Complex Numbers And Quadratic Equations question 726
Question: The number of real solutions of the equation $ |x{{|}^{2}} $ - $ 3|x|+2=0 $ are [IIT 1982, 89; MP PET 1997; DCE 2002; AMU 2000; UPSEAT 1999; AIEEE 2003]
Options:
A) 1
B) 2
C) 3
D) 4
Show Answer
Answer:
Correct Answer: D
Solution:
Given $ |x{{|}^{2}}-3|x|+2=0 $ Here we consider two cases $ viz.x<0 $ and $ x>0 $ Case I: $ x<0 $ This gives $ x^{2}+3x+2=0 $
Þ $ (x+2)(x+1)=0\Rightarrow x=-2,-1 $ Also $ x=-1,-2 $ satisfy $ x<0, $ so $ x=-1 $ , - 2 is solution in this case. Case II: $ x>0 $ . This gives $ x^{2}-3x+2=0 $
Þ $ (x-2)(x-1)=0\Rightarrow x=2,1 $ , so $ x=2 $ , 1 is solution in this case. Hence the number of solutions are four i.e. $ x=-1,1,2,-2 $ Aliter: $ |x{{|}^{2}}-3|x|+2=0 $
Þ $ (|x|-1)(|x|-2)=0 $
Þ $ |x|=1 $ and $ |x|=2 $
Þ $ x=\pm 1,x=\pm 2 $ .
BETA
