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

CAT 2023 Slot 1 Solved Paper

Question : 51 of 66

Marks: +1, -0

Let

α

and

β

be the two distinct roots of the equation

2 x^2-6 x+k=0

, such that

(α+β)

and

α β

are the distinct roots of the equation

x^2+p x+p=0

. Then, the value of

8( {\k}- {\p})

Solution:

Given

{\a}

and

{\b}

are the distinct roots of the equation

2 x^2-6 x+k=0

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

\;⇒ a b=k \/ 2

(Product of the roots)
Now,

( {\a}+ {\b})

and

{\ab}

are the roots of the quadratic equation

x^2+p x+p=0

\;⇒a+b+a b=-p⇒3+k \/ 2=-p--(1)

\;⇒(a+b)(a b)=p⇒3(k \/ 2)=p--(2)

\;3+\;{k}/{2}=-\;{3 k}/{2}⇒2 k=-3⇒k=-\;{3}/{2}

\;p=\;{3 k}/{2}=\;{3}/{2} (-\;{3}/{2}) =-\;{9}/{4}

\;⇒8(k-p)=8 (-\;{3}/{2}+\;{9}/{4}) =-12+18=6

Go to Question:

Prev Question

Next Question