(SEM I) THEORY EXAMINATION 2023-24 ENGINEERING MATHEMATICS-I

This question paper aims to assess the fundamental mathematical skills required for first-year engineering students. It covers linear algebra, vector calculus, partial differentiation, multiple integrals, transformation of variables, eigenvalues, eigenvectors, differential equations, Beta and Gamma functions, and Gauss divergence theorem.

The paper is divided into three sections (A, B, C) to evaluate conceptual clarity, analytical skills, proof-based understanding, and advanced problem-solving abilities.

SECTION A — Short Answer Questions (2 marks × 7 = 14)

This section checks the basic understanding of fundamental mathematical concepts.

Topics include:

1. Eigenvalues & Matrix Algebra

Product and sum of eigenvalues of a 2×2 matrix

2. Symmetry of Curves

Determining symmetry of an algebraic curve around coordinate axes

3. Error Analysis

Propagation of errors in a formula E=RIE = RIE=RI

4. Gamma Function

Evaluation of ratio Γ(14)Γ(34)\Gamma\left(\frac{1}{4}\right)\Gamma\left(\frac{3}{4}\right)Γ(41​)Γ(43​)

5. Beta Function Identity

Proof of relation:

• B(p,q)=B(p+1,q)+B(p,q+1)B(p,q)=B(p+1,q)+B(p,q+1)B(p,q)=B(p+1,q)+B(p,q+1)

6. Vector Calculus

Proof that a vector field is irrotational

7. Unit Normal Vector

Finding a normal vector to an implicitly defined surface at a point

This section tests definitions, theoretical relations, and quick computational skills.

SECTION B — Descriptive, Derivative & Analytical Questions (7 marks × choose any 3 = 21)

Students must solve any three problems involving deeper reasoning.

1. Homogeneous Linear Systems

Solving a system of four-variable linear equations

2. Partial Derivatives & Euler's Theorem

Proving relations involving first and second-order partial derivatives

3. Taylor Series in Two Variables

Expanding f(x,y)=excos⁡yf(x,y)=e^x \cos yf(x,y)=excosy about a given point

4. Double Integration & Change of Variables

Evaluating a double integral by transforming coordinates

5. Directional Derivative

Calculating derivative in the direction of tangent to a parametric curve

This section checks ability in:

Solving linear systems

Using chain rule

Multivariable Taylor expansion

Coordinate transformations

Vector direction calculus

SECTION C — Long Answer / Advanced Problem Solving (7 marks × 4 = 28)

This section tests higher-level understanding of linear algebra, differential equations, multivariable calculus, and vector calculus.

Question 3 — Matrix Algebra (Choose 1)

3(a)

Finding eigenvectors of a 3×3 matrix

3(b) Using Cayley–Hamilton Theorem

Calculating A−1,A−2,A−3A^{-1}, A^{-2}, A^{-3}A−1,A−2,A−3

This tests mastery over eigen-systems and CH theorem.

Question 4 — Advanced Differential Calculus (Choose 1)

4(a)

Proving recurrence relation

Evaluating yny_nyn​ at x=0x=0x=0

4(b) Chain Rule Transformation

Showing transformation identity for multi-variable function z=f(x,y)z=f(x,y)z=f(x,y)

This checks higher-order derivatives and multi-variable consistency relations.

Question 5 — Jacobian & Optimization (Choose 1)

5(a)

Finding Jacobian ∂(u,v)∂(x,y)\frac{\partial(u,v)}{\partial(x,y)}∂(x,y)∂(u,v)​ using given relations

5(b) Lagrange’s Multiplier Method

Finding maximum pressure on the surface of a unit sphere

Here, the focus is on:

Implicit functional differentiation

Constrained optimization

Question 6 — Multiple Integrals & Beta Function (Choose 1)

6(a)

Finding volume bounded by a three-variable surface and coordinate planes

6(b)

Proving Beta function property:

• B(m,n)=Γ(m)Γ(n)Γ(m+n)B(m,n)=\frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}B(m,n)=Γ(m+n)Γ(m)Γ(n)​

This tests calculus of 3D regions and special functions.

Question 7 — Gauss Divergence & Laplacian (Choose 1)

7(a)

Applying Gauss Divergence theorem to a cylindrical surface

7(b)

Proving Laplacian result for rnr^nrn:

• ∇2rn=n(n+1)rn−2\nabla^2 r^n = n(n+1)r^{n-2}∇2rn=n(n+1)rn−2

This section checks:

Vector calculus theorems

Surface integrals and divergence

Radial Laplacian identities