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CAT 2023 Slot 1 QA Question & Solution

AlgebraMedium

Question

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

Solution

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