CAT 2023 Slot 1 QA Question 6: Let α and β be the two distinct roots of the equation $2x^2 - 6x + k = 0$, such that $(\alpha + \beta)$ and $\alpha \beta$ are distinct roots of the equation $x^2 + px + p = 0$. Then, the value of $8(

Question

Let α and β be the two distinct roots of the equation $2x^2 - 6x + k = 0$, such that $(\alpha + \beta)$ and $\alpha \beta$ are distinct roots of the equation $x^2 + px + p = 0$. Then, the value of $8(

Accepted Answer

For $2x^2 - 6x + k = 0$, the sum and product of roots are:

$\alpha + \beta = \dfrac{6}{2} = 3$
$\alpha \beta = \dfrac{k}{2}$

These two values are the roots of $x^2 + px + p = 0$. Hence, their sum and product are $-p$ and $p$ respectively.

$3 + \dfrac{k}{2} = -p$
$3 	imes \dfrac{k}{2} = p$

Subst

Let α and β be the two distinct roots of the equation 2x2−6x+k=02x^2 - 6x + k = 02x2−6x+k=0, such that (α+β)(\alpha + \beta)(α+β) and αβ\alpha \betaαβ are distinct roots of the equation x2+px+p=0x^2 + px + p = 0x2+px+p=0. Then, the value of 8(k−p)8(k-p)8(k−p) is

Let α and β be the two distinct roots of the equation $2x^2 - 6x + k = 0$ , such that $(\alpha + \beta)$ and $\alpha \beta$ are distinct roots of the equation $x^2 + px + p = 0$ . Then, the value of $8(k-p)$ is