Sandwich Theorem

Q. $\underset{x\to 0}{\lim }x\sin \left(\frac{1}{x}\right)$ is equal to

1. \N
2. 1
3. $\frac{1}{2}$
4. does not exist

Q. Find $\underset{x\to 1}{lim}f\left(x\right),$ where $f\left(x\right)=\{\begin{array}{cc}{x}^2-1,& x\le 1\\ -x^2-1,& x>1\end{array}$

Q.

The value of $\underset{x\to 0}{lim}\left(\right[\frac{100x}{sinx}]+[\frac{99sinx}{x}\left]\right)$, where [.] denotes the greatest integer function, is

1. 198

2. 197

3. 199

4. Does not exist

Q. The maximum and minimum values of the function |sin 4x+3| are

1. 1, 2
2. 2, 4
3. 4, 2
4. – 1, 1

Q.

${lim}_{x\to 0}\frac{sin3\theta }{tan2\theta }$

Q.

${lim}_{x\to 0}\frac{3sin2x+2x}{3x+2tan3x}$

Q.

${lim}_{x\to 2}\frac{x^2-x-2}{x^2-2x+sin(x-2)}$

Q.

${lim}_{x\to 0}\frac{cosax-cosbx}{coscx-cosdx}$

Q. Let $I_1=\underset{0}{\overset{1}{\int }}\frac{e^{x}dx}{1+x}$ and $I_2=\underset{0}{\overset{1}{\int }}\frac{x^{2}dx}{e^{x^3}(2-x^3)}$, then $\frac{I_1}{I_2}$ is equal

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

Q. $\begin{array}{cccc}& ListI& & ListII\\ P.& \text{The number of polynomials}f\left(x\right)\text{with non-negative integer coefficients of degree}\le 2,\text{satisfying}f\left(0\right)=0\text{and}\underset{0}{\overset{1}{\int }}f\left(x\right)dx=1,\text{is}& 1.& 8\\ Q.& \text{The number of points in the interval}[-\sqrt{13},\sqrt{13}]\text{at which}f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)\text{attains its maximum value, is}& 2.& 2\\ R.& \underset{-2}{\overset{2}{\int }}\frac{3x^2}{1+e^x}dx\text{equals}& 3.& 4\\ S.& \frac{\left(\underset{-1/2}{\overset{1/2}{\int }}\cos 2x\log \left(\frac{1+x}{1-x}\right)dx\right)}{\left(\underset{0}{\overset{1/2}{\int }}\cos 2x\log \left(\frac{1+x}{1-x}\right)dx\right)}\text{equals}& 4.& 0\end{array}$

Which of the following option is correct?

1. $\left(P\right)\to \left(3\right),\left(Q\right)\to \left(2\right)\left(R\right)\to \left(4\right),\left(S\right)\to \left(1\right)$
2. $\left(P\right)\to \left(2\right),\left(Q\right)\to \left(3\right)\left(R\right)\to \left(4\right),\left(S\right)\to \left(1\right)$
3. $\left(P\right)\to \left(3\right),\left(Q\right)\to \left(2\right)\left(R\right)\to \left(1\right),\left(S\right)\to \left(4\right)$
4. $\left(P\right)\to \left(2\right),\left(Q\right)\to \left(3\right)\left(R\right)\to \left(1\right),\left(S\right)\to \left(4\right)$

Q. The value of $\underset{x\to \infty }{lim}\frac{2{\left(x\right)}^{1/2}+3{\left(x\right)}^{1/3}+\cdots +n{\left(x\right)}^{1/n}}{{(2x-3)}^{1/2}+{(2x-3)}^{1/3}+\cdots +{(2x-3)}^{1/n}}$ is

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

Q. If $I=\sum _{k=1}^{98}\underset{k}{\overset{k+1}{\int }}\frac{k+1}{x(x+1)}dx,$ then

1. $I>{\log }_{e}99$
2. $I<{\log }_{e}99$
3. $I<\frac{49}{50}$
4. $I>\frac{49}{50}$

Q.

${lim}_{x\to 0}\frac{a^{mx}-b^{nx}}{sinkx}$

Q. $\underset{1}{\overset{e}{\int }}(\frac{1}{\sqrt{x\ln x}}+\sqrt{\frac{\ln x}{x}})dx$ equals

1. $\sqrt{e}$
2. $2e$
3. $2\sqrt{e}$
4. $\sqrt{2e}$

Q. $\underset{6}{\overset{16}{\int }}\frac{{\log }_e{x}^2}{{\log }_e{x}^2+{\log }_e(x^2-44x+484)}dx$ is equal to

1. $6$
2. $10$
3. $5$
4. $8$

Q.

What is the maximum value of sin(sinx)

Q. Evaluate the following integrals:
$\underset{-\mathrm{\pi }/2}{\overset{\mathrm{\pi }/2}{\int }}\left\{\sin \left|x\right|+\cos \left|x\right|\right\}dx$

Q. If $\underset{x\to 0}{lim}\frac{a+e^{1/x}}{1+b{e}^{1/x}}=2$, then $(a,b)$=

Q. For $\alpha ϵR$ (the set of all real numbers), $a\ne -1$,
${lim}_{n\to \infty }\frac{(1^a+2^a+\dots +n^a)}{(n+1{)}^{a-1}[(na+1)+(na+2)+\dots +(na+n)]}=\frac{1}{60}$

1. $5$
2. $7$
3. $\frac{-15}{2}$
4. $\frac{-17}{2}$

Q. If $y=\underset{x^2}{\overset{x^3}{\int }}\frac{1}{\ln t}dt$ $(\text{where}x\in R^{+}-\{1\left\}\right)$, then the value of $\frac{dy}{dx}$ is

1. $x(x-1)(\ln x{)}^2$
2. $x(x-1)(\ln x{)}^{-2}$
3. $x(x-1)\left(\ln x\right)$
4. $x(x-1)(\ln x{)}^{-1}$

Q.

${lim}_{x\to 0}\frac{5^x+3^x+2^x-3}{x}$

Q. ${lim}_{x\to 1}\frac{x^4-3x^3+2}{x^3-5x^2+3x+1}$

Q.

${lim}_{x\to 0}\frac{(1-cos2x)sin5x}{x^{2}sin3x}=$

1. $\frac{5}{6}$

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

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

4. $\frac{6}{5}$

Q. ${\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4\frac{5\pi }{8}+{\cos }^4\frac{7\pi }{8}=\frac{m}{2}$. Find $m$

Q. Find the absolute maximum and minimum values of the function of given by

$f\left(x\right)={\cos }^{2}x+\sin x,x\in [0,\mathrm{\pi }]$

Q. Evaluate $\underset{\theta \to \frac{\pi }{4}}{lim}\frac{\sqrt{2}-\cos \theta -\sin \theta }{(4\theta -\pi {)}^2}$

1. $\frac{1}{16\sqrt{2}}$
2. $\frac{1}{4\sqrt{2}}$
3. $-\frac{1}{4\sqrt{2}}$
4. $-\frac{1}{16\sqrt{2}}$

Q. If $f\left(x\right)$ is continuous at $x=0$, where
$\begin{array}{c}f\left(x\right)=\frac{1-\cos kx}{x^2},forx\ne 0\\ =\frac{1}{2},forx=0\end{array}$
Find k.

Q. Prove that : ${cos10}^o{cos30}^o{cos50}^o{cos70}^o=\frac{3}{16}$

Q.

Evaluate the expression when $x=-\frac{4}{5}andy=\frac{1}{3}$

$2x+6y$

Q.

${lim}_{x\to 0}\frac{1-cosmx}{x^2}$