Quadratic Equations with Both Roots Common

Q.

if the equations $2x^2+3x+5\lambda =0$ and $x^2+2x+3\lambda =0$ have a common root, then λ =

1. 2, -1

2. 0

3. 0, -1

4. -1

Q.

If one root of the equations $x^2+px+q=0$ and $x^2+\alpha x+\beta =0$ is common, then its value will be ( where p ≠ α and q ≠ β )

1. $\frac{q-\beta }{\alpha -p}$

2. $\frac{p\beta -\alpha q}{q-\beta }$

3. None of these

4. $\frac{q-\beta }{\alpha -p}\text{or}\frac{p\beta -\alpha q}{q-\beta }$

Q. For two quadratic equations to have both roots common, the ratio of the corresponding coefficients must be equal.

1. False
2. True

Q. Consider the quadratic equation $4{\cos }^{2}x+2\cos x-1=0.$
Then which of the following quadratic equation has same roots as that of the given equation is

1. $4{\cos }^{2}2x+\cos 2x-1=0$
2. $4{\cos }^{2}2x-2\cos 2x+1=0$
3. $4{\cos }^{2}2x+2\cos 2x-1=0$
4. ${\cos }^{2}2x+2\cos 2x-4=0$

Q. Given that quadratic equation $a{x}^2+bx+c=0$ where $a,b,c\in Q$ has an irrational root and the other root is common with a root of quadratic equation$d{x}^2+ex+f=0$ for $d,e,f\in Q.$ Then the value of $\frac{a^{2}ef}{d^{2}bc}$ is

1. $1$
2. $\frac{3}{2}$
3. $\frac{1}{2}$
4. Any real number

Q. If $a,b,c$ be the sides of $△ABC$ and equations $a{x}^2+bx+c=0$ and $5x^2+12x+13=0$ have a common root, then $\angle C$ is

1. ${45}^{\circ }$
2. ${60}^{\circ }$
3. ${120}^{\circ }$
4. ${90}^{\circ }$

Q. Find the value of $m$ for which the equations $x^2+mx+1=0\&(b-c)x^2+(c-a)x+(a-b)=0$ have both roots common.

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

Q.

If $a,b,p,q$ are non-zero real numbers, then the two equations $2a^2{x}^2-2abx+b^2=0$ and $p^2{x}^2+2pqx+q^2=0$ have

1. No common root

2. One common root if $2a^2+b^2=p^2+q^2$

3. Two common roots if $3pq=2ab$

4. Two common roots if $3bq=2ap$

Q. If the equations $4x^2-x-1=0$ and $3x^2+(\lambda +\mu )x+\lambda -\mu =0$ have a common root, then the rational values of $\lambda$ and $\mu$ are

1. $\lambda =\frac{-3}{4},\mu =\frac{1}{4}$
2. $\lambda =\frac{-3}{4},\mu =\frac{3}{4}$
3. $\lambda =0,\mu =\frac{-3}{4}$
4. $\lambda =\frac{-3}{4},\mu =0$

Q. For two quadratic equations to have both roots common, the ratio of the corresponding coefficients must be equal.

1. False
2. True

Q. If equations $x^2-3x+4=0$ and $4x^2-2[3a+b]x+b=0(a,b\in R)$ have a common root, then the complete set of values of $a$ is
$($Here, $\left[K\right]$ denotes the greatest integer less than or equal to $K.)$

1. $(3,\frac{10}{3}]$
2. $[\frac{-10}{3},-3)$
3. $[\frac{-11}{3},\frac{-10}{3})$
4. $(\frac{-10}{3},-3]$

Q. The two equations $x^3+1=0$ and $a{x}^2+bx+c=0,a,b,c\in R$ have two roots in common. Then $a+b$ is equal to

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

Q. Given quadratic equations $x^2+5x+p=0\&2x^2+qx+12=0$ have both roots common, then $p+q=$

1. $22$
2. $16$
3. $10$

Q. For two quadratic equations to have both roots common, the ratio of the corresponding coefficients must be equal.

Q. Find the value of $m$ for which the equations $x^2+mx+1=0\&(b-c)x^2+(c-a)x+(a-b)=0$ have both roots common.

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

Q. If equation $(k-1)x^2+(k^2+1)x+6=0$ and $2x^2+10x+12=0$ have both roots common, then the value of $k$ is ___

Q. If the equations $k(6x^2+3)+rx+(2x^2-1)=0$ and $6k(2x^2+1)+px+(4x^2-2)=0,k\ne 0$ have both roots common, then the value of $\frac{p}{r}$ is

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

Q. If the equations $x^2 +3x+5=0$ and $ax^2 +bx+c=0;~a, b, c\in\mathbb N$ have a common root, then the least possible value of $a+b+c$ is

Q.

If $x^2+7x+1=0$ and $(a^2-4a)x+bx-3=0$ have both the roots in common, then find the value of $a$.

1. $4$

2. $3$

3. $1$

4. $2$

Q. The two equations $x^3+1=0$ and $a{x}^2+bx+c=0,a,b,c\in R$ have two roots in common. Then $a+b$ is equal to

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

Q. If equations $x^2-3x+4=0$ and $4x^2-2[3a+b]x+b=0(a,b\in R)$ have a common root, then the complete set of values of $a$ is
$($Here, $\left[K\right]$ denotes the greatest integer less than or equal to $K.)$

1. $(3,\frac{10}{3}]$
2. $(\frac{-10}{3},-3]$
3. $[\frac{-10}{3},-3)$
4. $[\frac{-11}{3},\frac{-10}{3})$

Q. Choose the correct pair of quadratic equations with both roots common.

1. $2x^2-6x-12=0$
2. $4x^2+2x=0$
3. $2x^2+4x+4=0$
4. $2x^2+x=0$
5. $-x^2+3x+6=0$
6. $x^2+2x+2=0$

Q. Given that quadratic equation $a{x}^2+bx+c=0$ where $a,b,c\in Q$ has an irrational root and the other root is common with a root of quadratic equation$d{x}^2+ex+f=0$ for $d,e,f\in Q.$ Then the value of $\frac{a^{2}ef}{d^{2}bc}$ is

1. $\frac{3}{2}$
2. $\frac{1}{2}$
3. Any real number
4. $1$

Q.

If both the roots of
$k(6x^2+3)+rx+2x^2-1=0$ and
$2(6k+2)x^2+px+2(3k-1)=0$
are same, then $2r-p$ is equal to

1. -1

2. 0

3. 1

4. 2

Q.

If $a,b,c\in R$ and equations $a{x}^2+bx+c=0$ and $2x^2+4x+6=0$ have a common root with $a+b+c=18$, the value of $a^{2}bc$ is

__

Q.

If $x^2+7x+1=0$ and $(a^2-4a)x+bx-3=0$ have both the roots in common, then find the value of $a$.

1. $1$

2. $4$

3. $2$

4. $3$

Q. If equation $(k-1)x^2+(k^2+1)x+6=0$ and $2x^2+10x+12=0$ have both roots common, then the value of $k$ is ___

Q.

If the equations $a{x}^2+bx+1=0$ and $2x^2+4x+3=0$ have both the roots common, then the value of $a+b$ is:

1. $1$

2. $3$

3. $2$

4. $4$

Q.

If $a, b, c\in \mathbb{R}$ and equations \( ax^2 + bx + c= 0\) and \(2x^2 + 4x + 6 = 0 \) have a common root with $a + b + c = 18$, the value of \(a^2bc\) is

__

Q. If the equations $x^2+2x+3=0$ and $a{x}^2+bx+c=0,a,b,c\in R$, have a common root, then $a:b:c$ is

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