Existence of Limit

Q.

$\sin \left[\left(\frac{1}{2}\right){\cos }^{-1}\left(\frac{4}{5}\right)\right]=$

1. $\frac{1}{\sqrt{10}}$

2. $-\frac{1}{\sqrt{10}}$

3. $\frac{1}{10}$

4. $-\frac{1}{10}$

Q.

$\sin {12}^o\sin {48}^o\sin {54}^o$ is equal to

1. $\frac{1}{16}$

2. $\frac{1}{32}$

3. $\frac{1}{8}$

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

Q.

Show that ${lim}_{x\to 0}{e}^{\frac{-1}{x}}$ does not exist.

Q.

If $\cos \left(\alpha +\beta \right)=0$, then $\sin \left(\alpha -\beta \right)$ can be reduced to

1. $\cos \left(\beta \right)$

2. $\cos \left(2\beta \right)$

3. $\cos \left(2\alpha \right)$

4. $\sin \left(2\alpha \right)$

Q.

The necessary condition to be maximum or minimum for the function is

1. $f\left(x\right)=0$and it is sufficient

2. $f\left(x\right)=0$and it is sufficient

3. $f\left(x\right)=0$but it is not sufficient

4. $f\left(x\right)=0$and $f\left(x\right)=-ve$

Q.

Write the value of ${lim}_{x\to 2}\frac{|x-2|}{x-2}.$

Q.

The function $f:\mathrm{ℝ}\to \mathrm{ℝ}$ defined by $f\left(x\right)=\left(x-1\right)\left(x-2\right)\left(x-3\right)$ is

1. One-One but not Onto

2. Onto but not One-One

3. both One-One and Onto

4. neither One-One nor Onto

Q.

${\sin }^{2}17.5°+{\sin }^{2}72.5°$ is equal to

1. ${\cos }^{2}90°$

2. ${\tan }^{2}45°$

3. ${\cos }^{2}30°$

4. ${\sin }^{2}45°$

Q.

${lim}_{x\to 1}[x-1],$ Where [.] is the greatest integer function, is equal to

1. 1

2. 2

3. 0

4. None of these

Q.

$\sin 36°\sin 72°\sin 108°\sin 144°=$

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

2. $\frac{1}{16}$

3. $\frac{3}{4}$

4. $\frac{5}{16}$

Q.

Find: $14÷\left(\frac{5}{6}\right)$

Q.

How do you solve the modulus of a quadratic equation?

Q.

Let the coefficients of third, fourth and fifth terms in the expansion of ${\left[x+\left(\frac{a}{x^2}\right)\right]}^n,x\ne 0$ be in the ratio $12:8:3$. Then the term independent of $x$ in the expansion is equal to ………

Q. log2^x+logx^2=(10/3)=log2^y+logy^2 then x+y=

Q.

The value of $\underset{x\to \infty }{\lim }\left(\frac{x^2+bx+4}{x^2+ax+5}\right)$ is

1. $\frac{b}{a}$

2. $0$

3. $1$

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

Q.

The function $f\left(x\right)=a\sin \left(x\right)+\frac{1}{3}\sin \left(3x\right)$ has maximum value at $x=\frac{\pi }{3}$. The value of $a$ is:

1. $3$

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

3. $2$

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

Q.

Evaluate :${\int }_0^{10\pi }|\sin x|dx$

1. $20$

2. $8$

3. $10$

4. $18$

Q. Let f(x) be a function defined by $f\left(x\right)=(\begin{array}{c}4x-5,\mathrm{if}x\le 2\\ x-k,\mathrm{if}x>2\end{array}$ If

$\underset{x\to 2}{\lim }f\left(x\right)$
exists, then the value of k is

Q.

If $f\left(x\right)=xsin\left(\frac{1}{x}\right),x\ne 0,$ then ${lim}_{x\to 0}f\left(x\right)=$

1. 1

2. 0

3. -1

4. does not exist

Q.

When should we look for RHL and LHL for a limit given?

Q.

$\underset{x\to 0}{\lim }\frac{(1-\cos 2x)\sin 5x}{x^2\sin 3x}=$

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

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

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

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

Q. If f is an odd function. $\underset{x\to 0}{\lim }f\left(x\right)$exists and is equal to

Q. $li{m}_{x\to 1}\frac{\sqrt{1-cos2(x-1)}}{x-1}$

1. Exists and it equals $\sqrt{2}$

2. Exists and it equals $\sqrt{2}$
3. Does not exist because $(x-1)\to 0$
4. Does not exist because the left hand limit is not equal to the right hand limit.

Q. Evaluate the limit:
$\underset{x\to -\infty }{lim}(\sqrt{4x^2-7x}+2x)$

Q.

If $\sin \left(2A\right)=2\sin \left(A\right)$. Find $A$?

Q.

$\underset{x\to 0}{\lim }\left(\frac{1}{x}\right){\sin }^{-1}\left(\frac{2x}{1+x^2}\right)=$

1. $-2$

2. $0$

3. $2$

4. $\infty$

Q. If the roots of the equation (x + 1) (x + 9) + 8 = 0 are a and b, then the roots of the equation (x + a) (x + b) -8 = 0 are

(1) 1 and 9 (2) -4 and -6 (3) 4 and 6 (4) Cannot be determined

Q.

Show that ${lim}_{x\to 2^{-}}\frac{x}{\left[x\right]}\ne {lim}_{x\to 2^{+}}\frac{x}{\left[x\right]}$.

Q.

$1$ radian equals to?

Q.

Let $f\left(x\right)=\sin x,g\left(x\right)=x^2,h\left(x\right)={\log }_{\mathrm{e}}x$. If $F\left(x\right)=\left(hogof\right)\left(x\right)$, then $F\left(x\right)$ is equal to

1. $a{\mathrm{cosec}}^{3}x$

2. $2\cot x^2-4x^2{\mathrm{cosec}}^2{x}^2$

3. $2\cot x^{\mathit{2}}$

4. $-2\cos e{c}^{2}x$