4 SEMESTER MATHS UNIT 5

degree 4th sem physics important questions

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4 Semester Maths Important questions

4 Semester Maths Most Important Questions

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4 Semester Physics important questions

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**Physics Question Paper**  

**Total Marks: 75**  

**Time: 3 Hours**

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### **Unit - 1: Gauss Theorem and Electrostatics**  

1. **Gauss Theorem**  

   a. State and mathematically express Gauss’s theorem in electrostatics. (2 Marks)  

   b. Using Gauss’s theorem, derive the electric field due to a uniformly charged sphere. (5 Marks)  

2. **Electricity and Magnetism**  

   a. Define electric flux and write its SI unit. (1 Mark)  

   b. Explain the concept of a uniformly charged sphere and derive the electric field inside and outside the sphere. (5 Marks)  

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### **Unit - 2: Dielectrics and Polarization**  

3. **D, E, and P Relationship**  

   a. Define electric displacement vector (D), electric field (E), and polarization (P). (2 Marks)  

   b. Derive the relationship between D, E, and P in a dielectric medium. (5 Marks)  

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### **Unit - 3: Hall Effect and Transformers**  

4. **Hall Effect**  

   a. Explain the Hall Effect and derive the expression for Hall voltage. (5 Marks)  

   b. Discuss the applications of the Hall Effect in modern technology. (5 Marks)  

5. **Transformers**  

   a. Explain the working principle of a transformer. (3 Marks)  

   b. Derive the expression for the efficiency of a transformer. (5 Marks)  

**OR**  

6. **LCR Series AC Circuit**  

   a. Derive the expression for impedance in an LCR series AC circuit. (5 Marks)  

   b. Explain the concept of resonance in an LCR circuit and derive the expression for resonant frequency. (3 Marks)  

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### **Unit - 4: Maxwell’s Equations and Semiconductors**  

7. **Maxwell’s Equations**  

   a. Write all four Maxwell’s equations in differential form and explain their significance. (5 Marks)  

   b. Define displacement current and explain its role in Maxwell’s equations. (5 Marks)  

8. **PN Junction Diode**  

   a. Explain the formation of a PN junction diode and its I-V characteristics. (5 Marks)  

   b. Discuss the applications of a Zener diode in voltage regulation. (5 Marks)  

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### **Unit - 5: Transistors and Logic Gates**  

9. **Transistor**  

   a. Explain the working of a transistor as an amplifier. (5 Marks)  

   b. Define De Morgan’s theorems and prove them using logic gates. (5 Marks)  

10. **Logic Gates**  

    a. Draw the circuit diagrams and truth tables for AND, OR, and NOT gates. (5 Marks)  

    b. Explain the working of a half adder and a full adder circuit. (5 Marks)  

11. **Universal Gates**  

    a. Prove that NAND and NOR gates are universal gates. (5 Marks)  

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### **Additional Questions**  

12. **Q-Factor**  

    a. Define Q-factor and derive its expression for an LCR circuit. (5 Marks)  

13. **Applications of Hall Effect**  

    a. Discuss any three practical applications of the Hall Effect. (5 Marks)  

14. **Displacement Current**  

    a. Explain the concept of displacement current and its significance in Maxwell’s equations. (5 Marks)  

15. **Zener Diode**  

    a. Explain the working principle of a Zener diode and its use as a voltage regulator. (5 Marks)  

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**End of Question Paper**  

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### **Mark Distribution**  

- **Unit 1:** 13 Marks  

- **Unit 2:** 7 Marks  

- **Unit 3:** 10 Marks  

- **Unit 4:** 15 Marks  

- **Unit 5:** 20 Marks  

- **Additional Questions:** 10 Marks  

This question paper is designed to cover all the topics comprehensively, ensuring students can aim for a perfect score of 75/75. Each question is weighted appropriately to balance the difficulty and importance of the topics.

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4 Semester Maths questions and answers

### **Unit 1: Vector Spaces**

1. Define a vector space and provide examples.

2. Prove that the set of all real-valued functions forms a vector space.

3. What is a subspace? Give examples and non-examples.

4. Prove that the intersection of two subspaces is a subspace.

5. Define linear independence and dependence of vectors. Provide examples.

6. Find a basis for the vector space \( \mathbb{R}^3 \).

7. State and prove the Rank-Nullity thereom

### **Unit 2: Linear Transformations**

1. Define a linear transformation and provide examples.

2. Prove that the kernel and image of a linear transformation are subspaces.

3. Find the matrix representation of a given linear transformation.

4. Determine whether a given transformation is injective, surjective, or bijective.

5. State and prove the Dimension Theorem for linear transformations.

6. Compute the rank and nullity of a given matrix.**Unit 3: Matrices and Determinants**

1. Define the determinant of a matrix and explain its properties.

2. Compute the determinant of a 3x3 or 4x4 matrix.

3. Prove that the determinant of a product of matrices is the product of their determinants.

4. Define the adjoint and inverse of a matrix. Find the inverse of a given matrix.

5. Solve a system of linear equations using Cramer's Rule.

6. Define eigenvalues and eigenvectors. Find the eigenvalues and eigenvectors of a given matrix.

Unit 4: Inner Product Spaces**

1. Define an inner product space and provide examples.

2. Prove the Cauchy-Schwarz inequality in an inner product space.

3. Define orthogonality and orthogonal projection. Find the orthogonal projection of a vector onto a subspace.

4. Apply the Gram-Schmidt process to orthogonalize a set of vectors.

5. Define a symmetric matrix and prove that its eigenvalues are real.

6. Diagonalize a given symmetric m

atrix.

### **Unit 5: Canonical Forms**

1. Define the Jordan canonical form of a matrix.

2. Find the Jordan canonical form of a given matrix.

3. Define the minimal polynomial of a matrix and explain its significance.

4. Prove that every square matrix satisfies its characteristic equation (Cayley-Hamilton Theorem).

5. Define the rational canonical form of a matrix.

# **Practice Problems**

1. Solve systems of linear equations using matrix methods.

2. Find the eigenvalues and eigenvectors of a given matrix.

3. Diagonalize a matrix, if possible.

4. Compute the determinant, rank, and inverse of a matrix.

5. Apply the Gram-Schmidt process to a set of vectors.

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