CAT 2023 Slot 3 Solved Paper

It is given that x2+bx+c=0 has two real roots. Let the roots of the equation be α,β⋅(α>β) Then, we can say that ‌

1

α

−‌

1

β

=‌

1

3

....Eq(1)
Similarly, ‌

1

α2

+‌

1

β2

=‌

5

9

....Eq(2)
Eq(2) can be written as (‌

1

α

−‌

1

β

)2+2⋅‌

1

α

⋅‌

1

β

=‌

5

9

‌⇒(‌

1

3

)2+2⋅‌

1

α

⋅‌

1

β

=‌

5

9

‌⇒‌

2

α⋅β

=‌

4

9

⇒‌

1

α⋅β

=‌

2

9

‌⇒α⋅β=‌

9

2

We know that the product of the roots is equal to c, which implies c=‌

9

2

We also know that the sum of the roots is equal to −b.
‌⇒‌

1

α2

+‌

1

β2

=(‌

1

α

+‌

1

β

)2−‌

2

αβ

=‌

5

9

‌⇒(‌

α+β

αβ

)2−‌

4

9

=‌

5

9

‌⇒(‌

α+β

αβ

)2=(1)2
‌⇒α+β=±αβ
Hence, the maximum value of b is ‌

9

2

.
Hence, the maximum value of (b+c) is 9