Question

Statement-I : If equations $a{x}^2+bx+c=0;(a,b,c\in R)$ and $2x^2+3x+4=0$ have a common root, then $a:b:c=2:3:4$.
Statement-II : If $p+iq$ is one root of a quadratic equation with real coefficients, then $p-iq$ will be the other root ; $p,q\in R,i=\sqrt{-1}$.

• A. Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I
• B. Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for statement-I
• C. Statement-I is true, Statement-II is false
• D. Statement-I is false, Statement-II is true

Answer 1

Answer: (B)

Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for statement-I

Given that $a{x}^2+bx+c=0$ and $2x^2+3x+4=0$ have a root in common.
but $\Delta =3^2-4\left(2\right)\left(4\right)=-23<0$
i.e. roots are imaginary, and one root is conjugate complex of the other.
If two imaginary roots are common then it follows that their conjugates also are common.
$\therefore$ Both roots are common.
$\Rightarrow a:b:c=2:3:4$
Statement-I is true, Statement-II is true; but Statement-II is not correct explanation for Statement-I.
Hence, option $B$ is correct.