Range

Q.

The equation$\sqrt{3}\sin x+\cos x=4$ has

1. infinitely many solutions

2. no solution

3. two solution

4. only one solution

Q. The range of $f\left(x\right)=x^3+x$ is

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

Q. The range of the function $f\left(x\right)={\log }_2(3-2x-x^2)$ is

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

Q. If $f:R\to R$ is a function defined as $f\left(x\right)=x^2-6x+16,$ then the range of $f$ is

1. $[9,\infty )$
2. $[7,\infty )$
3. $[16,\infty )$
4. $[11,\infty )$

Q. The range of $f\left(x\right)={\log }_e(3x^2-4x+5)$ is

1. $(-\infty ,\infty )$
2. $[{\log }_{e}5,\infty )$
3. $[{\log }_e\frac{11}{3},{\text{log}}_{e}5]$
4. $[{\log }_e\frac{11}{3},\infty )$

Q. The range of $f\left(x\right)=\frac{e^x}{1+\left[x\right]},x\ge 0$ is

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

Q.

The range of the function $y=\frac{x-1}{(x^2-3x+3)}$ is [a, b] where a, b are respectively

1. $\frac{1}{3},2$

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

3. $\frac{-1}{3},2$

4. $\frac{1}{3},1$

Q. The number of integral elements in the range of the function $f\left(x\right)=\left[\right\{2x+3\left\}\right]$ is
(where $[.]$ represents the greatest integer function and $\left\{x\right\}$ is the fractional part of $x)$

Q. Let $f\left(x\right)=x^2-12x-10.$ Then

1. domain of $f$ is $R$
2. domain of $f$ is $R-\left\{6\right\}$
3. range of $f$ is $R$
4. range of $f$ is $[-46,\infty )$

Q. If $f\left(x\right)=\frac{1}{|x-1|-x^2}$ and $D_f,R_f$ denote the domain and range of the function respectively, then

1. $D_f=R-\left\{\frac{-1\pm \sqrt{5}}{2}\right\}$
2. $D_f=R-\left\{\frac{-1\pm \sqrt{3}}{2}\right\}$
3. $R_f=(-\infty ,0)\cup [\frac{4}{5},\infty )$
4. $R_f=(-\infty ,-\frac{4}{3})\cup [\frac{4}{5},\infty )$

Q. The range of the function $f\left(x\right)=x^2-6x+7$ is

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

Q. Let $f\left(x\right)=\frac{x}{1+x^2}$. Then range of $f$ is

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

Q. The range of the function $f\left(x\right)=\sqrt{x^2-3x+5}$ is

1. $(-\infty ,-\frac{\sqrt{11}}{2}]\cup [\frac{\sqrt{11}}{2},\infty )$
2. $[-\frac{\sqrt{11}}{2},\frac{\sqrt{11}}{2})$
3. $[\frac{\sqrt{11}}{2},\infty )$
4. $[\sqrt{5},\infty )$

Q. If $x$ is real, then the value of the expression $\frac{x^2+14x+9}{x^2+2x+3}$ lies between

1. $4$ and $5$
2. $-4$ and $5$
3. $-5$ and $4$
4. None of these

Q. The range of the function $f\left(x\right)=\frac{x^2+x+2}{x^2+x+1},x\in R$ is

1. $(1,\infty )$
2. $(1,\frac{11}{7})$
3. $(1,\frac{7}{3}]$
4. $(1,\frac{7}{5})$

Q. The range of $f\left(x\right)=\frac{x+2}{x-3}$ is

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

Q. Let $A=\{x:x\text{is a prime factor of}30\}$ and $B=\{y:y\in N,-3\le y+3<8\}$. If $R$ is a relation from $A$ to $B$, then $R^{-1}$ can be

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

Q. If x is real , then value of the expression $\frac{x^2+14x+9}{x^2+2x+3}$ lies between

1. 5 and 4
2. 5 and –4
3. – 5 and 4
4. None of these

Q. The range of the function $f\left(x\right)=\frac{[x-3]}{\left[x\right]-3}$ in its domain is
($[.]$ represents the greatest integer function)

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

Q. If $A=\{x:|x|\le 5;x\in Z-\{0\left\}\right\}$, $B=\{x:x\le 100;x\in W\}$ and $f:A\to B$ is a function defined by $f\left(x\right)=x^2+1$, then the number of elements in the range of $f$ that lie in $[5,26)$ is

1. $5$
2. $3$
3. $4$
4. $2$

Q. Let $f\left(x\right)=\sqrt{4-\sqrt{2-x}}$ and $g\left(x\right)=(x-a)(x-a+3).$ If $g\left(f\right(x\left)\right)<0\forall x\in D_f,$ then the complete set of values of $a$ is
$[D_f$ denotes the domain of the function $f]$

1. $(2,5)$
2. $(2,3)$
3. $(0,3)$
4. $(0,5)$

Q. If a function $f:[-2,\infty )\to R$ is such that $f\left(x\right)=x^2+4x-|x^2-4|,$ then the value(s) $f\left(x\right)$ can have is (are)

1. $-6$
2. $-8$
3. $4$
4. $0$

Q. Let $f\left(x\right)=f_1\left(x\right)-2f_2\left(x\right)$,

where, $f_1\left(x\right)=\{\begin{array}{cc}min\{x^2,|x|\},& |x|\le 1\\ max\{x^2,|x|\},& |x|>1\end{array}$

and, $f_2\left(x\right)=\{\begin{array}{cc}min\{x^2,|x|\},& |x|>1\\ max\{x^2,|x|\},& |x|\le 1\end{array}$

and, $g\left(x\right)=\{\begin{array}{c}min\left\{f\right(t):-3\le t\le x,-3\le x<0\}\\ max\left\{f\right(t):0\le t\le x,0\le x\le 3\}\end{array}$

For $-3\le x\le -1$, the range of $g\left(x\right)$ is

1. $[-1,3]$
2. $[-1,-15]$
3. $[-1,9]$
4. None of these

Q. The domain and range of the function ${\text{cosec}}^{-1}\sqrt{{\log }_{\left(\frac{3-4\sec x}{1-2\sec x}\right)}2}$ are respectively

1. $R;(-\frac{\pi }{2},\frac{\pi }{2})$
2. $R^{+};(0,\frac{\pi }{2})$
3. $(2n\pi -\frac{\pi }{2},2n\pi +\frac{\pi }{2})-\left\{2n\pi \right\},n\in Z;(0,\frac{\pi }{2})$
4. $(2n\pi -\frac{\pi }{2},2n\pi +\frac{\pi }{2})-\left\{2n\pi \right\},n\in Z;(-\frac{\pi }{2},\frac{\pi }{2})-\left\{0\right\}$

Q. The range of the function $f\left(x\right)=|x^2+2x-15|$ is

1. $R^{+}$
2. $R-\left\{0\right\}$
3. $[0,\infty )$
4. $R$

Q. The range of the function $f\left(x\right)=\frac{1}{3-\sin 3x},x\in R$ is

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

Q. The range of $x$ satisfying $3^x+2^{2x}\ge 5^x$ is

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

Q. The range of the function $f\left(x\right)=\sqrt{x^2-3x+5}$ is

1. $(-\infty ,-\frac{\sqrt{11}}{2}]\cup [\frac{\sqrt{11}}{2},\infty )$
2. $[-\frac{\sqrt{11}}{2},\frac{\sqrt{11}}{2})$
3. $[\frac{\sqrt{11}}{2},\infty )$
4. $[\sqrt{5},\infty )$

Q.

If $f\left(x\right)=x^2+2bx+2c^2$ and $g\left(x\right)=-x^2-2cx+b^2$ such that in $f\left(x\right)>maxg\left(x\right)$, then the relation between $b$ and $c$, is

1. No real value of b &c

2. $0<c<b\sqrt{2}$

3. $|c|<|b|\sqrt{2}$

4. $|c|>|b|\sqrt{2}$

Q. Let $A$ be the set of first ten natural numbers and let $R$ be a relation on $A\times A$ defined by $R=\left\{\right(x,y):x+2y=10;x,y\in A\}.$ Then range of $R^{-1}$ is

1. $\{2,4,6,8\}$
2. $\{1,2,3,4\}$
3. $\{1,2,4,6\}$
4. $\{1,4,6,8\}$