Range of a Quadratic Expression

Q. Set of real values of $a$ for which$(a+4)x^2-2ax+2a-6>0$

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

Q. For the quadratic expression $y=a{x}^2+bx+c,a<0.$ The maximum value of $y$ occurs at

1. $x=-\frac{D}{4a^2}$
2. $x=-\frac{b}{2a}$
3. $x=\frac{D}{4a^2}$
4. $x=\frac{b}{2a}$

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

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

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

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

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

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

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

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

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

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

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

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

Q. The values of $m$ for which the quadratic expression $y=(m^2-9)x^2+(m-3)x-7$ is always greater than or equal to $-1$ is

1. $R$
2. $[-\frac{25}{9},3]$
3. $\varphi$
4. $(-\infty ,-3)\cup (3,\infty )$

Q. For any quadratic expression $f\left(x\right)=a{x}^2+bx+c,a>0$ and if $(v,f(v\left)\right)$ is it's vertex, then $y\in$

1. $\left[f\right(v),\infty )$
2. $(-\infty ,f(v\left)\right]$
3. $[-f(v),f(v\left)\right]$

Q. For the given quadratic equation $y=-x^2-5x-4,y<0$ for

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

Q. Range of rational expression $y=\frac{x^2-x+4}{x^2+x+4},x\in R$ is

1. $[-\frac{3}{5},\frac{3}{5}]$
2. $[-\frac{5}{3},\frac{5}{3}]$
3. $[\frac{2}{3},\frac{5}{3}]$
4. $[\frac{3}{5},\frac{5}{3}]$

Q. The range of $f\left(x\right)=x+\frac{1}{x}\forall x\ne 0$ is

1. $\left(-2,2\right)$
2. $(-\infty ,-1]\cup [1,\infty )$
3. $(-\infty ,-2]\cup [2,\infty )$

Q.

If the roots of the equation $a{x}^2+bx+a_1^2+b_1^2+c_1^2-a_1{b}_1-a_1{c}_1-b_1{c}_1=0$ are non real then

1. $2(b-a)+\sum (a_1+b_1{)}^2<0$

2. $2(a-b)+\sum (a_1+b_1{)}^2=0$

3. $2(b-a)+\sum (a_1+b_1{)}^2=0$

4. $2(a-b)+\sum (a_1-b_1{)}^2>0$

Q. Find the range of the rational expression $y=\frac{x^2+2x-4}{x^2-2x+3}\forall x\in R$

1. $[1,3]$
2. $[-3,3]$
3. $[-1,3]$
4. $[-3,1]$

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

1. $a$ >$-3$

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

3. $a\le 1$

4. $a\ge 1$

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. $25$
4. $16$

Q. $\text{Let}f\left(\alpha \right)={\alpha }^2-a\alpha +1,f\left(\beta \right)=b\beta -a{\beta }^2-\frac{3}{a},\text{where}a,b\in I^{-}\text{and}|f\left(\alpha \right)+f\left(\beta \right)|=|f\left(\alpha \right)|+|f\left(\beta \right)|\text{is satisfied}\forall \alpha ,\beta \in R,\text{then which of the following is/are true?}$

1. Sum of all values of $a$ is $-3$
2. Sum of all values of $b$ is $-6$
3. Sum of all values of $b$ is $-3$
4. Sum of all values of $a$ is $-2$

Q.

Given that, for all real ${}^{\prime }{x}^{\prime }$, the expression $\frac{x^2+2x+4}{x^2-2x+4}$ lies between $\frac{1}{3}$ and $3$. The values between which the expression $\frac{{9.3}^{2x}+{6.3}^x+4}{{9.3}^{2x}-{6.3}^x+4}$ lies are

1. $-1$ and $1$

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

3. $0$ and $2$

4. $-2$ and $0$

Q. If the roots of $x^2-2x-a^2+1=0$ lie between the roots of $x^2-2(a+1)x+a(a-1)=0$, then the range of $a$ is

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

Q. For any quadratic expression $y=a{x}^2+bx+c,a<0.$ It's range will be

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

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

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

Q. Let $x^2-(m-3)x+m,(m\in R)$ be a quadratic expression which is always greater than zero. Then the total number of intergral value(s) of $m$ is .

1. 7

Q. The range of the quadratic expression $y=-4x^2+13x-3$ is

1. $(-\infty ,\frac{121}{2}]$
2. $(-\infty ,\frac{121}{4}]$
3. $(-\infty ,\frac{121}{16}]$
4. $(-\infty ,\frac{121}{8})$

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

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

Q. Let the graph of \(f(x)=ax^2+bx+c\) passes through origin and makes an intercept of \(10\) units on \(x-\)axis. If the maximum value of $f(x)$ is $25$, then the least value of \(|a+b+c|\) is