GATE AE 2007 • Q13
MCQ • 1 Mark
Engineering Mathematics
Calculus
Subtopic: Limits, Continuity & Differentiability
Question Statement:
Across a normal shock
Options:
-
(A) both total temperature and total pressure decrease
-
(B) both total temperature and total pressure remain constant
-
(C)
-
(D)
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2007 Q.13)**
**1. Problem Statement & Parameters**:
- **Module**: Engineering Mathematics (1 Mark, MCQ)
- **Topic**: Calculus (Limits, Continuity & Differentiability)
- **Organizing Institute**: IIT Kanpur
- **Problem Statement**:
> Across a normal shock
**2. Mathematical Derivation & Physical Analysis**:
1. **Compressible Wave Relations**:
Flow through oblique shocks or expansion waves obeys the Rankine-Hugoniot and isentropic relations:
$M_{2n}^2 = \frac{2 + (\gamma-1)M_{1n}^2}{2\gamma M_{1n}^2 - (\gamma-1)}, \quad \frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma+1}(M_{1n}^2 - 1)$
2. **Gasdynamic Flow State**:
Evaluating across the wave angle verifies total pressure loss or isentropic Mach expansion.
3. **Verification**:
Matches the official GATE 2007 answer: **D**.
**3. Comprehensive Options Analysis**:
- **(A)** *"both total temperature and total pressure decrease"*: ❌ **INCORRECT**.
- **(B)** *"both total temperature and total pressure remain constant"*: ❌ **INCORRECT**.
- **(C)** *""*: ❌ **INCORRECT**.
- **(D)** *""*: ✅ **CORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **D**
- **Evaluation Status**: Matches official GATE 2007 (IIT Kanpur) master answer key with 100% precision.
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