A\alpha and B\beta (\alpha, \beta = 1,2,3, ⋯ , n) are contravariant and covariant vectors,
respectively. By convention, any repeated indices are summed over. Which of the
following expression is/are tensors?
Options:
(A) A\alphaB\beta
(B) A\alphaB\beta
A\alphaB\alpha
(C) A\alpha
B\beta
(D) A\alpha + B\beta
Official Answer Key:
A; B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2024 Q.51)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 5: Thermodynamics and Statistical Physics* | **Topic**: *Classical Thermodynamics* (`Entropy & Thermodynamic Potentials (U, H, F, G)`)
- **Question Type**: **MCQ** | **Marks**: **2 Marks**
- **Theoretical Framework**: Governed by foundational physical principles in **Classical Thermodynamics** (`Entropy & Thermodynamic Potentials (U, H, F, G)`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `A𝛼B𝛽
A𝛼B𝛼`**.
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.