Question

If a, b be roots of x² – 3x + a = 0 and g, d be roots of x² – 12x + b = 0 and a, b, g, d, (in order) form an increasing G.P. Then

• A. a = 3, b = 12
• B. a = 12, b = 3
• C. a = 2, b = 32
• D. a = 4, b = 16

Answer 1

Answer: (C)

a = 2, b = 32

Answer 2

since a, b be roots of x² – 3x + a = 0

\ a + b = 3, ab = a

since g, d be roots of x² – 12x + b = 0

\ g + d = 12, gd = b

Q $\frac { \alpha } { \beta }$ = $\frac { \gamma } { \delta }$

Ž $\frac { \alpha \beta } { ( \alpha + \beta ) ^ { 2 } }$= $\frac { \gamma \delta } { ( \gamma + \delta ) ^ { 2 } }$

$\frac { \mathrm { a } } { 9 }$ = $\frac { \mathrm { b } } { 144 }$

16a = b

Let r be the common ratio of a, b, g, d

then b = ar, g = ar² and d = ar³

Q a and b be roots of equation x² – 3x + a = 0

\ a + b = a(1 + r) = 3 .........(i)

ab = a(ar) = a ..........(ii)

Q g and d be roots of equation x² – 12x + b = 0

\ g + d = ar²(1 + r) = 12 ........(iii)

gd = (ar²) (ar³) = b Ž a² r⁵ = b ........(iv)

from (iii) ø (i) r² = 4 Ž r = 2

then from (i), a = 1 Ž a = 2

b = 2⁵ =32