Property 4

Q.

What is the integration of $\ln x$?

Q.

What is the derivative of $\cos ecx$?

Q. 87.Sin inverse sin (7pi/6)

Q.

If f(x) and g(x) are two functions of x, then the integration of product of f(x) and g(x) is:

1. 

2. 

3. 

4. 

Q. The value of $\underset{0}{\overset{2\pi }{\int }}\left[\sin 2x\right(1+\cos 3x\left)\right]dx$, where $\left[t\right]$ denotes the greatest integer function, is :

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

Q. The value of the definite integral $\underset{0}{\overset{\pi /2}{\int }}\sqrt{\tan x}dx$ is

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

Q.

Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.

Q. 18. what is integration of x sin inverse x

Q.

Let z be a complex number such that |z| = 5 . Find the maximum value of | z + 3 + 4i |

Q. Let $J_{n,m}=\underset{0}{\overset{1/2}{\int }}\frac{x^n}{x^m-1}dx,\forall n>m\text{and n},m\in N.$
Consider a matrix $A={\left[a_{ij}\right]}_{3\times 3}$ where
$a_{ij}=\{\begin{array}{cc}{J}_{6+i,3}-J_{i+3,3}& i\le j\\ 0& i>j\end{array}.$ Then $|adj{A}^{-1}|$ is

1. $(105{)}^2\times 2^{38}$
2. $(15{)}^2\times 2^{42}$
3. $(15{)}^2\times 2^{34}$
4. $(105{)}^2\times 2^{36}$

Q. Let $f:[–1,2]\to [0,\infty )$ be a continuous function such that $f\left(x\right)=f(1–x)$ for all $x\in [–1,2]$. Let $R_1=\underset{-1}{\overset{2}{\int }}xf\left(x\right)dx$, and $R_2$ be the area of the region bounded by $y=f\left(x\right),x=–1,x=2$, and the $x-$axis. Then

1. $R_1=2R_2$
2. $3R_1=R_2$
3. $2R_1=R_2$
4. $R_1=3R_2$

Q. Let $\left[t\right]$ denote the greatest integer $\le t$. Then the value of $8\cdot \underset{-\frac{1}{2}}{\overset{1}{\int }}\left(\right[2x]+|x|)dx$ is

Q.

The domain of $f\left(x\right)=\frac{1}{\sqrt{\left(\left|\cos x\right|+\cos x\right)}}$ is

1. $\left[\left(4n-1\right)\frac{\mathrm{\pi }}{2},\left(4n+1\right)\frac{\mathrm{\pi }}{2}\right]$

2. $\left[\left(4n+1\right)\frac{\mathrm{\pi }}{2},\left(4n-1\right)\frac{\mathrm{\pi }}{2}\right]$

3. $\left(\mathrm{n\pi },\left(\mathrm{n}+1\right)\mathrm{\pi }\right)$

4. $\left[2n\mathrm{\pi }-\left(\frac{\mathrm{\pi }}{2}\right),\left(\frac{\mathrm{\pi }}{2}\right)+2\mathrm{n\pi }\right]$

Q. The value of the definite integral ${\int }_0^1\frac{xdx}{x^3+16}$ lies in the interval [a, b]. The smallest such interval is.

1. 

2. [0, 1]

3. 
   

4. None of these

Q. The value of $\underset{0}{\overset{2\pi }{\int }}\frac{x{\sin }^{8}x}{{\sin }^{8}x+{\cos }^{8}x}dx$ is equal to :

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

Q. If $x=ct$ and $y=\frac{c}{t},$ find $\frac{dy}{dx}$ at $t=2.$

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

Q. Which of the following definite integrals reduces to $\frac{\pi }{2}$?

1. $\underset{0}{\overset{\pi }{\int }}\frac{dx}{1+(\sin x{)}^{\cos x}}$
2. $\underset{0}{\overset{\pi /2}{\int }}\frac{dx}{1+(\tan x{)}^5}$
3. $\underset{0}{\overset{\infty }{\int }}\frac{x^2+1}{x^4-x^2+1}dx$
4. $\underset{0}{\overset{\pi /2}{\int }}(\ln \left(\sec x\right))(e^{\ln \left(\ln 2\right)}{)}^{-1}dx$

Q. Let f(x) = sin x and g(x) = ln|x|. If the ranges of the composite functions fog and gof are $R_1$ and $R_2$ respectively, then

1. $R_1=\{u:-1<u<1\},R_2=\{v:-\infty >v<0\}$
2. $R_1=\{u:-1\le u\le 1\},R_2=\{v:-\infty <v\le 0\}$
3. $R_1=\{u:-1<u<1\},R_2=\{v:-\infty <v<0\}$
4. $R_1=\{u:-\infty <u\le 0\},R_2\{v:1\le v\le 1\}$

Q.

Evaluate : ${\int }_0^{\pi }{\cos }^{3}xdx$

1. $0$

2. $1$

3. $-1$

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

Q. The value of integral $\underset{0}{\overset{\pi /4}{\int }}[\sin x+[\cos x+[\tan x+[\sec x\left]\right]\left]\right]dx$ is equal to, (where $[\cdot ]$ denotes the greatest integer function)

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

Q. 43.The value of sin inverse (5/13) + cot inverse (3/4) is equal to (1) sin inverse (63/65) (2) sin inverse (12/13) (3) sin inverse (65/68) (4) sin inverse (5/12)

Q. The graph of $y=x^3+x-2$ is

1. 
2. 
3. 
4. None of these

Q.

If f(x) = sin[π^2]x+sin[-π^2]x, where [x]denotes greatest Integer less than or equal to x, then

Q. The area bounded by the curve $y=1+\frac{8}{x^2}$ with x-axis and the ordinates at x = 2, and x = 4 is

1. 2
2. 3
3. 4
4. 5

Q. Let $f$ and $g$ be continuous function on $[0,a]$ such that $f\left(x\right)=f(a-x)$ and $g\left(x\right)+g(a-x)=4,$ then $\underset{0}{\overset{a}{\int }}f\left(x\right)g\left(x\right)dx$ is equal to :

1. $\underset{0}{\overset{a}{\int }}f\left(x\right)dx$
2. $2\underset{0}{\overset{a}{\int }}f\left(x\right)dx$
3. $-3\underset{0}{\overset{a}{\int }}f\left(x\right)dx$
4. $4\underset{0}{\overset{a}{\int }}f\left(x\right)dx$

Q. $\text{Statement-I}:$ The value of the integral $\underset{\pi /6}{\overset{\pi /3}{\int }}\frac{dx}{1+\sqrt{\tan x}}$ is equal to $\frac{\pi }{6}$.
$\text{Statement-II}:\underset{a}{\overset{b}{\int }}f\left(x\right)d\left(x\right)=\underset{a}{\overset{b}{\int }}f(a+b-x)d\left(x\right)$.

1. Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
2. Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.
3. Statement-I is true; Statement-II is false.
4. Statement-I is false; Statement-II is true.

Q.

What is the equation of the curve whose subnormal is equal to constant $a$?

1. $y=ax+b$

2. $y^2=2ax+2b$

3. $a{y}^2-x^3=a$

4. None of these

Q. $\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{2}}{\int }}\frac{dx}{e^{\sin x}+1}$ is equal to

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

Q.

Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7^th and (m – 1)^th numbers is 5:9. Find the value of m.

Q.

By using properties of determinants, show that:

(i)

(ii)