GATE AE 2025 • Q16
MCQ • 1 Mark
Aerodynamics & Fluid Mechanics
Basic Fluid Mechanics
Subtopic: Conservation Equations of Mass, Momentum & Energy
Question Statement:
If i = √-1,
(i+1)3
i-1 = ________.
Options:
-
(A) 2
-
(B) -2
-
(C) (i + 1)
-
(D) (i - 1)
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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