Property 3

Q.

Let $f:R\to R$ be defined as $f\left(x\right)=x^4$. Choose the correct answer.
(a) f is one-one onto
(b) f is many-one onto
(c) f is one-one but not onto
(d) f is neither one-one nor onto

Q. Suppose $\text{det}\left[\begin{array}{cc}\sum _{k=0}^{n}k& \sum _{k=0}^n{}^n{C}_k{k}^2\\ \sum _{k=0}^n{}^n{C}_{k}k& \sum _{k=0}^n{}^n{C}_k{3}^k\end{array}\right]=0$
hold for some positive integer $n.\text{Then}\sum _{k=0}^n\frac{{}^n{C}_k}{k+1}$ equals

Q. Let $f\left(x\right)$ be a differentiable function defined on $[0,2]$ such that $f^{\prime }\left(x\right)=f^{\prime }(2-x)$ for all $x\in (0,2),$ $f\left(0\right)=1$ and $f\left(2\right)=e^2.$ Then the value of $\underset{0}{\overset{2}{\int }}f\left(x\right)dx$ is

1. $1+e^2$
2. $1-e^2$
3. $2(1-e^2)$
4. $2(1+e^2)$

Q. If $f\left(x\right)=x^{100}+x^{99}+....+x+1,\text{then}{f}^{{}^{\prime }}\left(1\right)$ is equal to

1. $5050$
2. $5051$
3. $50051$
4. $5049$

Q. Find the integral ${\int }_0^{\infty }{e}^{-x}dx$.

1. 0
2. 2
3. 1
4. $\infty$

Q.

The function$f\left(x\right)=\left[x\right]\cos \left[\right(2x-1)/2]\pi ,[.]$ denotes the greatest integer function is discontinuous at

1. All $x$

2. All integer points

3. No $x$

4. $x$ which is not an integer

Q. Let $\left[t\right]$ denote the greatest integer less than or equal to $t$. Then the value of the integral $\underset{-3}{\overset{101}{\int }}\left(\right[\sin \left(\pi x\right)]+e^{\left[\cos \right(2\pi x\left)\right]})dx$ is equal to

Q. The value of the integral $\underset{-2}{\overset{2}{\int }}\frac{{\sin }^{2}x}{\left[\frac{x}{\pi }\right]+\frac{1}{2}}dx$

(where $\left[x\right]$ denotes the greatest integer less than or equal to $x$ ) is:

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

Q. The value of $\underset{0}{\overset{\pi }{\int }}|\cos x{|}^{3}dx$ is :

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

Q. The value of integral $\underset{0}{\overset{\infty }{\int }}[n\cdot e^{-x}]dx$ is equal to $($where $[\cdot ]$ denotes the greatest integer function and $n\in N,n>1)$

1. $\ln \left(\frac{n^{n-1}}{n!}\right)$
2. $\ln \left(\frac{n^n}{n!}\right)$
3. $\ln \left(\frac{n^{n+1}}{n!}\right)$
4. $0$

Q. Let $f$ be a continuous function satisfying the equation $\underset{0}{\overset{x}{\int }}f\left(t\right)dt+\underset{0}{\overset{x}{\int }}tf(x-t)dt=e^{-x}-1.$ Then the value of $e^{10}f\left(10\right)$ is

Q. If $\underset{0}{\overset{1}{\int }}\frac{e^x}{1+x}dx=k,$ then $\underset{0}{\overset{1}{\int }}\frac{e^x}{(1+x{)}^2}dx$ is equal to

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

Q. The absolute value of $\frac{\underset{0}{\overset{\pi /2}{\int }}(x\cos x+1)e^{\sin x}dx}{\underset{0}{\overset{\pi /2}{\int }}(x\sin x-1)e^{\cos x}dx}$ is equal to

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

Q.

If f:R->R is defined as f(x)=x²-3x+2 .

Find f(f(x))

Q. Suppose $f\left(x\right)=\{\begin{array}{cc}a+bx,& x<1\\ 4,& x=1\\ b-ax,& x>1\end{array}$ and if $\underset{x\to 1}{lim}f\left(x\right)=f\left(1\right)$ what are possible values of $a$ and $b?$

Q. If $f\left(x\right)=x\text{sgn}(x-1),$ then

1. $f\left(x\right)$ is continuous at $x=1$
2. $f\left(x\right)$ is discontinuous at $x=1$
3. $\underset{x\to 1^{+}}{lim}x\text{sgn}(x-1)=-1$
4. $\underset{x\to 1^{-}}{lim}x\text{sgn}(x-1)=-1$
   

Q. The value of integral $\underset{0}{\overset{1}{\int }}[3x^2+1]dx$ is
(where $[.]$ denotes the greatest integer function)

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

Q. The value of $\alpha$ for which $4\alpha \underset{-1}{\overset{2}{\int }}{e}^{-\alpha |x|}dx=5$ is

1. $\ln \sqrt{2}$
2. $\ln 2$
3. $\ln \sqrt{8}$
4. $\ln \left(\frac{3}{2}\right)$

Q.

By using properties of definite integrals, evaluate the integrals
${\int }_0^{\pi }log(1+cosx)dx.$

Q. Integrate the function:
$\sqrt{x^2+4x-5}$

Q. Integrate the rational function: $\frac{(x^2+1)(x^2+2)}{(x^2+3)(x^2+4)}$

Q.

Find the value of x for which the following expressions attain minimum value:

a) x^2 +1

Q. If $\underset{0}{\overset{\infty }{\int }}\frac{dx}{(1+x^2{)}^4}=\frac{K\pi }{32}$, then the value of $K$ is:

Q. The value of $\underset{1}{\overset{3}{\int }}{e}^{\left\{x\right\}}dx$ is equal to
(where $\{.\}$ denotes fractional part function)

1. $e-1$
2. $2(e-1)$
3. $3(e-1)$
4. $4(e-1)$

Q. Integrate the function: $\frac{e^x}{(1+e^x)(2+e^x)}$

Q. Consider of function f (x) such that $(x^{2^{2019}-1}-1)f\left(x\right)=(x+1)(x^2+1)(x^4+1)(x^8+1)...(x^{2^{2018}}+1)-1$
then the value of f (2) is equal to

Q. The value of $\underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{dx}{\left[x\right]+\left[\sin x\right]+4},$ where $\left[t\right]$ denotes the greatest integer less than or equal to $t$, is :

1. $\frac{3}{20}(4\pi -3)$
2. $\frac{3}{10}(4\pi -3)$
3. $\frac{1}{12}(7\pi +5)$
4. $\frac{1}{12}(7\pi -5)$

Q.

All normals to the curve x = a cos t + at sin t, y = a sin t – at cos t

are at a distance a from the origin that is equal to….

1. a

2. 3a

3. 2a

4. 8a

Q. Integrate the function:$e^x(\frac{1}{x}-\frac{1}{x^2})$

Q. If $\underset{-1}{\overset{1}{\int }}\frac{3^x+3^{-x}}{1+3^x}dx=\frac{8}{\ln \left(k\right)},$ then which of the following is/are true?

1. $k=\underset{0}{\overset{3}{\int }}\left(3x^2\right)dx$
2. $k=\underset{-3}{\overset{3}{\int }}\left(3x^2\right)dx$
3. highest prime factor of $k$ is $3$
4. $k$ is not divisible by $6$