Signum Function

Q.

The sum of the infinite series $1+\frac{2}{3}+\frac{7}{3^2}+\frac{12}{3^3}+\frac{17}{3^4}+\frac{22}{3^5}+\dots$. is equal to

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

2. $\frac{15}{6}$

3. $\frac{13}{6}$

4. $\frac{11}{6}$

Q. The minimum value of $|x|+|x+\frac{1}{2}|+|x-3|+|x-\frac{5}{2}|$ is

1. 0
2. 2
3. 4
4. 6

Q.

If $f\left(x\right)=e^x\sin x$, then $f\left(x\right)$ is equal to

1. $e^{6x}\sin 6x$

2. $2e^x\cos x$

3. $8e^x\sin x$

4. $8e^x\cos x$

Q. The image of the interval $[-1,3]$ under the mapping $f\left(x\right)=4x^3-12x$ is

1. $[-2,0]$
2. $[-8,72]$
3. $[8,72]$
4. $[1,3]$

Q.

If $f\left(x\right)=\frac{x}{x-1},x\ne 1$, then ${\left(fofo...of\right)}_{\left(17times\right)}\left(x\right)$ is equal to

1. $\frac{x}{\left(x-1\right)}$

2. $x$

3. ${\left[\frac{x}{x-1}\right]}^{17}$

4. $\frac{17x}{x-1}$

Q. State, whether each of the following sets is a finite set or an infinite set:

$\text{{}x:x=3n-2,n\in W,n\le 8\text{}}$

Q.

Let $A=\{xϵR:x\ne 0,-4\le x\le 4\}$ and $f:A\to R$ be defined $f\left(x\right)=\frac{|x|}{x}$ for $xϵA$. Then A is

1. $\{x:0\le x\le 4\}$

2. $\left\{1\right\}$

3. $\{1,-1\}$

4. $\{x:-4\le x\le 0\}$

Q. The set of real values of '$x$' satisfying the equality $\left[\frac{3}{x}\right]+\left[\frac{4}{x}\right]=5$ (where $[.]$ denotes the greatest integer function) belongs to the interval $(a,\frac{b}{c}]$ where $a$, $b$, $c$ $\in N$ and $\frac{b}{c}$ is in its lowest form. Find the value of $a+b+c+abc$.

Q. Show that the signum function $f:R\to R$, given by
$f\left(x\right)=\left\{1,\text{if}x>00,\text{if}x=0\right\}$ is neither one - nor onto $-1$, if $x<0$.

Q.

Let $f:R\to R$ be the Signum function defined as $f\left(x\right)=\{\begin{array}{cc}1,& x>0\\ 0,& x=0\\ -1,& x<0\end{array}$ and $g:R\to R$ be the

greatest integer function given by g(x) =[x] is greatest integer less than or equal to x. Then, fog and gof coincide in (0, 1].

Q. If $f:R\to (-1,1)$ is defined by $f\left(x\right)=\frac{-x\left|x\right|}{1+x^2}$ then $f^{-1}\left(x\right)$ equals

1. $-sgn\left(x\right)\sqrt{\frac{\left|x\right|}{1-\left|x\right|}}$
2. $\sqrt{\frac{x}{1-x}}$
3. $\sqrt{\frac{\left|x\right|}{1-\left|x\right|}}$
4. None of these

Q. Consider $f\left(x\right)=sgn\left(\sin x\right)+\left[x\right];2\le x\le 4$ and $g\left(x\right)=-2+|x-3|$; where $[.]$ denotes greatest integer function. Then $\underset{x\to 3}{lim}gof\left(x\right)$ equal to

Q. Find the range of $f\left(x\right)=\mathrm{sgn}\left(x^2-2x+3\right)$ is

1. $\left\{0,1\right\}$
2. $\left\{-1,0,1\right\}$
3. $\left\{1\right\}$
4. $\left\{-1,-2,2,3\right\}$

Q. $sgn(x^3-4x^2+3x)=1,x\in Z$ and $x\in [-5,10]$, then number of possible values of $x$ is :

1. $7$
2. $13$
3. $10$
4. $8$

Q. The range of the function $f\left(x\right)=|x-1|+|x-2|,-1\le x\le 3$ is

1. $[1,3]$
2. $[1,5]$
3. $[3,5]$
4. none of these

Q. lf $x$ satisfies $|x-1|+|x-2|+|x-3|\ge 6$, then

1. $x\le -2$ or $x\ge 4$
2. $R$
3. $0\le$ $x$ $\le 1$
4. $x\le 0$ or $x\ge 4$

Q. Let $f:R\to R$ be defined as $f\left(x\right)=3^{-\left|x\right|}-3^x+sgn\left(e^{-x}\right)+2$ (Where sgn x denotes signum function of x). Then which one of the following is correct ?

1. f is injective but not surjective
2. f is surjective but not injective
3. f is injective as well as surjective
4. f is neither injective nor surjective

Q. Let $f\left(x\right)=\left[x\right]$ and $g\left(x\right)=sgn\left(x\right)$ (where $[\cdot ]$ denotes greatest integer function), then discuss the continuity of $f\left(x\right)\pm g\left(x\right)$, $f\left(x\right).g\left(x\right)$ and $\frac{f\left(x\right)}{g\left(x\right)}$ at $x=0$.

Q. Number of integral solutions of the inequation $x^2-10x+25sgn(x^2+4x-32)\le 0$

1. $infinite$
2. $6$
3. $7$
4. $8$

Q. Find the value of sgn (-1) + sgn (1) + sgn (5) + sgn (0), where sgn is the signum function.
___

Q. Consider the equation $||x-1|-2|=\lambda$ . Which of the following statement(s) is/are true?

1. If the given equation has two solutions, then $\lambda$ belongs to $(2,\infty )\cup \left\{0\right\}$ .
2. The number of integral values of $\lambda$ so that the given equation has four solutions, is $1.$
3. If the given equation has three solutions, then $\lambda$ belongs to $\left\{2\right\}$ .
4. If the given equation has two solutions, then $\lambda$ belongs to $(-\infty ,2)$ .

Q. The range of the function $f\left(x\right)=\mathrm{sgn}\left(\frac{{\sin }^{2}x+2\sin x+4}{{\sin }^{2}x+2\sin x+3}\right)$ is (where sgn(.) denotes signum function)

1. $\{-1,0,1\}$
2. $\{-1,0\}$
3. $\left\{1\right\}$
4. $\{0,1\}$

Q. The minimum value of $|x|+|x+\frac{1}{2}|+|x-3|+|x-\frac{5}{2}|$ is

1. $6$
2. $2$
3. $0$
4. $4$

Q. Let $f\left(x\right)=sgn(\cos 2x-2\sin x+3)$, where sgn (.) is the signum function, then f(x)

1. so continuous over its domain
2. has a missing point discontinuity
3. has isolated point discontinuity
4. irremovable discontinuity

Q.

Express the algebraic expression $\frac{4{\mathrm{x}}^2}{9{\mathrm{y}}^2}÷\frac{2\mathrm{x}}{3\mathrm{y}}$in its lowest form.

Q. Let $f\left(x\right)=\{\begin{array}{c}\text{sgn}(x^2-1);|x|>1\\ x;|x|\le 1\end{array}$, then which of the following option is correct

1. $f\left(x\right)$ is continuous and differentiable at $x=1$
2. $f\left(x\right)$ is continuous but not differentiable at $x=1$
3. $f\left(x\right)$ is continuous and differentiable at $x=-1$
4. $f\left(x\right)$ is continuous but not differentiable at $x=-1$