Application of DerivativeHard
Question
a, b, c are three distinct real numbers, which are in G.P. and a + b + c = xb, then -
Options
A.x < -1
B.-1 < x < 2
C.2 < x < 3
D.x > 3
Solution
a + b + c = xb
Divide by b,
+ 1 + 
= x
or
+ 1 + r = x where r is common ration of G.P.
⇒ r2 + r(1 - x) + 1 = 0
since r is real & distinct ⇒ D > 0
⇒ (1 - x)2 - 4 > 0 ⇒ x2 - 2x - 3 > 0
or (x + 1)(x - 3) > 0 ⇒ x > 3 or x < - 1
Divide by b,
+ 1 + = xor
+ 1 + r = x where r is common ration of G.P.⇒ r2 + r(1 - x) + 1 = 0
since r is real & distinct ⇒ D > 0
⇒ (1 - x)2 - 4 > 0 ⇒ x2 - 2x - 3 > 0
or (x + 1)(x - 3) > 0 ⇒ x > 3 or x < - 1
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