Theorems for Continuity

Q.

The domain of ${\sin }^{-1}\left[{\log }_3\left(\frac{x}{3}\right)\right]$ is

1. $\left[1,9\right]$

2. $\left[-1,9\right]$

3. $\left[-9,1\right]$

4. $\left[-9,-1\right]$

Q.

What is the answer for $2\sqrt{3}$ whole square?

Q.

If $\alpha$and $\beta$are roots of$4x^2+3x+7=0$, then the value of$\left(\frac{1}{{\alpha }^3}\right)+\left(\frac{1}{{\beta }^3}\right)=$

1. $\raisebox{1ex}{$-27$}\!\left/ \!\raisebox{-1ex}{$64$}\right.$

2. $\raisebox{1ex}{$225$}\!\left/ \!\raisebox{-1ex}{$343$}\right.$

3. $\raisebox{1ex}{$63$}\!\left/ \!\raisebox{-1ex}{$16$}\right.$

4. $\raisebox{1ex}{$63$}\!\left/ \!\raisebox{-1ex}{$64$}\right.$

Q.

$\sqrt{6}$ multiply with $\sqrt{27}$ is equal to

Q.

If $f\left(x\right)=\sqrt{\left(1+{\cos }^2\left(x^2\right)\right)}$, then $f\left(\frac{\sqrt{\mathrm{\pi }}}{2}\right)$ is

1. $\frac{\sqrt{\mathrm{\pi }}}{6}$

2. $-\sqrt{\frac{\mathrm{\pi }}{6}}$

3. $\frac{1}{\sqrt{6}}$

4. $\frac{\mathrm{\pi }}{\sqrt{6}}$

Q.

Convert 37 degree Celsius into Fahrenheit.

Q. If $f\left(x\right)={\log }_{[x-1]}\left(\frac{|x|}{x}\right),$ where $[\cdot ]$ is greatest integer function, then domain $D_f$ and range $R_f$ of $f\left(x\right)$ are

1. $D_f=[3,\infty )$
2. $R_f=\left\{0\right\}$
3. $R_f=R$
4. $D_f=R$

Q.

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

1. $\frac{\mathrm{\pi }}{6}$

2. $\frac{\mathrm{\pi }}{4}$

3. ${\sin }^{-1}\left(\frac{2}{\mathrm{\pi }}\right)$

4. ${\cos }^{-1}\left(\frac{2}{\mathrm{\pi }}\right)$

Q.

If $a=\frac{z-1}{z+1}$ is purely imaginary number $z\ne 1$, then $\left|\begin{array}{c}z\end{array}\right|$is

1. $1$

2. $2$

3. $3$

4. $4$

Q.

The principal value of ${\sin }^{-1}\left(\sin \frac{5\mathrm{\pi }}{3}\right)$is

1. $\frac{5\mathrm{\pi }}{3}$

2. $-\frac{5\mathrm{\pi }}{3}$

3. $-\frac{\mathrm{\pi }}{3}$

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

Q.

Find the principal value of${\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)$

Q. The inverse of the function $f\left(x\right)=\frac{e^x-e^{-x}}{e^x+e^{-x}}+2$ is given by

1. $lo{g}_e{\left(\frac{x-2}{x-1}\right)}^{\frac{1}{2}}$
2. $lo{g}_e{\left(\frac{x-1}{3-x}\right)}^{\frac{1}{2}}$
3. $lo{g}_e{\left(\frac{x}{2-x}\right)}^{\frac{1}{2}}$
4. $lo{g}_e{\left(\frac{x-1}{x+1}\right)}^{-2}$

Q.

The domain of the real-valued function$f\left(x\right)=\sqrt{1-2x}+2{\sin }^{-1}\left[\frac{3x-1}{2}\right]$ is

1. $\left[-\frac{1}{3},1\right]$

2. $\left[\frac{1}{2},1\right]$

3. $\left[-\frac{1}{2},\frac{1}{3}\right]$

4. $\left[-\frac{1}{3},\frac{1}{2}\right]$

Q.

If $\left|\frac{z-25}{z-1}\right|=5$, then find the value of $\left|z\right|$

1. $3$

2. $4$

3. $5$

4. $6$

Q. Let $f\left(x\right)=\sin x+\cos ax$ be periodic function. Then

1. ${}^{\prime }{a}^{\prime }$ is real number
2. ${}^{\prime }{a}^{\prime }$ is rational number
3. ${}^{\prime }{a}^{\prime }$ is any irrational number
4. $a=0$

Q.

If ${\sin }^2\left(\theta \right)-2\cos \left(\theta \right)+\frac{1}{4}=0$, then the general value of $\left(\theta \right)$ is

1. $n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{3}$

2. $2n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{3}$

3. $2n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{6}$

4. $n\mathrm{\pi }\pm \frac{\mathrm{\pi }}{6}$

Q. If $f:R\to [-1,1]$ where $f\left(x\right)=\sin \left(\frac{\pi }{2}\right[x\left]\right)$ $\left(\right[.]$ denotes greatest integer function$)$, then $f\left(x\right)$ is:

1. an onto function
2. an into function
3. a periodic function
4. many-one function

Q. The values of $K$ for which the matrix

$A=\left[\begin{array}{ccc}1& 0& -K\\ 2& 1& 3\\ K& 0& 1\end{array}\right]$ is invertible, is:

1. No value of $K$
2. All the real values of $K$
3. $K\in [-1,1]$
4. $K\in \{-1,1\}$

Q.

For all real values of $x$, increasing function $f\left(x\right)$ is

1. $x^{-1}$

2. $x^2$

3. $x^3$

4. $x^4$

Q.

Prove that the functions f(x) =5x-3 is continuous at x = 0, at x = -3 and at x=5.

Q.

The value of ${\left(0.05\right)}^{{\log }_{\sqrt{20}}\left(0.1+0.01+0.001+...\right)}$ is

1. $81$

2. $\frac{1}{81}$

3. $20$

4. $\frac{1}{20}$

Q. Let $f:N\to Z$ be a function defined as $f\left(x\right)=x-1000.$ Then which of the following is true?

1. $f\left(0\right)=-1000$
2. $f\left(100\right)=0$
3. $f$ is a many-one function
4. None of these

Q.

The complex number $z=x+iy$ which satisfies the equation $\left|\frac{\left(z-3i\right)}{\left(z+3i\right)}\right|=1$ lie on which of the following.

1. The $X$- axis

2. The straight line $y=3$

3. A circle passing through origin

4. None of these

Q.

If $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ is defined by $\mathrm{f}\left(\mathrm{x}\right)=\left[\mathrm{x}-3\right]+\left[\mathrm{x}-4\right]$ for $\mathrm{x}\in \mathrm{R}$, then $\underset{\mathrm{x}\to 3^{-}}{\lim }\mathrm{f}\left(\mathrm{x}\right)=$

1. $-2$

2. $-1$

3. $0$

4. $1$

Q.

Find the value of $\mathrm{x}$ if $\frac{2x}{3}=18.$

Q. Let $R_1=\left\{\right(a,b)\in N\times N:|a-b|\le 13\}$ and $R_2=\left\{\right(a,b)\in N\times N:|a-b|\ne 13.\}$ Then on $N:$

1. $R_1$ is an equivalence relation but $R_2$ is not
2. Neither $R_1$ nor $R_2$ is an equivalence relation
3. Both $R_1$ and $R_2$ are equivalence relations
4. $R_2$ is an equivalence relation but $R_1$ is not

Q.

Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

Q.

If $f\left(x+y\right)=f\left(x\right)·\left(y\right)$ and $\sum _{x=1}^{\infty }f\left(x\right)=2$. $x,y\in \mathrm{\mathbb{N}}$, where $\mathrm{\mathbb{N}}$ is the set of all natural numbers, then the value of $\frac{f\left(4\right)}{f\left(2\right)}$ is

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

2. $\frac{1}{9}$

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

4. $\frac{4}{9}$

Q. If $f:N\to N,f\left(x\right)=x+3$, then $f^{-1}\left(x\right)=$_____

1. $3-x$
2. $x+3$
3. does not exist
4. $x-3$

Q.

The domain of the function $f\left(x\right)=\frac{\sqrt{4-x^2}}{{\sin }^{-1}\left(2-x\right)}$ is

1. $\left[0,2\right]$

2. $[0,2)$

3. $[1,2)$

4. $\left[1,2\right]$