GATE AE 2025 • Q16 MCQ • 1 Mark Aerodynamics & Fluid Mechanics

Aerodynamics & Fluid Mechanics

Basic Fluid Mechanics

Subtopic: Conservation Equations of Mass, Momentum & Energy

Question Statement:

If $i = \sqrt{-1}$, then $\frac{(i+1)^3}{i-1} =$ ________.

Options:

  • (A) 2
  • (B) -2
  • (C) (i + 1)
  • (D) (i - 1)

Official Answer Key:

A

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2025 Q.16)** **1. Intuitive Physical Setup & Given Parameters**: We are asked to simplify the complex algebraic expression $\frac{(i + 1)^3}{i - 1}$, where $i = \sqrt{-1}$. **2. Step-by-Step Mathematical Derivation**: 1. **Expand the Numerator $(1 + i)^3$**: Using binomial expansion: (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i (1 + i)^3 = (1 + i)^2 (1 + i) = 2i(1 + i) = 2i + 2i^2 = -2 + 2i 2. **Factor Numerator in terms of Denominator**: Notice that the numerator $-2 + 2i$ can be factored as: -2 + 2i = 2(i - 1) 3. **Division**: \frac{(i + 1)^3}{i - 1} = \frac{2(i - 1)}{i - 1} = 2 Alternatively, converting to polar form: $1 + i = \sqrt{2}e^{i\pi/4} \implies (1+i)^3 = 2\sqrt{2}e^{i3\pi/4}$. Denominator: $i - 1 = -1 + i = \sqrt{2}e^{i3\pi/4}$. Dividing gives $\frac{2\sqrt{2}e^{i3\pi/4}}{\sqrt{2}e^{i3\pi/4}} = 2$. **3. Comprehensive Options & Distractor Analysis**: - **(A) 2**: ✅ **CORRECT**. Factoring $(1+i)^3 = 2(i-1)$ cancels the denominator cleanly, leaving real scalar 2. - **(B) -2**: ❌ **INCORRECT**. Sign error from inverting $(1 - i)$ vs $(i - 1)$. - **(C) (i + 1)**: ❌ **INCORRECT**. Retains imaginary component. - **(D) (i - 1)**: ❌ **INCORRECT**. Incorrect cancellation. **4. Official Master Key & Exam Takeaway**: - **Official Answer Key**: **A** - **Exam Takeaway**: Remember the powers of $(1+i)$: $(1+i)^2 = 2i$, $(1+i)^4 = -4$. Factoring $2(i-1)$ in the numerator avoids tedious rationalization by complex conjugates.
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