Location of Roots when Compared with a constant 'k'

Q. Find all the values of the parameter $d$ for which both roots of the equation $x^2-6dx+(2-2d+9d^2)=0$ exceed the number $3.$

1. $(-\infty ,1)$
2. $(\frac{11}{9},\infty )$
3. $(-\infty ,1)$
4. $(1,\frac{11}{9})$

Q. Find the value of $k$ for which one root of the equation $x^2-(k+1)x+k^2+k-8=0$ exceed $2$ and other is smaller than $2$.

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

Q. If at least one of the roots of the equation $x^2-(k+2)x+\frac{7k}{4}=0$ is negative, then $k$ lies in the interval

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

Q. If both roots of $a{x}^2+bx+c=0,a>0$ are greater than $k$ then $-\frac{b}{2a}>k.$

1. False
2. True

Q. The set of values of $k$ for which roots of the quadratic equation $-x^2-2(k-1)x-(k+5)=0$ are less than \(1)\ is

1. $R-1$
2. $[4,\infty )$
3. $[2,\frac{3+\sqrt{27}}{2}]$
4. $R$

Q. Set of values of $a$ for which both the roots of the quadratic polynomial $f\left(x\right)=a{x}^2+(a-3)x+1$ lie on one side of the $y-$axis is

1. $(0,1)\cup (3,\infty )$
2. $(0,1]\cup [9,\infty )$
3. $[0,1)\cup [3,\infty )$
4. $[0,1]\cup [9,\infty )$

Q. Find the value of $m$ such that roots of the quadratic equation $x^2-(m-3)x+m=0$ $(m\in R)$ are equal in magnitude but opposite in sign.

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

Q. The values of $m$ for which roots of the quadratic equation $-x^2+(2m+3)x-m^2=0$ lies on either side of $3.$

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

Q. If $k\in R$ lies between the roots of $a{x}^2+bx+c=0,a<0$. Consider $f\left(x\right)=a{x}^2+bx+c=0$, then $f\left(k\right)$ $\&$

1. $D\ge 0$
2. $>0$
3. $<0$
4. $D>0$

Q. For any quadratic expression $f\left(x\right)=a{x}^2+bx+c;a<0$, having zeros as $\alpha ,\beta$ and if $k$ is any real number such that $k<\beta <\alpha ,$ then select the correct statement(s).

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

Q. Find $k$ for which one root of the equation $x^2-(k+1)x+k^2+k-8=0$ is greater than 2 and other is less than 2

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

Q. If both the roots of the equation $x^2+2(k+1)x+9k-5=0$ are negative, then the least positive integral value of $k$ is

Q. If both the roots of $x^2+2ax+a=0$ are less than $2$, then the set values of ${}^{\prime }{a}^{\prime }$ is

1. $(-2,0]\cup [1,\infty )$
2. $(-\frac{4}{5},0]\cup [1,\infty )$
3. $(-\infty ,0]\cup [1,\infty )$
4. $[-\frac{4}{5},0)\cup (1,\infty )$

Q. '$af\left(k\right)<0$' is the necessary and sufficient condition for a particular real number $k$ to lie between the roots of a quadratic equation $f\left(x\right)=0$, where $f\left(x\right)=a{x}^2+bx+c$. If $f\left(k_1\right)f\left(k_2\right)<0$, then exactly one of the roots will lie between $k_1$ and $k_2$.

If $a(a+b+c)<0<c(a+b+c)$, then

1. both the roots lie in $(0,1)$
2. exactly one of the roots lies in $(0,1)$
3. one root is less than $0$, the other is greater than $1$
4. at least one of the roots lies in $(0,1)$

Q. If both the roots of $x^2+2(k+1)x+9k-5=0$ are less than $0\&k<100,$ then the number of integral values of $k$ is

Q. Let S be the set of values of 'a' for which 2 lie between the roots of quadratic equation $x^2+(a+2)x-(a+3)=0$. Then S is given by

1. $[5,\infty )$
   
   
2. $(-\infty ,-5)$
3. $(5,\infty )$
4. $(-\infty ,-5]$

Q. Find the value of $m$ such that roots of the quadratic equation $x^2-(m-3)x+m=0$ $(m\in R)$ are opposite in sign.

1. $(-\infty ,9)$
2. $(-\infty ,1)\cup (9,\infty )$
3. $(-\infty ,1)$
4. $(-\infty ,0)$

Q. The range of $k$ for which both the roots of the quadratic equation $(k+1)x^2-3kx+4k=0$ are greater than $1$, is

1. $(-\infty ,-1)$
2. $[-\frac{16}{7},-1]$
3. $[-\frac{16}{7},-1)$
4. $(-\frac{16}{7},-1)$

Q.

If the roots of $x^2-(a-3)x+a=0$ are such that at least one of the root(s) is greater than $2,$ then find the range of $a.$

1. $[9,\infty )$

2. $[7,9]$

3. $[7,\infty )$

4. $(7,9)$

Q. If $\gamma ,\delta$ are the roots of $x^2-3x+a=0,aϵR$, and $\gamma <1<\delta$ then

1. $aϵ(-\infty ,\frac{9}{4})$
2. none of these
3. $aϵ(-\infty ,2)$
4. $aϵ(2,\frac{9}{4})$

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

1. $(-\infty ,1)$
2. $[1,3)$
3. $(0,1]$
4. None of the above

Q. The set of values of $a$ for which $6$ lies between the roots of the equation $x^2+2(a-3)x+9=0$, is

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

Q. Consider $f\left(x\right)=a{x}^2+bx+c$ with $a>0,$
If both roots of the quadratic equation are smaller than any constant $k,$ then

1. $f\left(k\right)>0$
2. None of these
3. $-\frac{b}{2a}>k$
4. $f\left(k\right)<0$

Q. If both roots of $a{x}^2+bx+c=0,a>0$ are greater than $k$ then $-\frac{b}{2a}>k.$

1. False
2. True

Q. Find a for which one root is positive and other root is negative for $-x^2-(3a-2)x+a^2+1=0$

1. $\varphi$
2. $R-0$
3. None of the above
4. $R$

Q. Find $a$ for which one root is positive and other root is negative for $-x^2-(3a-2)x+a^2+1=0$

1. $R-0$
2. $\varphi$
3. None of the above
4. $R$

Q. The least positive integer value of $a$ for which both roots of the quadratic equation $(a^2-6a+5)x^2+\left(\sqrt{a^2+2a}\right)x+(6a-a^2-8)=0$ lie on either side of origin, is -

1. $2$
2. $1$
3. $6$
4. $3$

Q. Find the value of $m$ of the quadratic equation $x^2-(m-3)x+m=0$ $(m\in R)$ such that one root is smaller than $2$ and other root is greater than $2$

1. $(10,\infty )$
2. $(1,10)$
3. None of the above.
4. $(9,10)$

Q. If the roots of $x^2-6kx+(2-2k+9k^2)=0$ are greater than $3$, then the range of $k$ is

1. $(-\infty ,1)\cup (\frac{11}{9},\infty )$
2. $(1,\infty )$
3. $(\frac{11}{9},\infty )$
4. $(1,\frac{11}{9})$

Q. The range of $k$ for which both the roots of the quadratic equation $(k+1)x^2-3kx+4k=0$ are greater than $1$ is:

1. $(-\infty ,-1)$
2. $[-\frac{16}{7},-1)$
3. $(-\frac{16}{7},-1)$
4. $[-\frac{16}{7},-1]$