Range of Quadratic Expression

Q. If $x$ is real, then maximum value of $\frac{3x^2+9x+17}{3x^2+9x+7}$ is

1. $82$
2. $41$
3. $\frac{41}{4}$
4. $\frac{17}{3}$

Q. The largest natural number ${}^{\prime }{a}^{\prime }$ for which the maximum value of $f\left(x\right)=a-1+2x-x^2$ is always smaller than the minimum value of $g\left(x\right)=x^2-2ax+10-2a$ is

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

Q. If the equation $x^2+9y^2-4x+3=0$ is satisfied for all real values of $x$ and $y,$ then

1. $x\in [1,3]$
2. $y\in [-\frac{1}{3},\frac{1}{3}]$
3. $x\in (\frac{1}{3},3]$
4. $y\in (-3,-\frac{1}{3})$

Q. For all real number $x$, $2^x+3^x-4^x+6^x-9^x$ is always less than

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

Q. The range of $f\left(x\right)=\frac{1}{-x^2+4x+5}$, $x\in R-\{-1,5\}$ is

1. $(-\infty ,0]\cup [\frac{1}{9},\infty )$
2. $(-\infty ,0)\cup [\frac{1}{9},\infty )$
3. $(-\infty ,0]\cup (\frac{1}{9},\infty )$
4. $(-\infty ,0)\cup (\frac{1}{9},\infty )$

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

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

Q. The least integral value of $k$ for which $(k-1)x^2+8x+k+5$ is always positive $\forall x\in R,$ is

Q. If least value of $f\left(x\right)=x^2+bx+c$ be $-\frac{1}{4}$ and maximum value of $g\left(x\right)=-x^2+bx+2$ occurs at $\frac{3}{2}$, then $c$ is equal to

Q. Range of the rational expression $y=\frac{x+3}{2x^2+3x+9},x\in R$ is

1. $[-\frac{1}{21},\frac{1}{3}]$
2. $[-\frac{1}{3},\frac{1}{3}]$
3. $[-\frac{1}{21},\frac{1}{21}]$
4. $[\frac{1}{21},\frac{1}{3}]$

Q.

The equation whose roots are the squares of the roots of the equation $2x^2+3x+1=0$, is:

1. $4x^2+5x+1=0$

2. $4x^2-x+1=0$

3. $4x^2-5x-1=0$

4. $4x^2-5x+1=0$

Q. The values of $m$ for which $y=\frac{m{x}^2+3x-4}{-4x^2+3x+m}$ has range $R$ is

1. $(1,7)$
2. $[1,7]$
3. $(2,5)$
4. $[2,5]$

Q. The maximum value of ${\left(\frac{1}{2}\right)}^{x^2-3x+2}$ is
$(x\in R)$

1. $\frac{1}{4}$
2. not defined
3. $\sqrt[4]{42}$
4. $0$

Q. The inequality $\frac{|x{|}^2-|x|-2}{2|x|-|x{|}^2-2}>0$ holds if and only if

1. $-1<x<-\frac{-2}{3}\text{or}\frac{2}{3}<x<1$
2. $-2<x<2$
3. $\frac{2}{3}<x<1$
4. none of these

Q.

The maximum value of the expression $y=2(a-x)(x+\sqrt{x^2+b^2})$ is

1. $2a^2+b^2$

2. $|2a^2-b^2|$

3. $a^2+b^2$

4. $a^2+2b^2$

Q. The maximum value of $f\left(x\right)=(x+3)(4-x)+3$ is

1. $\frac{61}{2}$
2. $\frac{61}{4}$
3. $3$
4. $15$

Q. Number of positive integral value of '$p$' for which the equation $p{.2}^{e^x+e^{-x}}-{8.2}^{\frac{e^x+e^{-x}}{2}}+p=0$ has atleast one solution is

Q.

Complete set of values of a such that $\frac{x^2-x}{1-ax}$ attains all real values is

1. [1, 4]

2. [0, 4]

3. $[1,\infty )$

4. $[4,\infty )$

Q.

If '$a$' and '$b$' are the non-zero distinct zeros of $x^2+ax+b$, then the least value of quadratic polynomial $x^2+ax+b$ is

1. $\frac{2}{3}$
2. $\frac{9}{4}$
3. $-\frac{9}{4}$
4. $1$

Q. Number of real solutions of the equation $|||x^2-1|-1|+3|=1$ is

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

Q. The range of $f\left(x\right)=-x^2+7x+60$ in $x\in [-3,2]$ is

1. $[30,70]$
2. $[30,\frac{289}{4}]$
3. $[70,\frac{289}{4}]$
4. $[30,\frac{289}{4})$

Q. If the expression $(n-2)x^2+8x+(n+4)$ is negative $\forall x\in R$, then $n$ lies in

1. $(-\infty ,-6)$
2. $(-\infty ,2)$
3. $(-\infty ,-6)\cup (4,\infty )$
4. $(4,\infty )$

Q. For every possible $x\in R$, If $\frac{x^2+2x+a}{x^2+4x+3a}$ can take all real values, then

1. $a\in (0,1)$
2. $a\in [-1,1]$
3. $a\in [0,1]$
4. $a\in (-1,1)$

Q. Range of the rational expression $y=\frac{x+3}{2x^2+3x+9},x\in R$ is

1. $[\frac{1}{21},\frac{1}{3}]$
2. $[-\frac{1}{21},\frac{1}{3}]$
3. $[-\frac{1}{3},\frac{1}{3}]$
4. $[-\frac{1}{21},\frac{1}{21}]$

Q. If the minimum value of $f\left(x\right)=a{x}^2+2x+5,a>0$ is equal to the maximum value of $g\left(x\right)=3+2x-x^2$, then the value of ${}^{\prime }{a}^{\prime }$ is

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

Q. The number of intergral values of $a$ for which $y=\frac{x^2-ax+11}{x^2-5x+4}$ can take all real values is

Q. Consider the function $f\left(x\right)=x^2-4x+17.$ If $M$ and $m$ are the maximum and minimum values of $f$ in $[0,3]$ respectively, then the value of $2M-m$ is

Q. The minimum integral value of $a$ for which $\frac{x^2-x}{1-ax}$ attains all real values, is

Q. For $x\in R-\left\{b\right\},$ if $y=\frac{(x-a)(x-c)}{x-b}$ will assume all real values, then

1. $a<b<c$
2. $b<c<a$
3. $a=b=c$
4. $c<a<b$

Q.

The set of all real numbers x for which $x^2-|x+2|+x>0$ is

1. $(-\infty ,-\sqrt{2})\cup (\sqrt{2},\infty )$

2. $(\sqrt{2},\infty )$

3. $(-\infty ,-1)\cup (1,\infty )$

4. $(-\infty ,-2)\cup (2,\infty )$

Q. If $a{x}^2+bx+6=0$ does not have distinct real roots, then the least value of $3a+b=$