Location of Roots when Compared with a constant 'k'

Q.

The values of $a$ for which $2x^2-2(2a+1)x+a(a+1)=0$ may have one root less than $a$ and other root greater than $a$ are given by

1. -1 < $a$ < 0

2. 1 > $a$ > 0

3. $a\ge 0$

4. $a$ > 0 or $a$ < - 1

Q.

For the equation $x^2-2ax+a^2-1=0$, the values of ${}^{\prime }{a}^{\prime }$ for which $3$ lies in between the roots of given equation is

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

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

3. $(2,4)$

4. $[3,4]$

Q. If the roots of the quadratic equation $(4p-p^2-5)x^2-(2p-1)x+3p=0$ lie on either side of unity, then the number of integral values of $p$ is

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

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 $a{x}^2+bx+c=0$ are negative and $b<0$, then which of the following statements is always true?

1. $a<0,c<0$
2. $a>0,c>0$
3. $a>0,c<0$
4. $a<0,c>0$

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

Q. Consider $f\left(x\right)=a{x}^2+bx+c$ with $a>0$,
If both roots of the quadratic equation are greater than any constant $k$. The necessary and sufficient condition for this are :

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

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

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

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

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

1. $92$
2. $94$
3. $95$
4. $93$

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

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

1. $[7,9)$
2. $[7,9]$
3. $[7,\infty )$
4. $[9,\infty )$

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

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

Q. If both the roots of $a{x}^2+bx+c=0$ are negative and $b<0$, then which of the following statements is always true?

1. $a>0,c>0$
2. $a<0,c>0$
3. $a>0,c<0$
4. $a<0,c<0$

Q.

The values of $a$ for which $2x^2-2(2a+1)x+a(a+1)=0$ may have one root less than $a$ and other root greater than $a$ are given by

1. $a\ge 0$

2. $a$ > 0 or $a$ < - 1

3. -1 < $a$ < 0

4. 1 > $a$ > 0

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

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

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

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

1. both the roots lie in $(0,1)$
2. one root is less than $0$, the other is greater than $1$
3. one root lies in $(0,1)$ and the other in $(1,\infty )$
4. one root lies in $(-\infty ,0)$ and the other in $(0,1)$

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

1. $95$
2. $92$
3. $94$
4. $93$

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

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

Q. If both the roots of the quadratic equation $x^2-6nx+(9n^2-2n+2)=0$ are greater than 3, then the range of $n$ is:

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

Q. If both the roots of $x^2+2(a+2)x+9a-1=0$ are negative, then ${}^{\prime }{a}^{\prime }$ lies in

1. $(-2,\infty )$
2. $[\frac{1}{9},1]\cup [4,\infty )$
3. $(\frac{1}{9},1]\cup [4,\infty )$
4. $(\frac{1}{9},\frac{5-\sqrt{5}}{2}]\cup [\frac{5+\sqrt{5}}{2},\infty )$

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

Q. The set of all real values of $\lambda$ for which the quadratic equation, $({\lambda }^2+1)x^2-4\lambda x+2=0$ always have exactly one root in the interval $(0,1)$ is:

1. $(0,2)$
2. $(2,4]$
3. $(-3,-1)$
4. $(1,3]$

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