Complex Numbers And Quadratic Equations question 767
Question: The equation $ {x^{(3/4){{({\log_2}x)}^{2}}+({\log_2}x)-5/4}}=\sqrt{2} $ has [IIT 1989]
Options:
A) At least one real solution
B) Exactly three real solutions
C) Exactly one irrational solution
D) All the above
Show Answer
Answer:
Correct Answer: D
Solution:
For the given equation to be meaningful we must have $ x>0 $ . For $ x>0 $ the given equation can be written as $ \frac{3}{4}{{({\log_2}x)}^{2}}+{\log_2}x-\frac{5}{4}={\log_{x}}\sqrt{2}=\frac{1}{2}{\log_{x}}2 $
Þ $ \frac{3}{4}t^{2}+t-\frac{5}{4}=\frac{1}{2}( \frac{1}{t} ) $ By putting $ t={\log_2}x $ so that $ {\log_{x}}2=\frac{1}{t} $ because $ {\log_2}x{\log_{x}}2=1 $ .
Þ $ 3t^{3}+4t^{2}-5t-2=0\Rightarrow (t-1)(t+2)(3t+1)=0 $
Þ $ {\log_2}x=t=1,-2,-\frac{1}{3} $
Þ $ x=2,{2^{-2}},{2^{-1/3}} $ or $ x=2,\frac{1}{4},\frac{1}{{2^{1/3}}} $ Thus the given equation has exactly three real solutions out of which exactly one is irrational namely $ \frac{1}{{2^{1/3}}} $ .
BETA
