Basic Mathematics & Logarithm
Exponential and Modulus Equations
Grade Class 11
Question:
<p>Let \(S\) be the set of all real roots of \(3^{3x+2} = x+2-|3^x-1|+|3^x-4|\). Then \(S\): [JEE Main 2020]</p>
is an empty set
contains exactly two elements
contains more than two elements
is a singleton
Step-by-Step Solution
Key Concept: Substitute t = 3^x. Analyse cases based on whether t < 1, 1 \leq t < 4, or t \geq 4. Show the equation has a unique solution.
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. Let $t=3^x>0$. Rewrite the RHS by cases: Case 1 $t<1$: RHS $=x+2-(1-t)+(4-t)=x+5+2t-2=x+3+2t$. Case 2 $1\leq t<4$: RHS $=x+2+(t-1)+(4-t)=x+5$. Case 3 $t\geq4$: RHS $=x+2+(t-1)+(t-4)=x-3+2t$. In Case 2: $3^{3x+2}=x+5$, i.e., $9t^3=x+5$. Analysing shows exactly one solution. $S$ is a singleton. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: D