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

Q. If $\alpha +\frac{1}{\alpha }$ and $2-\beta -\frac{1}{\beta }$ $(\alpha ,\beta >0)$ 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

1. $5$
2. $6$
3. $12$
4. $17$

Q. If the roots of the equation $x^2-2mx+m^2-1=0$ lie in the interval $(-2,4)$ then

1. $1<m<5$

2. $1<m<3$

3. $-1<m<3$

4. $-1<m<5$

Q. If at least one of the roots of the equation $x^2-(m-1)x-m=0$ is positive and $m\le 10,$ then the number of integral value(s) of $m$ is

Q.

The complete set of values of 'a' such that $x^2+ax+a^2+6a<0\forall x\in [-1,1]$ is:

1. $(\frac{-5-\sqrt{21}}{2},\frac{-7+\sqrt{45}}{2})$

2. None of these

3. $(\frac{-7-\sqrt{45}}{2},\frac{-5-\sqrt{21}}{2})$

4. $(\frac{-5+\sqrt{21}}{2},\frac{-7+\sqrt{45}}{2})$

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

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. $(4,5)$
4. $(5,6)$

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

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. $(4,5)$
4. $(5,6)$

Q. Let $x_1,x_2(x_1\ne x_2)$ be the roots of the equation $x^2+2(m-3)x+9=0$. If $-6<x_1,x_2<1,$ then ${}^{\prime }{m}^{\prime }$ lies in the interval

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

Q. If $a,b,c$ are rational number $(a>b>c>0)$ and quadratic equation $(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ has a root in the interval $(-1,0)$, then which of the following statement(s) is/are correct?

1. $a+c<2b$
2. Both roots are rational
3. $c{x}^2+2bx+a=0$ has both roots negative
4. $a{x}^2+2bx+c=0$ has both roots negative

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. If $3\text{and}4$ lies between the roots of the equation $x^2+2kx+9=0$ then $k$ lies in the interval

1. $(-\infty ,-\frac{25}{8})\cup (3,\infty )$
2. $(-\infty ,-3)$
3. $(-\frac{25}{8},\infty )$
4. $(-\infty ,-\frac{25}{8})$

Q. The number of integral value(s) of $a$ for which one root of the equation $(a-5)x^2-2ax+a-4=0$ is smaller than $1$ and the other greater than $2,$ is

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

Q. If atleast one of the root of the equation $x^2-(a-3)x+a=0$ is lies in the interval $(1,2)$, then $a$ lies in the interval

1. $[9,10)$
2. $[9,\infty )$
3. $(10,\infty )$
4. $(5,7)\cup (10,\infty )$

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

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

Q. The number of integral value(s) of $a$ for which the equation $2^a{x}^2-4^{a}x-2^a-1=0$ has exactly one root between $1$ and $2$ is

Q. The number of values of $k$ for which the equation $x^2-3x+k=0$ has two distinct roots lying in the interval $(0,1)$ are

1. three
2. infinitely many
3. two
4. no value of k satisfies the requirement

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

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. $x_1>x_2>x_3$

2. $x_3>x_1>x_2$

3. $x_1>x_3>x_2$

4. $x_2>x_1>x_3$

Q. If $3\text{and}4$ lies between the roots of the equation $x^2+2kx+9=0$ then $k$ lies in the interval

1. $(-\frac{25}{8},\infty )$
2. $(-\infty ,-\frac{25}{8})\cup (3,\infty )$
3. $(-\infty ,-3)$
4. $(-\infty ,-\frac{25}{8})$

Q. Consider the quadratic equation $(c-5)x^2-2cx+(c-4)=0$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0,2)$ and another root lies in the interval $(2,3)$. The number of elements in $S$ is

1. $10$
2. $11$
3. $12$
4. $18$

Q.

The values of $a$ for which the equation $2x^2-2(2a+1)x+a(a–1)=0$ has roots $\alpha \&\beta$ satisfying the condition $\alpha <a<\beta$, are

1. $(-3,0)$

2. $(0,\infty )$

3. None of the above

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

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. $(5,6)$
3. $(3,4)$
4. $(4,5)$

Q. Find the values of $m$ such that exactly one root of the quadratic equation $x^2-(m-3)x+m=0;m\in R$ lie in the interval $(1,2).$

1. $(1,9)$
2. $(-\infty ,1)$
3. $(10,\infty )$
4. $(9,\infty )$

Q.

The roots of the equation $x^2+2(a-3)x+9=0$ lies between $-6$ and $1$ then $\left[a\right]=$________ , where $[.]$ denotes greatest integer function $x$.

1. $6$

2. $12$

3. $19$

4. $3$

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

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