Inverse of a Function

Q. Length of latus rectum of the parabola whose parametric equations are $X=t^2+t+1,y=t^2-t+1$ where $tϵR$, is equal to

1. 4
2. 1
3. 3
4. 

Q. If $f\left(x\right)=sinx+cosx,g\left(x\right)=x^2-1theng\left(f\right(x\left)\right)$ in invertible in the Domain

1. 
2. 
3. 
4. 

Q.

If f(x) = $x^n$ then the graph of g(x) =$x^{\frac{1}{n}}$ is the mirror image of f(x) about y = x.

1. False

2. True

Q.

Values of 'm' for which both the roots of equation $x^2$ - 2mx + $m^2$ - 1=0 are less than 4 are

1. (-, -3]

2. (-, 3)

3. (3, )

4. [-3, )

Q. If $R$ is a relation from $\{11,12,13\}$ to $\{8,10,12\}$ defined by $y=x-3.$ Then

1. $(11,8)\in R^{-1}$
2. $(10,13)\in R$
3. $(10,13)\in R^{-1}$
4. $(11,8)\in R$

Q. Inverse exists for a function which is

1. Injective
2. Surjective
3. Bijective
4. Many-one

Q. Let f(x) be a function given by $f:[0,2]\to [\frac{1}{7},\frac{2}{7}]\cup [1,4)$ and satisfies 3x -f(x) +1 = 0 for $0\le x\le 1$ and x - 7 f(x) = 0 for $1\le x\le 2$, there the sum of solutions of the equation $f\left(x\right)=f^{-1}\left(x\right)$ is

Q. Which of the following is a singleton set?

1. $\{x:x\text{is a point where a tangent touches a circle}\}$
2. $\{y:y\text{is a prime number},20\le y<23\}$
3. $\{a:a\in N,a^2+5a+6=0\}$
4. $\{b:b\in N,3b^2-5b-2=0\}$
5. $\text{Set of all people born in May 2000}$

Q.

${}^n{C}_r{÷}^n{C}_{r-1}=$

[MP PET 1984]

1. 

2. 

3. 

4. 

Q. Let $f:(4,6)\to (6,8)$ be a function defined by $f\left(x\right)=x+\left[\frac{x}{2}\right]$ (where [.] denotes the greatest integer function),
then $f^{-1}\left(x\right)$ is equal to

1. $X-\left[\frac{x}{2}\right]$
2. $-x-2$
3. $x-2$
4. $\frac{1}{x+\left[\frac{\pi }{2}\right]}$

Q. Let $T_r$ be the $r^{th}$ term of an A.P for r$=$ 1, 2, 3, ......... If for some positive integers m, n we have $T_m=\frac{1}{n}$ and $T_n=\frac{1}{m}$ , then $T_{mn}$ is equal to

1. $\frac{1}{mn}$
2. $\frac{1}{m}+\frac{1}{n}$
3. 1
4. 0

Q. If $f\left(x\right)=sinx+cosx,g\left(x\right)=x^2-1$ then $g\left(f\right(x\left)\right)$ in invertible in the Domain

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

Q.

If the function 'a' is$x^3$. which of the following could be the function b?

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

2. x+$x^3$

3. $x^{\frac{1}{5}}$

4. $x^3$+$x^{\frac{1}{3}}$

Q. Let $A=\{1,2,3\},B=\{1,3,5\}.$ A relation $R$ is defined from $A$ to $B$ as $R=\left\{\right(1,3),(1,5),(2,1\left)\right\}.$ Then $R^{-1}=$

1. $\left\{\right(1,2),(3,1),(1,3),(1,5\left)\right\}$
2. $\left\{\right(1,2),(3,1),(2,1\left)\right\}$
3. $\left\{\right(1,2),(5,1),(3,1\left)\right\}$
4. $\left\{\right(1,3),(1,5),(2,1\left)\right\}.$

Q.

Let A = {1, 2, 3}, B = {1, 3, 5}. A relation R:A $\to$ B is

defined by R = {(1, 3), (1, 5), (2, 1)}. Then $R^{-1}$ is defined by

1. {(1, 2), (3, 1), (1, 3), (1, 5)}

2. {(1, 2), (3, 1), (2, 1)}

3. {(1, 2), (5, 1), (3, 1)}

4. None of these