Definition of Functions

Q.

Let f be a function satisfying $2f\left(x\right)-3f\left(\frac{1}{x}\right)=x^2$ for any $x\ne 0$, then the value of f(2) is

1. $-\frac{7}{8}$
2. -2
3. 4
4. $\frac{-7}{4}$

Q. A real-valued function $f\left(x\right)$ satisfies the functional equation $f(x-y)=f\left(x\right)f\left(y\right)-f(a-x)f(a+y)\forall x,y\in R$, where $a$ is a given constant and $f\left(0\right)=1$. Then, $f(2a-x)$ is equal to

1. $f\left(x\right)$
2. $-f\left(x\right)$
3. $f\left(x\right)$
4. $f\left(a\right)+f(a-x)$

Q. If 'x' satisfies equation [x+0.19] + [x+0.20] + -----+ [x +0.91] =542 then [100x] is (where [.] repersents greatest integer function)

Q. If $f\left(x\right)$ satisfies the relation $f(x+y)=f\left(x\right)+f\left(y\right)$ for all $x,y\in R$ and $f\left(1\right)=5$, then

1. $f\left(x\right)$ is an odd function
2. $f\left(x\right)$ is an even function
3. $\sum _{r=1}^{m}f\left(r\right)=5{}^{m+1}{C}_2$
4. $\sum _{r=1}^{m}f\left(r\right)=\frac{5m(m+2)}{3}$

Q. The number of values of $x$ satisfying the equation $\frac{|x-5||x-2|}{|x+9|}=4$ is

Q.

Find the point where the graph of the function Sgn (lnx) breaks (or becomes discontinuous)
(Sgn is the Signum function)

1. -1

2. 0

3. 1

4. e

Q. Let $A$ and $B$ be two smallest sets such that $A\cup \left\{1\right\}=\{1,2,3,4\}$ and $B\cup \left\{5\right\}=\{4,5,6,7,8\}.$ If $P=A-B$ and $Q=B-A,$ then the number of relations from $P$ to $Q$ is

1. $2^{12}$
2. $2^{10}$
3. $2^9$
4. $2^6$

Q. Let the function be $f:R\to R$ such that$f\left(x\right)=3x^2+8x+5$, then the correct OPTIONS is/are

1. $f\left(x\right)$ is one one function
2. Range of $f\left(x\right)$ is $[-\frac{1}{3},\infty )$
3. $f\left(x\right)$ is many one function
4. Range of $f\left(x\right)$ is $R$

Q.

Which of the following relations between two sets are functions?

1. 

2. 

3. 

4. 

Q. If $f\left(x\right)=ax+b$, where $a$ and $b$ are integers, $f(–1)=–5$ and $f\left(3\right)=3$, then $a$ and $b$ are equal to , respectively.

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

Q. Let f(x) = sin^–1(x – 4), then domain of f(x) is
माना f(x) = sin^–1(x – 4), तब f(x) का प्रांत है

1. [–3, 5]
2. [3, 5]
3. [–5, 3]
4. [–5, –3]

Q. If $f\left(x\right)=4x-x^2,x\in R$, then $f(a+1)-f(a-1)$ is equal to

1. $2(4-a)$
2. $4(2-a)$
3. $4(2+a)$
4. $2(4+a)$

Q. Which of the following correspondences can be called a function?

1. $f:\{-1,0,1\}\to \{0,1,2,3\}$ defined by $f\left(x\right)=x^3$
2. $f:\{0,1,4\}\to \{-2,-1,0,1,2\}$ defined by $f\left(x\right)=\pm \sqrt{x}$
3. $f:\{0,1,4\}\to \{-2,-1,0,1,2\}$ defined by $f\left(x\right)=\sqrt{x}$
4. $f:\{0,1,4\}\to \{-2,-1,0,1,2\}$ defined by $f\left(x\right)=-\sqrt{x}$

Q.

If the relation f is defined by $f\left(x\right)=\{\begin{array}{cc}{x}^2,& 0\le x\le 3\\ 3x,& 3\le x\le 10\end{array}$
and the relation g is defined by $g\left(x\right)=\{\begin{array}{cc}{x}^2,& 0\le x\le 2\\ 3x,& 2\le x\le 10\end{array}$
then,

1. 'f' is a function and 'g' is not a function

2. 'g' is a function and 'f' is not a function

3. Both 'f' and 'g' are functions

4. neither 'f' nor 'g' are functions

Q. Let a function $f$ be defined from $R\to R$ by $f\left(x\right)=\{\begin{array}{cc}x+m,& x\le 1\\ 2mx-1,& x>1\end{array}$
If the function $f$ is surjective, then $m\in$

1. $[-2,2]$
2. $R-\{-2,1,2\}$
3. $R-\left\{1\right\}$
4. $(0,2]$

Q. If ${}^n{\text{C}}_4,{}^n{\text{C}}_5$ and ${}^n{\text{C}}_6$ are in A.P., then $n$ can be :

1. $11$
2. $14$
3. $12$
4. $9$

Q. If $f\left(x\right)=x^2$ and $g\left(x\right)=2^x$, then the solution set of the equation $f\left(2^x\right)=g\left(x^2\right)$ is

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

Q. What is the range of $f\left(x\right)=\tan \left(\sqrt{\frac{{\pi }^2}{4}-x^2}\right)$ ?

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

Q. Let A and B be any two sets such that n(B) = p, n(A) = q then the total number of functions f : A → B is equal to

1. $p^q$
2. $q^p$
3. $pCq$
4. $pq$

Q. Value of $si{n}^{-1}\left(sin\frac{3\pi }{4}\right)$

1. $\frac{\pi }{4}$
2. $\frac{-\pi }{4}$
3. $\frac{3\pi }{4}$
4. None

Q. Let $f:X\to Y$ be a function such that $f\left(x\right)=\sqrt{x-2}+\sqrt{4-x}$. If $f\left(x\right)$ is both injective as well as surjective, then set $X$ and $Y$ is

1. $[2,4]$ and $[\sqrt{2},2]$
2. $[3,4]$ and $[\sqrt{2},2]$
3. $[2,4]$ and $[1,2]$
4. $[2,3]$ and $[1,2]$

Q. The domain of the function $f\left(x\right)={\sin }^{-1}x+\cos x$ is

1. $[-1,1]$
2. $(-\infty ,-1)\cup (1,\infty )$
3. $[-1,\pi +1]$
4. $(-\infty ,\infty )$

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

1. an injective function
2. an onto function
3. a bijecive function
4. neither a one one nor an onto function

Q. Let $A=\{a,b,c,d,e\}$ and $B=\{1,2,3,4,5\}$. The number of relations which are not functions from $A$ to $B$ is

1. $2^{25}-5^5$
2. $2^{20}-5^5$
3. $5^5-25$
4. $2^{25}-5\times 3$

Q.

$sin\left(2si{n}^{-1}\sqrt{\frac{63}{65}}\right)=$

1. $\frac{2\sqrt{126}}{65}$
   

2. $\frac{4\sqrt{63}}{65}$
   

3. $\frac{8\sqrt{63}}{65}$
   

4. $\frac{\sqrt{65}}{63}$
   

Q. Let $(1+x)(1+x+x^2)(1+x+x^2+x^3)......(1+x+x^2+....+x^{30})=a_0+a_{1}x+a_2{x}^2+........+a_{465}{x}^{465}$

then sum of $a_0+a_2+a_4+........+a_{464}$ is

1. $\left(31\right)!$
2. $\frac{\left(31\right)!}{2}$
3. $\left(30\right)!$
4. $\frac{\left(60\right)!}{2}$

Q. Which of the following statements is not a tautology?

1. $(p\wedge q)\Rightarrow p$
2. $p\wedge q\Rightarrow (\sim p)\vee q$
3. $(p\vee q)\Rightarrow (p\vee (\sim q\left)\right)$
4. $p\Rightarrow (p\vee q)$

Q. Let $f\left(x\right)=x+2|x+1|+2|x-1|$. If $f\left(x\right)=k$ has exactly one real solution, then the value of $k$ is

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

Q. Each entry of $\text{List I}$ is to be matched with one entry of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& 100(\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{99\cdot 100})\text{equals}& \text{(P)}& 7\\ \text{(B)}& \text{If}x\text{is the arithmetic mean between two real numbers}a\text{and}b\text{,}& \text{(Q)}& 9\\ & y=a^{2/3}\cdot b^{1/3}\text{and}z=a^{1/3}\cdot b^{2/3},\text{then}\frac{y^3+z^3}{xyz}\text{equals}& & \\ \text{(C)}& \text{If}198\text{arithmetic means are inserted between}\frac{1}{4}\text{and}\frac{3}{4}\text{, then}& \text{(R)}& 99\\ & \text{the sum of these arithmetic means is}& & \\ \text{(D)}& \text{If}n\text{is a positive integer such that}n\text{,}\frac{n(n-1)}{2}\text{and}& \text{(S)}& 100\\ & \frac{n(n-1)(n-2)}{6}\text{are in A.P., then the value of}n\text{is}& & \\ & & \text{(T)}& 2\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(A)}\to \text{(Q)},\text{(B)}\to \text{(P)}$
2. $\text{(A)}\to \text{(R)},\text{(B)}\to \text{(T)}$
3. $\text{(A)}\to \text{(S)},\text{(B)}\to \text{(P)}$
4. $\text{(A)}\to \text{(T)},\text{(B)}\to \text{(R)}$

Q.

Let f(x) be a function defined on [0, 1] such that $f\left(x\right)=\{\begin{array}{ccc}x,& ifxϵQ& \\ 1-x,& ifx\text{/}ϵQ& \end{array}$
Then, for all $xϵ[0,1],f\left(f\right(x\left)\right)=$

1. constant

2. $1+x$

3. $x$

4. none of these