5$ implies that

1. $x\in \left(2,\infty \right)$

2. $x\in \left(0,\frac{1}{5}\right)\cup \left(5,\infty \right)$

3. $x\in \left(1,\infty \right)$

4. $x\in \left(1,5\right)$

Q.

If $a=\sum _{n==0}^{\infty }\frac{x^{3n}}{3n!}$, $b=\sum _{n=1}^{\infty }\frac{x^{(3n-2)}}{(3n-2)!}$ and $c=\sum _{n=1}^{\infty }\frac{x^{(3n-1)}}{(3n-1)!}$.Then the value of $a^3+b^3+c^3-3abc$ is

1. $1$

2. $0$

3. $-1$

4. $2$

Q. Let F(x) = f(x) + $f\left(\frac{1}{x}\right)$, where f(x) = ${\int }_1^x\frac{logt}{1+t}dt$ . Then F(e) =

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

Q.

If $\int \frac{dx}{\sqrt{{\sin }^{3}x\cos x}}=g\left(x\right)+C$, then $g\left(x\right)=$

1. $-\frac{2}{\sqrt{\cot x}}$

2. $-\frac{2}{\sqrt{\tan x}}$

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

4. $\frac{2}{\sqrt{\tan x}}$

Q.

The value of $\sum _{n=1}^{100}{\int }_{n-1}^n{e}^{x-\left[x\right]}dx$, where $\left[x\right]$ is the greatest integer $\le x$, is

1. $100\left(e-1\right)$

2. $100e$

3. $100\left(1-e\right)$

4. $100\left(1+e\right)$

Q.

What is the value of $cot\left(-315°\right)$?

Q.

$\frac{\sin (B+A)+\cos (B-A)}{\sin (B-A)+\cos (B+A)}=$

1. $\frac{(\cos B+\sin B)}{(\cos B–\sin B)}$

2. $\frac{(\cos A+\sin A)}{(\cos A–\sin A)}$

3. $\frac{(\cos A–\sin A)}{(\cos A+\sin A)}$

4. None of these

Q.

If$f\left(x\right)=\left({\log }_{\cos }\tan x\right){({\log }_{\tan x}cotx)}^{-1}+{\tan }^{-1}\frac{x}{\sqrt{4-x^2}}$ then $f\left(1\right)$ is

1. $0$

2. $-2$

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

4. $\sqrt{3}$

Q. If ω is a complex cube root of unity, then value of expression $cos\left[\right\{(1-w)(1-w{)}^2+....+(10-w\left)\right(10-w{)}^2\left\}\frac{\pi }{900}\right]$

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

Q. If $f\left(x\right)=\sin x+\cos x$, then the number of solution of $f\left(x\right)=\left[f\left(\frac{\pi }{10}\right)\right]$, for $x\in (0,4\pi )$ is
(where [.] represent greatest integer function)

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

Q. 1) Let $z_1=2-i,z_2=-2+i,$ find
i) $Re\left(\frac{z_1{z}_2}{{\overline{z}}_1}\right)$

2) Let $z_1=2-i,z_2=-2+i,$ find
ii) $Im\left(\frac{1}{z_1{\overline{z}}_1}\right)$

Q.

The value of${\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{2}}{e}^x\left(\log \left(\sin \right(x\left)\right)+\cot \left(x\right)\right)dx=$

1. $e^{\frac{\mathrm{\pi }}{4}}\log \left(2\right)$

2. $-e^{\frac{\mathrm{\pi }}{4}}\log \left(2\right)$

3. $\left(\frac{1}{2}\right)e^{\frac{\mathrm{\pi }}{4}}\log \left(2\right)$

4. $\left(-\frac{1}{2}\right)e^{\frac{\mathrm{\pi }}{4}}\log \left(2\right)$

Q.

If $ta{n}^{-1}\frac{1}{a-1}=ta{n}^{-1}\frac{1}{x}+ta{n}^{-1}\frac{1}{a^2-x+1}$, then x is

1. a³
2. a² - a + 1
3. a² + a - 1
4. a/2