Conditions for Common Roots

If the equations $x^2-11x+k=0$ and $x^2-14x+2k=0$ have a common root, find the sum of possible values of $k$.

If one root of the equations $a{x}^2+bx+c=0$ and $b{x}^2+cx+a=0\left(a, b, c\in R\right)$ is common, then find the value of ${\left(\frac{a^3+b^3+c^3}{abc}\right)}^3$

If $x^2+3x+5=0$ and $a{x}^2+bx+c=0$ have common root / roots and $a, b, c\in N.$ then the minimum value of $a+b+c$ is 

If all the equations ${\mathrm{x}}^2+(2\mathrm{a}+3\mathrm{b})\mathrm{x}+60=0, {\mathrm{x}}^2+\mathrm{ax}+10=0$ and ${\mathrm{x}}^2+\mathrm{bx}+8=0$ where $\mathrm{a}, \mathrm{b}\in \mathrm{R},$ have a common root, then value of $|\mathrm{a}-\mathrm{b}|$ is

If the equation $x^2+px+2q=0$ and $x^2+qx+2p=0(\mathrm{p}\ne \mathrm{q})$ have a common root
then the absolute value of $(p+q)$ is

If all the equations $x^2+(2a+3b)x+60=0,x^2+ax+10=0$ and $x^2+bx+8=0$ where $a,b\in R,$have a common root, then value of $|\mathrm{a}-\mathrm{b}|$ is

If $b{x}^2+ax+c=0, a{x}^2+bx+c=0$ and $a{x}^2+cx+b=0,$ each equation has equal roots, then $\left|\begin{array}{lll}3& a& bc\\ 3& b& ac\\ 3& c& ab\end{array}\right|$ is equal to

For next two question please follow the same  

 If the quadratic equations a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 have exactly one common root, then the relation between their coefficients is ${\left({\text{c}}_1{\text{a}}_2-{\text{c}}_2{\text{a}}_1\right)}^2=\left({\text{b}}_1{\text{c}}_2-{\text{b}}_2{\text{c}}_1\right)\left({\text{a}}_1{\text{b}}_2-{\text{a}}_2{\text{b}}_1\right)$. If both the roots are common, then the relation between their coefficient is  $\frac{{\text{a}}_1}{{\text{a}}_2}=\frac{{\text{b}}_1}{{\text{b}}_2}=\frac{{\text{c}}_1}{{\text{c}}_2}$.

 If the equations $a{x}^3+3b{x}^2+3cx+d=0$ and  $a{x}^2+2bx+c=0$  have a common root, then which of the following option is correct ?

If equations $x^2+ax+b=0(a,b\in R) \& x^3+3x^2+5x+3=0$ have two common roots, then value of $\frac{2b}{a}$ is equal to

If the quadratic equations $a{x}^2+2bx+3c=0$ and $3x^2+8x+15=0$ have a common root (where $a,b,c$ are the lengths of sides of a $∆ABC$), then $\left({\sin }^{2}A +{\sin }^{2}B+{\sin }^{2}C\right)=$_____ 

If the quadratic equations $a{x}^2+bx+c=0$ and $b{x}^2+cx+a=0$ have a common root, (where $a\ne 0$ ) then $\frac{a^3+b^3+c^3}{abc}=$____

If $a, b, c$ are in A.P. and if $\left(b-c\right)x^2+\left(c-a\right)x+\left(a-b\right)=0$ and $2\left(c+a\right)x^2+\left(b+c\right)x=0$ have common root then which one is correct

Consider the cubic equation: $2a{x}^3+b{x}^2+cx+d=0$ and quadratic equation $2a{x}^2+3bx+4c=0$ having a common root and ${\left(\lambda bc+ad\right)}^2=\frac{9}{2}\left(\mu bd+4c^2\right)\left(m{b}^2-nac\right)$ then the value of the expression $\lambda +\mu +m+n$ is

If equations $a{x}^2+bx+c=0; a, b, c\in R \& c\ne 0$ and $2x^2+3x+4=0$ have a common root, then $a:b:c$ equals 

If the quadratic equations $k\left(6x^2+3\right)+rx+2x^2-1=0$ and  $6k\left(2x^2+1\right)+px+4x^2-2=0$ have both the roots common, then  $2r-p$ is equal to

If the equations $a{x}^2+2bx+c=0$ and $d{x}^2+2ex+f=0$ have a common root and $a, b$ and $c$ are in geometric progression, then $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in:

If quadratic equations $x^2-bx+c=0$ and $x^2-cx+b=0$ and $b\ne c$ have a common root then $b+c-3$ is equal to

If the equations $2x^2-7x+9=0$ and ${ax}^2+bx+18=0$ have a common root, then $\left(a,\mathrm{}b\in R\right)$

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

The equations $k{x}^2+x+k=0$ and $k{x}^2+kx+1=0$ have exactly one root in common for