Range of a Quadratic Expression

Q. $x^2+y^2+xy=1$ for all $x,y\in R,$ the minimum value of $x^{3}y+x{y}^3+4$ is
(correct answer + 1, wrong answer - 0.25)

1. $3$
2. $2$
3. $1$
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. Sum of values of $x$, satisfying the equation $\sqrt{3x^2+6x+7}+\sqrt{5x^2+10x+14}=4-2x-x^2,$ is

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

Q. For the given quadratic expression $f\left(x\right)=x^2-2x+3.$ If $M,m$ are the maximum and minimum values of the $f\left(x\right)$ in $[-1,2]$. Then the value of $M+4m$ is

1. $-7$
2. $10$
3. $7$
4. $-10$

Q. $f:R\to R,$ $f\left(x\right)=\frac{3x^2+mx+n}{x^2+1}.$ If the range of $f\left(x\right)$ is $[-4,3],$ then

1. $m=0$
2. $m=2$
3. $n=4$
4. $n=-4$

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

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

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

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

Q. The range of the quadratic expression $y=-x^2+3x+3$ is

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

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

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

Q. Select the correct statement(s) for the equation $y=x^2.$

1. $y$ increases as we move $x$ from $0\text{to}\infty$
2. $y$ decreases as we move $x$ from $-\infty \text{to}0$
3. $y$ increases as we move $x$ from $-\infty \text{to}0$
4. $y=0$ at $x=0$

Q. Let $f\left(x\right)=x^2+2bx+2c^2$ and $g\left(x\right)=-x^2-2cx+b^2,b,c\ne 0$. If the minimum of $f\left(x\right)$ is greater than maximum value of $g\left(x\right)$, then the range of $\frac{b}{c}$ is

1. $(0,\frac{1}{\sqrt{2}})$
2. $(-\frac{1}{\sqrt{2}},0)$
3. $(\frac{1}{\sqrt{2}},\infty )$
4. $(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$

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

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

Q.

The values of ‘a’ for which $(a^2-1)x^2+2(a-1)x+2$ is positive for any $x$ are

1. $a\ge 1$

2. $a\le 1$

3. $a$ < $-3$ or $a$ >$1$

4. $a$ >$-3$

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

Q. Let $f:R\to R$ be defined by $f\left(x\right)=\frac{3x^2+mx+n}{x^2+1}.$ If the range of $f$ is $[-4,3),$ then the value of $m^2+n^2$ is

1. $18$
2. $84$
3. $16$
4. $25$

Q. If $5,5r,5r^2$ are the lengths of the sides of a triangle, then $r$ cannot be equal to :

1. $\frac{5}{4}$
2. $\frac{3}{2}$
3. $\frac{7}{4}$
4. $\frac{3}{4}$

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

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

Q. The number of integral values of $a$ for which $y=\frac{a{x}^2-7x+5}{5x^2-7x+a}$ can take all real values, where $x\in R$ (wherever the function is defined), is

Q. The range of the quadratic function $f\left(x\right)=x^2+x+1$ is given by

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

Q.

If $x$ is real, find the range of $y$ from the equation $x^2(y-1)-2x+(2y-1)=0$

1. $(-\infty ,0]$

2. $[0,\frac{7}{2}]$

3. $[0,2]$

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

Q. If the quadratic equation $x^2+[a^2-5a+b+4]x+b=0$ has roots $-5$ and $1$, then maximum value of $\left[a\right]$ (where $\left[a\right]$ denote the greatet integer function) is

Q. The range of \(f(x)=\dfrac1{-x^2+4x+5}\), \(x\in \mathbb R-\{-1, 5\}\) is

Q.

The values of a for which( $a^2-1$)$x^2$ + 2(a-1)x + 2 is positive for any x are

1. a < -3 or a > 1

2. a ≥ 1

3. a > -3

4. a ≤ 1

Q.

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

1. $|2a^2-b^2|$

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

3. $a^2+b^2$

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

Q.

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

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

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

3. no relation

4. $|c|<\frac{|b|}{\sqrt{2}}$

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

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

Q. If $x$ is real, the expression $\frac{x+2}{2x^2+3x+6}$ takes all value in the interval

1. $[-\frac{1}{13},\frac{1}{3}]$
2. $(\frac{1}{13},\frac{1}{3})$
3. None of these
4. $(-\frac{1}{3},\frac{1}{‘3})$

Q.

If $x^2+2ax+10-3a>0$ for each $x\in R$, then

1. $a>5$

2. $a<-5$

3. $-5<a<2$

4. $2<a<5$

Q. The range of the quadratic expression $y=3x^2+8x-3$ is

1. $\left(-\infty ,-\frac{25}{3}\right)$
2. $\left(-\frac{25}{3},\infty \right)$
3. $\left(-\frac{100}{3},\infty \right)$
4. $\left(-\frac{100}{12},\infty \right)$

Q. The number of real solutions of the equation ${\left(\frac{9}{10}\right)}^x=-3+x-x^2$ is

1. none
2. more than two
3. two
4. one