Functions without Antiderivatives as Known Combination of Basic Functions

Q. Let $A=\left\{\right(x,y)\in R\times R|2x^2+2y^2-2x-2y=1\},$
$B=\left\{\right(x,y)\in R\times R|4x^2+4y^2-16y+7=0\}$ and
$C=\left\{\right(x,y)\in R\times R|x^2+y^2-4x-2y+5\le r^2\}$.
Then the minimum value of $|r|$ such that $A\cup B\subseteq C$ is equal to:

1. $\frac{2+\sqrt{10}}{2}$
2. $\frac{3+2\sqrt{5}}{2}$
3. $1+\sqrt{5}$
4. $\frac{3+\sqrt{10}}{2}$

Q. If $\underset{x\to 0}{lim}\frac{x(1+a\cos x)-b\sin x}{x^3}=1$ then the value of $|a+b|$ is

Q. If $f\left(x\right)=cosx,0\le x\le \frac{\pi }{2}$, then the real number ‘c’ of the mean value theorem is

1. $\frac{\pi }{6}$
2. $co{s}^{-1}\left(\frac{2}{\pi }\right)$
3. $\frac{\pi }{4}$
4. $si{n}^{-1}\left(\frac{2}{\pi }\right)$

Q. Let $g\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt$, where $f$ is such that $\frac{1}{2}\le f\left(x\right)\le 1$ for $t\in [0,1]$ and $0\le f\left(t\right)\le \frac{1}{2}$ for $t\in [1,2]$ Then, $g\left(2\right)$ satisfies the inequality

1. $-\frac{3}{2}\le g\left(2\right)<\frac{1}{2}$
2. $0\le g\left(2\right)<2$
3. $\frac{1}{2}\le g\left(2\right)\le \frac{3}{2}$
4. $2<g\left(2\right)<4$

Q. Antiderivative of all the continuous functions can be written in terms of elementary functions

1. False
2. True

Q. Show that $\underset{x\to 1}{lim}\frac{x+x^2+x^3+....+x^n-n}{x-1}$ is $\frac{n(n+1)}{2}$.

Q. If $f\left(x\right)=sgn\left(x^5\right)$ then which of the following is/are false (where sgn denotes signum function)

1. $f^{\prime }\left(0^{+}\right)=1$
2. $f^{\prime }\left(0^{-}\right)=1$
3. f is continuous but not differentiable at x = 0
4. f is discontinuous at x = 0

Q. If $f\left(x\right)$ and $g\left(x\right)$ are inverse function of each other such that $f\left(1\right)=3$ & $f\left(3\right)=1$, then ${\int }_1^3(g\left(x\right)+\frac{x}{f^{\prime }\left(g\right(x\left)\right)})dx$ is equal to

Q. If, $A_1=\underset{n}{\overset{n+1}{\int }}(min\{|x-n|,|x-(n+1)|\left\}\right)dx$,
$A_2=\underset{n+1}{\overset{n+2}{\int }}(|x-n|-|x-(n+1\left)|\right)dx{A}_3=\underset{n+2}{\overset{n+3}{\int }}(|x-(n+4)|-|x-(n+3\left)|\right)dx$ and $g\left(x\right)=A_1+A_2+A_3,$ where $n\in N,$ then

1. $A_1+A_2+A_3=9$
2. $A_1+A_2+A_3=\frac{9}{4}$
3. $\sum _{n=1}^{100}g\left(x\right)=\frac{900}{4}$
4. $\sum _{n=1}^{100}g\left(x\right)=300$

Q. $\underset{x\to \infty }{lim}(\frac{x^2}{3x-2}-\frac{x}{3})$ is equal to

1. $\frac{1}{3}$
2. $\frac{2}{3}$
3. $\frac{-2}{3}$
4. $\frac{-2}{9}$
5. $\frac{2}{9}$

Q. The value of the definite integral $\underset{0}{\overset{\sqrt{\ln \left(\frac{\pi }{2}\right)}}{\int }}2x{e}^{x^2}\cos \left(e^{x^2}\right)dx$

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

Q. $\underset{\frac{\pi }{3}}{\overset{\frac{\pi }{6}}{\int }}\ln \left(\sin x\right)dx-\underset{-\ln 2}{\overset{\frac{1}{2}\ln \left(\frac{3}{4}\right)}{\int }}{\sin }^{-1}{e}^{x}dx$ is

1. $\frac{\pi }{6}\ln \left(\frac{3}{2}\right)$
2. $\frac{\pi }{3}\ln \left(\frac{3}{4}\right)$
3. $\frac{\pi }{6}\ln \left(\frac{2}{3}\right)$
4. $\pi \ln \left(\frac{4}{3}\right)$

Q.

$\underset{x\to 2}{lim}\frac{sin(e^{x-2}-1)}{log(x-1)}$

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

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

Q. $\underset{x\to 0}{lim}\frac{{\log }_e(1+x)}{3^x-1}=$

1. $1$
2. ${\log }_{3}e$
3. ${\log }_{e}3$
4. $0$

Q. Antiderivative of all the continuous functions can be written in terms of elementary functions

1. False
2. True

Q. If $\overline{a}=2\overline{i}+\overline{k},\overline{b}=\overline{i}+\overline{j}+\overline{k},\overline{c}=4\overline{i}-3\overline{j}+7\overline{k}$, then the vector $\overline{r}$ satisfying $\overline{r}\times \overline{b}=\overline{c}\times \overline{b}$ and $\overline{r}.\overline{a}=0$ is

1. $\overline{i}+8\overline{j}+2\overline{k}$
2. $\overline{i}-8\overline{j}+2\overline{k}$
3. $\overline{i}-8\overline{j}-2\overline{k}$
4. $-\overline{i}-8\overline{j}+2\overline{k}$

Q.

Consider $f:1,2,3\to$ {a, b, c}given by f(1)=a, f(2)=b and f(3)=c. Find the $(f^{-1}{)}^{-1}$ of $\left(f^{-1}\right)$ . Show that $(f^{-1}{)}^{-1}=f.$

Q. Antiderivative of all the continuous functions can be written in terms of elementary functions

1. False
2. True

Q. $\underset{x\to 1}{lim}\frac{1-x^{\frac{1}{3}}}{1-x^{\frac{1}{4}}}=$

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

Q. $\underset{x\to 0}{lim}\frac{cosx}{\pi -x}$

Q. The area bounded by $y=\sin \left(\frac{\pi }{2}x\right)$, $x=0,y=0$ and $x=\frac{4}{3}$ is

1. $\frac{6}{\pi }$
2. $\frac{3}{\pi }$
3. $\frac{2}{\pi }$
4. $\frac{9}{\pi }$

Q. Find the value of $a$?

Q. $\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}f\left(x\right)=\{\begin{array}{c}x;x\le 1\\ x^2+bx+c;x>1\end{array}\text{and}{f}^{\prime }\left(x\right)\text{exist finitely for}x\in R,& \text{(P)}& -1\\ & {\text{then}}^{\prime }b{c}^{\prime }\text{is equal to}& & \\ \text{(B)}& \text{The number of tangents to the}\text{curve}{y}^2-2x^3-4y+8=0\text{that}& \text{(Q)}& 1\\ & \text{pass through}(1,2)\text{is}& & \\ \text{(C)}& f\left(x\right)=2x^3-3a{x}^2+\frac{4}{3}{a}^{2}x+1(a>0)\text{attains its maxima and}& \text{(R)}& 2\\ & \text{minima at}x=k\text{and}x=k^2\text{respectively,}\text{then}a\text{is}& & \\ \text{(D)}& \text{If}f\left(x\right)=\underset{-1}{\overset{1}{\int }}\frac{\sin x}{1+t^2}dt\text{and}{f}^{\prime }\left(\frac{\pi }{3}\right)=\frac{\pi }{k},{\text{then}}^{\prime }{k}^{\prime }\text{is equal to}& \text{(S)}& 3\\ & & \text{(T)}& 4\\ & & \text{(U)}& 6\end{array}$

Which of the following option is CORRECT ?

1. $\left(A\right)\to \left(Q\right)$
2. $\left(B\right)\to \left(U\right)$
3. $\left(C\right)\to \left(S\right)$
4. $\left(D\right)\to \left(T\right)$

Q. Let $J=\underset{0}{\overset{1/2}{\int }}{(\frac{1}{4}-x^2)}^{4}dx$ and $K=\underset{0}{\overset{1/2}{\int }}{x}^4(1-x{)}^{4}dx.$ Then

1. $K=\frac{1}{1260}$
2. $J=\frac{1}{2}\underset{0}{\overset{1}{\int }}{x}^4(1-x{)}^{4}dx$
3. $J-K=0$
4. $\frac{J}{K}=2$

Q. $\underset{\frac{\pi }{3}}{\overset{\frac{\pi }{6}}{\int }}\ln \left(\sin x\right)dx-\underset{-\ln 2}{\overset{\frac{1}{2}\ln \left(\frac{3}{4}\right)}{\int }}{\sin }^{-1}{e}^{x}dx$ is

1. $\frac{\pi }{6}\ln \left(\frac{2}{3}\right)$
2. $\frac{\pi }{6}\ln \left(\frac{3}{2}\right)$
3. $\frac{\pi }{3}\ln \left(\frac{3}{4}\right)$
4. $\pi \ln \left(\frac{4}{3}\right)$

Q. If $f\left(x\right)$ and $g\left(x\right)$ are inverse function of each other such that $f\left(1\right)=3$ & $f\left(3\right)=1$, then ${\int }_1^3(g\left(x\right)+\frac{x}{f^{\prime }\left(g\right(x\left)\right)})dx$ is equal to

Q. $0<a<b<\frac{\pi }{2};f(a,b)=\frac{\tan b-\tan a}{b-a}$ then

1. $f(a,b)<\frac{1}{2}$
2. $f(a,b)\ge 1$
3. $f(a,b)\le 1$
4. $f(a,b)<0$

Q. If $f\left(x\right)=x-\left[x\right]$ and $g\left(x\right)=\underset{0}{\overset{x}{\int }}f(t+n)dt\forall n\in N,$ then $g^{\prime }\left(\frac{5}{2}\right)$ is equal to

1. $f\left(\frac{1}{2}\right)$
2. $\frac{1}{2}$
3. $f\left(\frac{3}{2}\right)$
4. $f\left(\frac{5}{2}\right)-f\left(5\right)$

Q. If $y=\frac{x}{(x+5)}$, then $\frac{dx}{dy}$ equals-

1. $\frac{5}{(1-y{)}^2}$
2. $\frac{5}{(1+y{)}^2}$
3. $\frac{1}{(1+y{)}^2}$
4. None of these