Definition of Function

Q.

Let $A=\left\{a,b,c\right\}$ and $B=\left\{1,2,3,4\right\}$. Then the number of elements in the set $C=\left\{f:A\to B|2\in f\left(A\right)\text{and}f\text{is not one-one}\right\}$ is

Q.

What is the maximum possible length of an identifier in python?

Q.

What Is the formula of ${\left(a+b-c\right)}^2$?

Q. If the function $f:R-\{1,-1\}\to A$ defined by $f\left(x\right)=\frac{x^2}{1-x^2}$, is surjective, then $A$ is

1. $R-[-1,0)$
2. $R-\{-1\}$
3. $R-(-1,0)$
4. $[0,\infty )$

Q.

Write the following cubes in expanded form: ${\left(2x+1\right)}^3$

Q. A polynomial $p\left(x\right)$ when divided by $(x-3),$ leaves a remainder of $5.$ But when $p\left(x\right)$ is divided by $(x-5),$ the remainder is $3.$ Let $r\left(x\right)$ be the remainder when $p\left(x\right)$ is divided by $(x-3)(x-5).$ Then the value of $r\left(1\right)$ is

1. $0$
2. $3$
3. $5$
4. $7$

Q.

$f:R\to R$ defined by $f\left(x\right)=1+x^2$. Is it one one , onto or bijective?

Q.

What is the value of $\log \left(16\right)$?

Q.

The equation ${\left(x-5\right)}^2+\left(x-5\right)\left(y-6\right)-2{\left(y-6\right)}^2=0$ represents

1. A circle

2. Two straight lines passing through the origin

3. Two straight lines passing through the point $(5,6)$

4. None of these

Q. Which among the following relations between $x,y$ doesn't represent a function

1. $(x-1{)}^2+(y-3{)}^2=5^2,x\in [-4,6],y\in R$
2. $y^2=4x,x\in [0,\infty ),y\in R$
3. $x^2=4y,x,y\in R$
4. $y=2x+5^2,x,y\in R$

Q.

The product of first $n$ odd natural numbers equals to

1. ${}^{2n}C_n\times {}^n{P}_n$

2. ${\left(\frac{1}{2}\right)}^n{}^{2n}C_n\times {}^n{P}_n$

3. ${\left(\frac{1}{4}\right)}^n{}^{2n}C_n\times {}^{2n}{P}_n$

4. none of these

Q. The sum of absolute maximum and absolute minimum values of the function $f\left(x\right)=|2x^2+3x-2|+\sin x\cos x$ in the interval $[0,1]$ is

1. $3+\frac{\sin \left(1\right){\cos }^2\left(\frac{1}{2}\right)}{2}$
2. $3+\frac{1}{2}(1+2\cos (1\left)\right)\sin \left(1\right)$
3. $2+\sin \left(\frac{1}{2}\right)\cos \left(\frac{1}{2}\right)$
4. $5+\frac{1}{2}\left(\sin \right(1)+\sin (2\left)\right)$

Q.

Expand the following, using suitable identities: ${(2x-y+z)}^2$

Q.

If $f\left(x\right)=\frac{4^x}{4^x+2}$, then $f\left(\frac{1}{97}\right)+f\left(\frac{2}{97}\right)+.......+f\left(\frac{96}{97}\right)$ is equal to:

1. $1$

2. $48$

3. $-48$

4. $-1$

Q.

Find the value of $\sin 35°\cos 55°+\cos 35°\sin 55°$.

Q.

Given $f\left(x\right)=\frac{1}{[1-x]}$, $g\left(x\right)=f(f(x\left)\right)$ and $h\left(x\right)=f(f(f\left(x\right)\left)\right)$. Then the value of $f\left(x\right).g\left(x\right).h\left(x\right)$ is

1. $-1$

2. $1$

3. $2$

4. $3$

Q.

What is $\sin 2x$ formula in terms of $\tan$.

Q. Which of the following is not a function?

1. 
2. 
3. 
4. 

Q. Find the coordinates of the focus, axis, the equation of the directrix and latus rectum of the parabola $y^2=8x.$

Q. The number of different matrices can be formed using all the letters of word $TOMATO,$ is

1. $180$
2. $360$
3. $720$
4. $900$

Q.

What is $5i$ equal to?

Q. Let $A=\{1,3,5,7\},B=\{2,4,6,8\}$ and $f:A\to B,$ then number of functions $f$ that can be defined from $A$ to $B$ is

1. If $i+f\left(i\right)<10,\forall i=\{1,3,5,7\},$ then number of functions is $12$
2. If $i+f\left(i\right)<10,\forall i=\{1,3,5,7\},$ then number of functions is $24$
3. If $f\left(i\right)-i>2,\forall i=\{1,3,5,7\},$ then number of functions is $0$
4. If $f\left(i\right)-i>2,\forall i=\{1,3,5,7\},$ then number of functions is $12$

Q. If $A=\{1,2,3,4\}$ then which among the following relations are not functions from $A$ to itself

1. $f_1=\left\{\right(x,y)|y=x+1,\forall x,y\in A\}$
2. $f_2=\left\{\right(x,y)|x+y>4,\forall x,y\in A\}$
3. $f_3=\left\{\right(x,y)|y<x,\forall x,y\in A\}$
4. $f_4=\left\{\right(x,y)|x+y=5,\forall x,y\in A\}$
   

Q. Let $f\left(x\right)=\frac{e^x}{1+x^2}$ and $g\left(x\right)=f^{\prime }\left(x\right)$ then

1. $g\left(x\right)$ has four points of local extremum
2. $g\left(x\right)$ has two points of local extremum
3. $g\left(x\right)$ has a point of local minimum at $x=1$
4. $g\left(x\right)$ has a point of local maximum at some $x\in (-1,0)$

Q. If $g=\left\{\right(1,1),(2,3),(3,5),(4,7\left)\right\}$ is a function defined as $g\left(x\right)=\alpha x+\beta$, then $\alpha -\beta =$

Q.

Which one is the correct statement about the function $f\left(x\right)=\sin 2x$

1. $f\left(x\right)$ is increasing in $\left(0,\frac{\mathrm{\pi }}{2}\right)$ and decreasing in $\left(\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)$

2. $f\left(x\right)$ is decreasing in $\left(0,\frac{\mathrm{\pi }}{2}\right)$ and increasing in $\left(\frac{\mathrm{\pi }}{2},\mathrm{\pi }\right)$

3. $f\left(x\right)$ is increasing in $\left(0,\frac{\mathrm{\pi }}{4}\right)$ and decreasing in $\left(\frac{\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{2}\right)$

4. The statements (a), (b) and (c) are all correct

Q. If all the letters of the word $RANK$ are rearranged to form $4$ letter words and arranged in ascending order as in a dictionary, then the rank of the word $RANK$ is

1. $22$
2. $19$
3. $20$
4. $21$

Q. Which of the following is/are a function?

1. 
2. 
3. 
4. 

Q. If $a_r=(\cos 2r\pi +i\sin 2r\pi {)}^{\frac{1}{9}}$ , then the value of $\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right|$ is

1. $1$
2. $-1$
3. $0$
4. $2$

Q.

What is $e$ in complex numbers?