Many One Function

Q.

Find the roots of $x+\frac{1}{x}=3;x\ne 0$.

Q.

Find the number of all onto functions from the set $\left\{1,2,3,....,n\right\}$ to itself.

Q. The smallest positive root of the equation $\tan x-x=0$ lies in

1. $(0,\pi /2)$
2. $(\pi ,\frac{3\pi }{2})$
3. $(\frac{3\pi }{2},2\pi )$
4. $(\pi /2,\pi )$

Q. Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as
$R_1=\left\{\right(x,y)\in N\times N:2x+y=10\}\text{and}{R}_2=\left\{\right(x,y)\in N\times N:x+2y=10\}.\text{Then}$

1. Both $R_1$ and $R_2$ are transitive relations.
2. Range of $R_2$ is $\{1,2,3,4\}$.
3. Range of $R_1$ is $\{2,4,8\}$.
4. Both $R_1$ and $R_2$ are symmetric relations.

Q. Which among the following pair(s) of function are identical.

1. $f\left(x\right)=\frac{\sec x}{\cos x}-\frac{\tan x}{\cot x},g\left(x\right)=\frac{\cos x}{\sec x}+\frac{\sin x}{cosecx}$
2. $f\left(x\right)=sgn(x^2-6x+10),g\left(x\right)=sgn({\cos }^{2}x+{\sin }^2(x+\frac{\pi }{3}))$ where sgn denotes signum function.
3. $f\left(x\right)=e^{ln(x^2-5x+6)},g\left(x\right)=x^2-5x+6$
4. $f\left(x\right)=\frac{\sin x}{\sec x}+\frac{\cos x}{cosecx},g\left(x\right)=\frac{2{\cos }^{2}x}{\cot x}$

Q.

If $R$ is a relation on the set $N$, defined by $\left\{\left(x,y\right):2x-y=10\right\}$, then $R$ is

1. Reflexive

2. Symmetric

3. Transitive

4. None of these

Q.

If $f\left(x\right)=x+2$, then $f\left(f\left(x\right)\right)$ at $x=4$ is

1. $8$

2. $1$

3. $4$

4. $5$

Q. Which one of the following function is not invertible?

1. $f:R\to R,f\left(x\right)=3x+1$
2. $f:R\to [0,\infty ),f\left(x\right)=x^2$
3. $f:R^{+}\to R^{+},f\left(x\right)=\frac{1}{x^3}$
4. $Noneoftheabove$

Q. For constant number $a,$ consider the function $f\left(x\right)=ax+\cos 2x+\sin x+\cos x$ on $R$ such that $f\left(u\right)<f\left(v\right)$ for $u<v.$ If the range of $a$ for any real number $x$ is $(\frac{m}{n},\infty )$ where $m,n\in N,$ then the minimum value of $(m+n)$ is

Q.

If $f\left(x\right)=\frac{x}{x-1}$, then $\frac{f\left(a\right)}{f\left(a+1\right)}$ is equal to

1. $f\left(a^2\right)$

2. $f\left(\frac{1}{a}\right)$

3. $f\left(-a\right)$

4. $f\left(\frac{-a}{a-1}\right)$

Q.

The equation $x^2-3\left|x\right|+2=0$ has

1. No real root

2. One real root

3. Two real root

4. Four real root

Q. Let $f:(0,1)\to R$ be defined by $f\left(x\right)=\frac{b-x}{1-bx}$ where $b$ is a constant such that $0<b<1$. Then

1. $f^{–1}$ is differentiable on $(0,1)$
2. $f\ne f^{-1}$ on $(0,1)$ and -----
3. $f=f^{-1}$ on $(0,1)$ and $f^{\prime }\left(b\right)=\frac{1}{f^{\prime }\left(0\right)}$
4. $f$ is not invertible on $(0,1)$

Q.

What are the Properties of determinants?

Q. Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.

Q.
From the given graph if $f\left(x\right)=\cos x,$
then $g\left(x\right)=$

1. $-\cos x-2$
2. $\cos x-2$
3. $\cos x+2$
4. $\cos x+2$

Q. Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as
$R_1=\left\{\right(x,y)\in N\times N:2x+y=10\}$ and
$R_2=\left\{\right(x,y)\in N\times N:x+2y=10\}.$ Then

1. Range of $R_1$ is $\{2,4,8\}$.
2. Range of $R_2$ is $\{1,2,3,4\}$.
3. Both $R_1$ and $R_2$ are transitive relations.
4. Both $R_1$ and $R_2$ are symmetric relations.

Q. If $f\left(x\right)=3\sin \sqrt{\frac{{\pi }^2}{16}-x^2},$ then its range is

1. $[-\frac{3}{\sqrt{2}},\frac{3}{\sqrt{2}}]$
2. $[0,\frac{3}{\sqrt{2}}]$
3. $[-\frac{3}{\sqrt{2}},0]$
4. $[-1,0]$

Q. $f:R\times R\to R$ such that f(x + iy) = $\sqrt{x^2+y^2}$. Then, f is

1. many – one and into
2. one-one and onto
3. many – one and onto
4. one – one and into

Q. A unit vector perpendicular to both the vectors $2\hat{i}+\hat{j}+3\hat{k}$ and $2\hat{i}-\hat{j}+\hat{k}$ is

1. $\frac{(\hat{i}+\hat{j}-\hat{k})}{\sqrt{3}}$
2. $\frac{(\hat{i}-\hat{j}+\hat{k})}{\sqrt{3}}$
3. $\hat{i}+\hat{j}-\hat{k}$
4. $\frac{4(\hat{i}+\hat{j}-\hat{k})}{\sqrt{3}}$

Q. If f(x) is continuous for all real values of x, then ${\sum }_{r=1}^n{\int }_0^{1}f(r-1+x)dx=$

1. ${\int }_0^{1}f\left(x\right)dx$
2. $n{\int }_0^{1}f\left(x\right)dx$
3. ${\int }_0^{n}f\left(x\right)dx$
4. $(n-1){\int }_0^{1}f\left(x\right)dx$

Q.

All the functions are either one- one or many-one.

1. True

2. False

Q.

Find the roots of the following quadratic equation:

$\sqrt{2}{x}^2-4\sqrt{3}x+4\sqrt{2}=0$

Q. Relation R in the set A of human beings in a town at a particular time given by $R=\left\{\right(x,y):xandyliveinthesamelocality\}$

Q. The graph of $y=\cot \left(\frac{x}{2}\right)$ is given. Find the period and range of the function.

1. Period: $2\pi$
   
   Range: $R$
2. Period: $\frac{\pi }{2}$
   
   Range: $R$
3. Period: $2\pi$
   
   Range: $R-\left\{0\right\}$
4. Period: $\frac{\pi }{2}$
   
   Range: $R-\left\{0\right\}$

Q. The line $3x-4y+8=0$ is rotated through an angle $\frac{\pi }{4}$ in the clockwise direction about the point $(0,2)$. The equation of the line in its new position is

1. $7y+x-14=0$
2. $7y+x-2=0$
3. $7y-x=0$
4. $7y-x-14=0$

Q.

If $f\left(x\right)=g\left(x\right){(x-a)}^2$, where $g\left(a\right)\ne 0$ and $g$ is continuous at $x=a$, then $f\left(x\right)$:

1. $f$ is increasing in the nbd. of $a$ if $g\left(a\right)>0$ and $f$ is decreasing in the nbd. of $a$ if $g\left(a\right)<0$.

2. $f$ is increasing in the nbd. of $a$ if $g\left(a\right)<0$.

3. $f$ is decreasing in the nbd. of a if $g\left(a\right)>0$.

4. $f$ is increasing in the nbd. of $a$ if $g\left(a\right)<0$ and $f$ is decreasing in the nbd. of $a$ if $g\left(a\right)>0$.

Q. Let $f:R\to R,f\left(x\right)=max.\{|{\tan }^{-1}x|,{\cot }^{-1}x\}$. Consider the following statements:
I. Function is continuous and derivable $\forall x\in R$
II. Range of function $\left[\frac{\pi }{4},\pi \right]$
III. $f\left(x\right)$ is many one-into
Identify the correct option:

1. All 3 statements are wrong.
2. Exactly one of the above statement is correct.
3. Exactly two of the above statement is correct.
4. All 3 statement are correct.

Q. Let $f:R\to R$ be a mapping, such that $f\left(x\right)=\frac{x^2}{1+x^2}$ Then, f is

1. Many – one into
2. One – one
3. Into
4. Onto

Q. Let $f\left(x\right)=$$\{\begin{array}{c}4x^2+2\left[x\right]x,-\frac{1}{2}\le x<0\\ a{x}^2-bx,0\le x<\frac{1}{2}\end{array}$ where $\left[x\right]$ denotes the greatest integer function. Then which of the following options is correct

1. $f\left(x\right)$ is continuous and differentiable in $(-\frac{1}{2},\frac{1}{2})$ for all real $a$, provided $b=2$
2. $f\left(x\right)$ is continuous and differentiable in $(-\frac{1}{2},\frac{1}{2})$ for all real $b$, provided $a=2$
3. $f\left(x\right)$ is continuous and differentiable in $(-\frac{1}{2},\frac{1}{2})$ iff $a=4,b=2$
4. For no real value of $a$ and $b,f\left(x\right)$ is differentiable in$(-\frac{1}{2},\frac{1}{2})$

Q.

Evaluate the expression.

$-9^2.3$