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)=excosyf(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
