Complex Numbers And Quadratic Equations question 727
Question: The number of real roots of the equation $ {e^{\sin x}}-{e^{-\sin x}}-4 $ $ =0 $ are [IIT 1982; Pb. CET 2000]
Options:
A) 1
B) 2
C) Infinite
D) None
Show Answer
Answer:
Correct Answer: D
Solution:
Given equation $ {e^{\sin x}}-{e^{-\sin x}}-4=0 $ Let $ {e^{\sin x}}=y $ , then given equation can be written as $ y^{2}-4y-1=0 $
Þ $ y=2\pm \sqrt{5} $ But the value of $ y={e^{\sin x}} $ is always positive, so $ y=2+\sqrt{5}(\because 2<\sqrt{5}) $
Þ $ {\log_{e}}y={\log_{e}}(2+\sqrt{5}) $
Þ $ \sin x={\log_{e}}(2+\sqrt{5})>1 $ which is impossible, since $ \sin x $ cannot be greater than 1. Hence we cannot find any real value of x which satisfies the given equation.
BETA
