GATE AE 2025 • Q9 MCQ • 2 Marks General Aptitude

General Aptitude

Quantitative Aptitude

Subtopic: Arithmetic & Numerical Estimation

Question Statement:

If a real variable $x$ satisfies $3^{x^2} = 27 \times 9^x$, then the value of $\frac{2^{x^2}}{(2^x)^2}$ is:

Options:

  • (A) $2^{-1}$
  • (B) $2^0$
  • (C) $2^3$
  • (D) $2^{15}$

Official Answer Key:

C

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2025 Q.9)** **1. Intuitive Physical Setup & Given Parameters**: We are given that real variable $x$ satisfies: 3^{x^2} = 27 \times 9^x We need to find the value of: E = \frac{2^{x^2}}{(2^x)^2} **2. Step-by-Step Mathematical Derivation**: 1. **Solve for $x$ from the Base-3 Equation**: Express all terms in base 3: 27 = 3^3, \quad 9^x = (3^2)^x = 3^{2x} 3^{x^2} = 3^3 \times 3^{2x} = 3^{x^2} = 3^{2x + 3} Equating exponents: x^2 = 2x + 3 x^2 - 2x - 3 = 0 (x - 3)(x + 1) = 0 \implies x = 3 \quad \text{or} \quad x = -1 2. **Evaluate the Target Expression $E$**: Simplify $E$ using exponent laws: E = \frac{2^{x^2}}{(2^x)^2} = \frac{2^{x^2}}{2^{2x}} = 2^{x^2 - 2x} 3. **Substitute $x^2 - 2x = 3$**: From the quadratic equation above, notice that: x^2 - 2x = 3 for both roots ($x = 3$ and $x = -1$). Therefore: E = 2^{x^2 - 2x} = 2^3 **3. Comprehensive Options & Distractor Analysis**: - **(A) 2^-1**: ❌ **INCORRECT**. Result of algebraic sign errors. - **(B) 2^0**: ❌ **INCORRECT**. Occurs if exponents are mistakenly canceled to 0. - **(C) 2^3**: ✅ **CORRECT**. Since $x^2 - 2x = 3$, $2^{x^2 - 2x} = 2^3$. - **(D) 2^15**: ❌ **INCORRECT**. Exponent multiplication error. **4. Official Master Key & Exam Takeaway**: - **Official Answer Key**: **C** - **Exam Takeaway**: Notice algebraic relationships before computing roots directly: $x^2 - 2x = 3$ directly transforms the denominator exponent without needing to track individual roots.
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