Common Roots

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 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 $a, b, c$ 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

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 the equations $x^4+ax=1$ and $x^5+a{x}^2+1=0$ have a common root, then

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

A monic quadratic trinomial P(x) is such that P(x) = 0 and $\text{P}\left(\text{P}\left(\text{P}\left(\text{x}\right)\right)\right)=0$ have a common root, then

If $a,b,c$ are in $\mathrm{AP}$ and if the equations ${(b-c)x}^2+\left(c-a\right)x+\left(a-b\right)=0 \dots \left(1\right)$ and $2{(c+a)x}^2+\left(b+c\right)x=0 \dots \left(2\right)$ have a common root, then

If $a{x}^2+2bx+c=0$ and $p{x}^2+2qx+r=0$ have one and only one root in common and $a, b, c, p, q, r$ being rational then $b^2-ac$ and $q^2-pr$ are

If $a, b, c$ are in G.P. the equations ${ax}^2+2bx+c=0$ and ${dx}^2+2ex+f=0$ have a common root if $d/a,e/b,f/c$ are in -

The value of $a$ for which the equations $x^2-3x+a=0$ and $x^2+ax-3=0$ have a common root is -

If the quadratic equations ${\mathrm{a}\mathrm{x}}^2+2\mathrm{c}\mathrm{x}+\mathrm{b}=0$ and ${\mathrm{a}\mathrm{x}}^2+2\mathrm{b}\mathrm{x}+\mathrm{c}=0$ have a common root then $\mathrm{a}+4\mathrm{b}+4\mathrm{c}$ is equal to

If a non-zero root of the equation $x^2+2x+3\lambda =0$ and $2x^2+3x+5\lambda =0$ is common, then the value of $\mathrm{\lambda }$ will be-

If $f\left(x\right)=4x^2+3x-7$ and $\alpha$ is a common root of equations $x^2- 3x+2=0$ and $x^2+2x-3=0$, then $f\left(\alpha \right)\mathit{=}$

If $\mathrm{a},\mathrm{ }\mathrm{b},\mathrm{ }\mathrm{c}$ are in G.P. then the equation ${\mathrm{a}\mathrm{x}}^2+2\mathrm{b}\mathrm{x}+\mathrm{c}=0$ and ${\mathrm{d}\mathrm{x}}^2+2\mathrm{e}\mathrm{x}+\mathrm{f}=0$ have a common root if $\mathrm{d}/\mathrm{a},\mathrm{e}/\mathrm{b},\mathrm{f}/\mathrm{c}$ are in :

If the quadratic equations ${ax}^2+2cx+b=0$ and ${ax}^2+2bx+c=0 \left(b\ne c\right)$ have a common root, then $a+4 b+4c$ is ;

Let $p$ be the sum of all possible determinants of order $2$ having $0, 1, 2$ and $3$as their four elements. Then the common root $\alpha$ of the equations

$x^2 + ax + \left[m + 1\right]=0$

$x^2 + bx + \left[m + 4\right]=0$

$x^2- cx + \left[m + 15\right]=0$

Such that $\alpha > p$ , where $a + b + c=0$ and $m =\underset{n\to \infty }{\lim }\frac{1}{n}{\sum }_{r=1}^{2n}\frac{r}{\sqrt{n^2+r^2}}$ . (where $\left[.\right]$ denotes the greatest integer function)

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