Location of Roots when Compared to two constants 'k1' & 'k2'

Q.

The range of the function $f\left(x\right)=\ln \left(3x^2-4x+5\right)$

1. $\left[\ln \left(\frac{11}{3}\right),\infty \right)$

2. $\left[\ln \left(10\right),\infty \right)$

3. $\left[\ln \left(\frac{11}{6}\right),\infty \right)$

4. $\left[\ln \left(\frac{11}{12}\right),\infty \right)$

Q. Consider $f\left(x\right)=a{x}^2+bx+c$ with $a>0$,
If exactly one root of the quadratic equation $f\left(x\right)=0$ lies between $k_1$ and $k_2$ where $k_1<k_2$. The necessary and sufficient condition(s) for this are :

1. $a.f\left(k_2\right)>0$
2. $D>0$
3. $f\left(k_1\right).f\left(k_2\right)<0$
4. $-\frac{b}{2a}>k_1$

Q. If $\alpha ,\beta$ are the roots of the quadratic equation $x^2+2(k-3)x+9=0$ $(\alpha \ne \beta )$. If $\alpha ,\beta \in (-6,1)$ then find the value of $k.$

1. $2,9$
2. $(6,\frac{27}{4})$
3. $6,\infty$
4. $(-2,\frac{27}{4})$

Q. If $x^2+2ax+a<0,\forall x\in [1,2]$ then the values of ${}^{\prime }{a}^{\prime }$ lies in the interval

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

Q. The number of positive integral values of $a$ for which the root(s) of the equation $(a-2)x^2+2ax+(a+3)=0$ lies in $(-2,1)$ is

Q. The least value of $k$ which makes the roots of the equation $x^2+5x+k=0$ imaginary is

1. $6$
2. $7$
3. $4$
4. $5$

Q. Find the set of values of $m$ for which exactly one root of the equation $x^2+mx+(m^2+6m)=0$ lie in $(-2,0)$

1. $(0,6)-\left\{2\right\}$
2. $(-6,0)-\{-2\}$
3. $(-6,0)$
4. $(-\infty ,-6)\cup (0,\infty )$

Q. Values of ${}^{\prime }{m}^{\prime }$ such that the roots of the equation $2x^2-x-1=0$ lie inside the roots of the equation $x^2+(2m-m^2)x-2m^3=0,$ is

1. $2$
2. $\frac{1}{8}$
3. $\frac{3}{4}$
4. $\frac{2}{3}$

Q. If $A$ and $G$ are the arithmetic and geometric means, respectively, of the roots of a quadratic equation then the equation is ___

Q. If $\alpha ,\beta$ are roots of the equation, $a{x}^2+bx+c=0$, then $\frac{1}{a\alpha +b}+\frac{1}{a\beta +b}=$

Q. Find the range of values of $a$ for which at least one root lies in $(0,\frac{1}{2})$ for the quadratic equation $4x^2-4(a-2)x+(a-2)=0\forall a\in R.$

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

Q.

Let $x^2-(m-3)x+m=0,m\in R$ be a quadratic equation. The values of $m$ for which both roots lie in between $1$ and $2$ is given by

1. $m\in \varphi$

2. $m\in [1,9]$

3. $m\in (5,7)$

4. $m<10$

Q. The least positive value of $a$ for which $4^x-a\cdot 2^x-a+3\le 0$ is satisfied by atleast one real value of $x$ is

Q. If both roots of the equation $x^2+ax+2=0$ lie in the interval $(0,3),$ then the range of values of $a$ is

1. $(-6,0)$
2. $(-\frac{11}{3},\infty )$
3. $(-\frac{11}{3},-2\sqrt{2}]$
4. $(-\infty ,-2\sqrt{2}]\cup [2\sqrt{2},\infty )$

Q. For any quadratic polynomial $f\left(x\right)=x^2+\frac{b}{a}x+\frac{c}{a};a\ne 0,$
if $\alpha ,\beta$ are the roots of $f\left(x\right)=0$ and
$k_1,k_2$ be two numbers such that $\alpha <k_1,k_2<\beta ,$ then select the correct statement.

1. $f\left(k_1\right)<0$
2. $f\left(k_1\right)f\left(k_2\right)>0$
3. $f\left(k_2\right)<0$
4. $D>0$

Q. If $\alpha +\frac{1}{\alpha }\&2-\beta -\frac{1}{\beta }(\alpha ,\beta >0)$ are the roots of the quadratic equation $x^2-2(a+1)x+a-3=0$, then the sum of integral values of $a$ is

Q. Find the values of $m$ such that both roots of the quadratic equation $x^2-(m-3)x+m=0$ $(m\in R)$ are smaller than $2$

1. $(7,10)$
2. $(-\infty ,1]$
3. None of the above
4. $(-\infty ,7)$

Q. If one of the root of the qudratic polynomial $f\left(x\right)=a{x}^2+bx+c;a>0$ is greater than $k_1$ and other root is less than $k_2$. Then select the correct statement for $k_1<k_2$.

1. $a\cdot f\left(k_1\right)<0$
2. $f\left(k_1\right)f\left(k_2\right)<0$
3. $a\cdot f\left(k_2\right)>0$
4. $D<0$

Q. Find the values of $m$ such that both the roots of the quadratic equation $x^2-(m-3)x+m=0$ $(m\in R)$ lies in the interval $(1,2).$

1. $\varphi$
2. None of the above.
3. $(5,7)$
4. $(9,\infty )$

Q. If both the roots of the quadratic equation $x^2-mx+4=0$ are real and distinct and they lie in the interval $[1,5]$, then $m$ lies in the interval :

1. $(4,5]$
2. $(4,5)$
3. $(5,6)$
4. $(3,4)$

Q. The values of $m$ such that exactly one root of $x^2+2(m-3)x+9=0$ lies between $1$ and $3$, is

1. $(6,\infty )$
2. $(-2,0)$
3. $(-2,6)$
4. $(-\infty ,0)$

Q. If one root of $x^2-2p(x-4)-15=0$ is less than $1$ and the other root is greater than $2,$ then the range of $p$ is

1. $(-\infty ,\frac{7}{3})$
2. $R$
3. $(0,7)$
4. $(\frac{7}{3},\infty )$

Q. Values of ${}^{\prime }{m}^{\prime }$ such that the roots of the equation $2x^2-x-1=0$ lie inside the roots of the equation $x^2+(2m-m^2)x-2m^3=0,$ is

1. $\frac{1}{8}$
2. $\frac{2}{3}$
3. $2$
4. $\frac{3}{4}$

Q.

$abc\ne 0$ & $a,b,c\in R$. If $x_1$ is a root of $a^2{x}^2+bx+c=0,x_2$ is a root of $a^2{x}^2-bx-c=0$ and $x_1>x_2>0$, then the equation $a^2{x}^2+2bx+2c=0$ has a root $x_3$ such that

1. $x_3>x_1>x_2$

2. $x_2>x_1>x_3$

3. $x_1>x_3>x_2$

4. $x_1>x_2>x_3$

Q. If both the roots of the quadratic equation $x^2-mx+4=0$ are real and distinct and they lie in the interval $[1,5]$, then $m$ lies in the interval :

1. $(3,4)$
2. $(4,5)$
3. $(5,6)$
4. $(4,5]$

Q. If one of the root of the qudratic polynomial $f\left(x\right)=a{x}^2+bx+c;a>0$ is greater than $k_1$ and other root is less than $k_2$. Then select the correct statement(s) for $k_1<k_2$.

1. $a\cdot f\left(k_1\right)<0$
2. $D>0$
3. $f\left(k_1\right)f\left(k_2\right)<0$
4. $a\cdot f\left(k_2\right)<0$

Q.

Both roots of $(a^2-1)x^2+2ax+1=0$ belong to the interval $(0,1)$ then exhaustive set of values of ${}^{\prime }{a}^{\prime }$ is :

1. None of these

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

3. $(-\infty ,-2)\cup (0,\infty )$

4. $(-2,\frac{-1+\sqrt{5}}{2})$

Q. Consider $f\left(x\right)=a{x}^2+bx+c$ with $a>0$,
If exactly one root of the quadratic equation $f\left(x\right)=0$ lies between $k_1$ and $k_2$ where $k_1<k_2$. The necessary condition for this is:

1. $a.f\left(k_2\right)<0$
2. $-\frac{b}{2a}>k_1$
3. $f\left(k_1\right).f\left(k_2\right)>0$
4. $D<0$

Q. If the roots of the equation $x^2+a^2=8x+6a$ are real, then

1. $aϵ(2,8)$
2. $aϵ[2,8]$
3. $aϵ(-2,8)$
4. $aϵ[-2,8]$

Q. If one root of $x^2-2p(x-4)-15=0$ is less than $1$ and the other root is greater than $2,$ then the range of $p$ is

1. $(-\frac{7}{3},\frac{7}{3})$
2. $(-\infty ,\frac{7}{3})$
3. $R$
4. $(\frac{7}{3},\infty )$