cat 2023 Question 6

Question

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

Accepted Answer

Given a and b are the distinct roots of the equation $2x^{2} - 6x + k = 0$

⇒ a + b = -(-6/2) = 3 (Sum of the roots)

⇒ ab = k/2 (Product of the roots)

Now, (a+b) and ab are the roots of the quadratic equation $x^{2} + px + p = 0$

⇒ a + b + ab = -p ⇒ 3 + k/2 = -p ---(1)

⇒ (a + b)(ab) = p ⇒ 3(k/2) = p ---(2)

$3+\dfrac{k}{2}=-\dfrac{3k}{2}$ ⇒ 2k = -3 ⇒ k = $-\dfrac{3}{2}$

p = $\dfrac{3k}{2}=\dfrac{3}{2}\left(-\dfrac{3}{2}\right)=-\dfrac{9}{4}$

⇒ 8(k-p) = $8\left(-\frac{3}{2}+\frac{9}{4}\right)=-12+18=6$

Question 6.

Question Explanation