Exams Joint Entrance Examination Mathematics Complex Numbers Question 286 Question 286 of 455 Medium MCQ
The number of real roots of 5 + ∣ 2 x − 1 ∣ = 2 x ( 2 x − 2 ) 5+|2^x-1|=2^x(2^x-2) 5 + ∣ 2 x − 1∣ = 2 x ( 2 x − 2 ) is:
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Let t = 2 x − 1 t=2^x-1 t = 2 x − 1 : 5 + ∣ t ∣ = ( t + 1 ) ( t − 1 ) = t 2 − 1 ⟹ ∣ t ∣ = t 2 − 6 5+|t|=(t+1)(t-1)=t^2-1 \implies |t|=t^2-6 5 + ∣ t ∣ = ( t + 1 ) ( t − 1 ) = t 2 − 1 ⟹ ∣ t ∣ = t 2 − 6 .
For t > 0 t>0 t > 0 : t 2 − t − 6 = 0 ⟹ t = 3 t^2-t-6=0 \implies t=3 t 2 − t − 6 = 0 ⟹ t = 3 or − 2 -2 − 2 (reject); t = 3 ⟹ 2 x = 4 ⟹ x = 2 t=3 \implies 2^x=4 \implies x=2 t = 3 ⟹ 2 x = 4 ⟹ x = 2 .
For t < 0 t<0 t < 0 : t 2 + t − 6 = 0 ⟹ t = − 3 , 2 t^2+t-6=0 \implies t=-3,2 t 2 + t − 6 = 0 ⟹ t = − 3 , 2 , both rejected.
So there is 1 real root. Answer the question to unlock