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Given $x 1, x 2$ are the roots of the equation $x 2 − 3 x + A = 0$ and $x 3, x 4$ are the roots of the equation $x 2 − 12 x + B = 0$ such that $x 1, x 2, x 3, x 4$ form an increasing GP. Then find the value of $A + B$ .

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answer is C.

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Detailed Solution

Let the terms

x 1, x 2, x 3, x 4

forms a G.P be

a, a r, a r 2, a r 3

. Given

x 2 − 3 x + A = 0

… (1) and

x 2 − 12 x + B = 0

… (2).
We know that,

Sum of roots = − Coefficient of x Coefficient of x 2

.
By above formula, for (1), we have

x 1 + x 2 = − (− 3) 1

a + a r = 3

… (3)
For (2), and we have

x 3 + x 4 = 12 a r 2 + a r 3 = 12

a r 2 (1 + r) = 12

… (4)
To find r, divide (4) by (3), and we have

a r 2 (1 + r) a (1 + r) = 123 r 2 = 4 r = ± 2

Thus, the value of r is 2. (Neglect the negative value) Substitute this value in the equation (3), and we have

a × (1 + 2) = 3 a × 3 = 3 a = 1

Substitute the value of a and r in the terms of G.P, we get

x 1, x 2, x 3, x 4 = 1, 2, 4, 8

.
We know that,

Product of roots = Constant term Coefficient of x 2

.
For (1), we have

x 1 x 2 = A 1

x 1 x 2 = A

… (5)
For (2), we have

x 3 x 4 = B 1

x 3 x 4 = B

… (6)
Then, the value of

A + B

is,

= x 1 x 2 + x 3 x 4 = (1 × 2) + (4 × 8) = 2 + 32 = 34

Hence, the correct option is 3.

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