Range of a Quadratic Expression

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

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

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

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 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

Q. The maximum value of $f\left(x\right)=\frac{8}{9x^2-6x+5}$ in its domain is

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

Q.

The maximum value of $\left(2\right(a-x\left)\right(x+\sqrt{x^2+b^2})\forall x,a,b\in R)$ is :

1. $a^2-b^2$

2. $a^2+b^2$

3. $a-b$

4. $a+b$

Q.

Let $(x,y,z)$ be points with integer coordinates satisfying the system of homogeneous equations:
$3x-y-z=0-3x+z=0-3x+2y+z=0$
Then the number of such points for which
$x^2+y^2+z^2\le 100$ is ___

Q.

Minimum value of $x^2+2xy+3y^2-6x-2y$ $x,y,ϵR$ is

1. - 10

2. - 12

3. - 11

4. - 9

Q. The maximum value of $y=6x-x^2-5$ is

1. $5$
2. $7$
3. $10$
4. $4$

Q. The graph of $y=\ln (x^2+1)$ is

1. 
2. 
3. 
4. 

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

Q.

If $x^2+5=2x-4cos(a+bx)$ where $a,b\in (0,5)$ is satisfied for alteast one real $x$, then the minimum value of $a+bx$ is

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

2. $\frac{\pi }{2}$

3. $0$

4. $\pi$

Q.

Minimum value of $x^2+2xy+3y^2-6x-2y$ $x,y,ϵR$ is

1. - 9

2. - 12

3. - 11

4. - 10

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

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

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

Q. Consider the logarithmic inequality $1+{\log }_5(x^2+1)\ge {\log }_5(a{x}^2+4x+a)$ for all real values of $x.$ The number of integers which $a$ cannot take, is

1. $2$
2. $12$
3. $6$
4. $8$

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. $(-\frac{1}{3},\frac{1}{13})$

4. $[-\frac{1}{3},\frac{1}{13}]$

Q.

If $2^{2x+2}-a\cdot 2^{x+2}+5-4a\ge 0$ has atleast one real solution, Then a $ϵ$

1. $(\frac{-8}{7},-1]$

2. $[1,\infty )$

3. $(-\infty ,1]$

4. $(-3,1]$

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

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

Q. The range of $k$, for which the inequality $k{x}^2-3kx+3<0$, $($where $x\in R)$ holds true is

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

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

Q.

Find the range of rational expression $y=\frac{x^2+34x-71}{x^2+2x-7}$ if $x$ is real

1. $[5,9]$

2. $[–5,9]$

3. $(-\infty ,5]\cup [9,\infty )$

4. $(-\infty ,-5]\cup [9,\infty )$

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. $-10$
4. $7$

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

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

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

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

Q. Let $f\left(x\right)=a{x}^2+bx+c$ and $g\left(x\right)=A{x}^2+bx+\lambda ,$ where $a\ne 0,A\ne 0,a,b,c,A,\lambda \in R$. Roots of $f\left(x\right)=0$ and $g\left(x\right)=0$ are imaginary, then which of the following may be correct?

1. $\frac{f\left(x\right)}{a}+\frac{g\left(x\right)}{A}>0\forall x\in R$
2. Roots of equation $af\left(x\right)+Ag\left(x\right)=0$ are real
3. $\frac{f\left(x\right)}{a}+\frac{g\left(x\right)}{A}>0$ for some $x$
4. $f\left(x\right)+g\left(x\right)=0$ for some $x$

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

Q. The minimum and maximum value of $y=\frac{x^2-x+4}{x^2+x+4}$ will be and respectively.

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

Q. The range of $a$ for which $x^2-ax+1-2a^2$ is always positive for all real values of $x$, is

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