Common Roots

Q. If the equations $x^2+ax+12=0,x^2+bx+15=0$ and $x^2+(a+b)x+36=0$ have a common positive integral root, then the value of $a-b$ is

1. $1$
2. $-1$
3. $15$
4. $-15$

Q. If $x^2-ax+b=0$ and $x^2-px+q=0$ have one root common and the second equation has equal roots, then $b+q$ is equal to

1. $p+a$
2. $\frac{p}{2}$
3. $\frac{ap}{2}$
4. $\frac{a}{2}$

Q. The number of distinct common root(s) of the equations $x^5-x^3+x^2-1=0$ and $x^4-1=0$ is

Q. The positive value of $\lambda$ for which the equations $x^2-x-12=0$ and $\lambda x^2+10x+3=0$ have one root in common, is

Q. If $x^2+x+1=0$ and $x^2+ax+b=0$ have a common root, then the minimum value of $(x-a{)}^2+2b$ is

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

Q. The value of $a$ for which $x^3+ax+1=0$ and $x^4+a{x}^2+1=0$ have exactly one common root is

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

Q. If one root of $a{x}^2+bx+c=0$ is reciprocal of one root of $a_1{x}^2+b_{1}x+c_1=0$, then which of the following condition is correct?

1. $(a{a}_1-c{c}_1)=(a_{1}b-b_{1}c)(b_{1}a-b{c}_1)$
2. $(b_{1}a-b{c}_1{)}^2=(a{a}_1-c{c}_1\left)\right(a_{1}b-b_{1}c)$
3. $(a{a}_1-c{c}_1{)}^2=(a_{1}b-b_{1}c\left)\right(b_{1}a-b{c}_1)$
4. $(a_{1}b-b_{1}c{)}^2=(a{a}_1-c{c}_1\left)\right(b_{1}a-b{c}_1)$

Q. If equations $x^2+bx+c=0$ and $b{x}^2+cx+1=0$ have a common root then

1. $b+c+1=0$
2. $b^2+c^2+1=bc$
3. $(b-c{)}^2+(c-1{)}^2+(b-1{)}^2=0$
4. $b+c+1=-bc$

Q.

Taking $\mathrm{U}=\left[1,5\right],\mathrm{A}=\left\{\mathrm{x}/\mathrm{x}\in \mathrm{N},{\mathrm{x}}^2-6\mathrm{x}+5=0\right\},\mathrm{A}=?$

1. $\left\{1,5\right\}$

2. $\left(1,5\right)$

3. $\left[1,5\right]$

4. $\left[-1,-5\right]$

Q. If $p{x}^2+5x+2=0$ and $3x^2+10x+q=0$ have both roots in common, then

1. $p=4$
2. $q=\frac{3}{2}$
3. $p=\frac{3}{2}$
4. $q=4$

Q. If $2x^2+5x+7=0$ and $a{x}^2+bx+c=0$ have at least one root common such that $a,b,c\in \{1,2,3,\dots ,100\}$, then the difference between the maximum and the minimum possible value of $a+b+c$ is

Q. If every pair of equations $x^2+ax+bc=0,$ $x^2+bx+ac=0,$ $x^2+cx+ab=0$ has a common root, then product of these common roots is

1. $a^2{b}^2{c}^2$
2. $abc$
3. $\frac{ab}{c}$
4. $\frac{c}{ab}$

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

Q. If every pair among the equations $x^2+ax+bc=0,x^2+bx+ca=0$ and $x^2+cx+ab=0$ have a common root, then which of the following is/are correct?

1. Sum of the common roots is $-\frac{1}{2}(a+b+c)$
2. Sum of the common roots is $\frac{1}{2}(a+b+c)$
3. Product of the common roots is $abc$
4. Product of the common roots is $-abc$

Q. The quadratic equations $x^2-6x+a=0,$ $x^2-cx+6=0$ have one root in common. The other roots of the first and second equations are integers in the ratio $4:3$ . Then common root is :

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

Q. If $2x^2+5x+2b=0$ and $2x^3+7x^2+5x+1=0$ have atleast one common root for three values of $b$, then the sum of all three values of $b$ is

1. $\frac{1}{2}$
2. $0$
3. $\frac{5}{2}$
4. $\frac{3}{2}$

Q. If $b{x}^2+acx+b^{2}c=0$ and $c{x}^2+abx+b^{2}c=0$ have a common root (where $a,b,c$ are non zero distinct real numbers), then which of the following is/are correct?

1. $\frac{a}{c}+\frac{b}{a}+\frac{a}{b}=0$
2. $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=0$
3. $\frac{1}{b{c}^2}+\frac{1}{a^{2}c}+\frac{1}{c{b}^2}=0$
4. $\frac{1}{a{c}^2}+\frac{1}{b{a}^2}+\frac{1}{c{b}^2}=0$

Q. If $x^2+ax+bc=0,$ $x^2+bx+ac=0,$ $a\ne b$ have one root in common, then their other roots satisfy the equation

1. $x^2+ax+b=0$
2. $x^2+bx+ab=0$
3. $x^2+cx+ab=0$
4. $x^2+abx+bc=0$

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

Q.

If two roots of the equation $x^3-3x+2=0$ are same, then the roots will be

1. 2, 2, 3

2. 1, 1, -2

3. - 2, 3, 3

4. -2, -2, 1

Q. Let $a,b\in R,a\ne 0,$ such that the equation, $a{x}^2-2bx+5=0$ has a repeated root $\alpha ,$ which is also a root of the equation $x^2-2bx-10=0.$ If $\beta$ is the other root of this equation, then ${\alpha }^2+{\beta }^2$ is equal to:

1. $24$
2. $25$
3. $26$
4. $28$

Q. If the three equations $x^2+ax+12=0,$ $x^2+bx+15=0,$ $x^2+(a+b)x+36=0$ have a common possible root. Then, the sum of roots is

1. $24$
2. $-24$
3. $20$
4. $-20$

Q. The sum of possible real values of $b$ for which the equations $2017x^2+bx+7102=0$ and $7102x^2+bx+2017=0$ have a common root, is

Q. If one root of the equation $x^2+ax+b=0$ is a root of the equation $x^2+cx+d=0$, prove that the other roots satisfy the equation $x^2+x(2a-c)+(a^2-ac+d)=0$.

Q. lf every pair from among the equations $x^2+ax+bc=0,x^2+bx+ca=0$ and $x^2+cx+ab=0$ has common root, then the sum of the three common roots is

1. $abc$
2. $2abc$
3. $-\frac{(a+b+c)}{2}$
4. $-\frac{(a+b+c)}{3}$

Q. If $a,b,c$ are non-zero real numbers and $a{x}^2+bx+c=0,$ $b{x}^2+cx+a=0$ have one root in common, then $\frac{a^3+b^3+c^3}{abc}=$

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

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

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

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

Q. If $(x-2)$ is common factor of expressions $x^2+ax+b$ and $x^2+cx+d$, then $\frac{b-d}{c-a}=$
$(a\ne c)$

Q. If $x^2+bx-a=0$ and $x^2-ax+b=0$ have only one common root, then

1. $a+b=0$
2. $a=b$
3. $a-b=1$
4. $a+b=1$