Basics & LogarithmsmediumPYQ · JEE Main · 10 Apr 2019 · Shift 2 (Afternoon)Free

Basics & Logarithms: Number Real Roots Equation (JEE Main 2019)

JEE Maths question with a full step-by-step solution.

Question

The number of real roots of the equation

5+|2^{x}-1|=2^{x}(2^{x}-2)

2

1

correct

3

4

Solution

Step 1: Let

t=2^{x}

t>0

. The equation becomes:

5+|t-1|=t(t-2)=t^{2}-2t

Step 2: Case I:

t\ge 1

(i.e.,

2^{x}\ge 1

, so

x\ge 0

). Then

|t-1|=t-1

5+t-1=t^{2}-2t\;\Rightarrow\;t^{2}-3t-4=0\;\Rightarrow\;(t-4)(t+1)=0

t=4

(accepted, since

t\ge 1

) or

t=-1

(rejected).

t=4\;\Rightarrow\;2^{x}=4\;\Rightarrow\;x=2

. One solution. Step 3: Case II:

0<t<1

(i.e.,

x<0

). Then

|t-1|=1-t

5+1-t=t^{2}-2t\;\Rightarrow\;t^{2}-t-6=0\;\Rightarrow\;(t-3)(t+2)=0

t=3

(rejected, not

<1

) or

t=-2

(rejected, not

>0

). No solutions. Step 4: Only one real root:

x=2

. Answer: (2)

1

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