BAS103 Engineering Mathematics-I Exam Solutions March 2025 - Math Traders

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BAS103 Engineering Mathematics-I Exam Solutions March 2025

BAS103 Engineering Mathematics-I Exam Solutions March 2025

BAS103 Engineering Mathematics-I | Complete Exam Solutions (March 2025)

Below are detailed, step-by-step solutions for every question in the BAS103 Engineering Mathematics-I examination paper (March 2025). This post is structured for exam preparation, concept clarity, and easy understanding.

SECTION A (2 × 7 = 14)

Q1(a) Eigenvalues of matrix

Given:
A = [[cosθ, -sinθ], 
     [-sinθ, -cosθ]]

Characteristic equation:
|A - λI| = | cosθ-λ  -sinθ |
            | -sinθ  -cosθ-λ |

= (cosθ-λ)(-cosθ-λ) - sin²θ
= λ² - 1

Eigenvalues:
λ = ±1

Q1(b)

u = (x² + y²) / (x + y)

Homogeneous of degree 1. By Euler’s theorem:
x∂u/∂x + y∂u/∂y = u
= (x² + y²) / (x + y)

Q1(c) Difference between Partial and Total Derivatives

  • Partial derivative: Rate of change with respect to one variable, keeping others constant.
  • Total derivative: Accounts for interdependence of variables.

Q1(d) Applications of Jacobians

  • Change of variables in multiple integrals
  • Thermodynamics
  • Coordinate transformations
  • Fluid mechanics

Q1(e) Liouville’s Theorem

The Jacobian of a canonical transformation is unity.

Q1(f) Evaluate

∫₁³ ∫₁² x² y² dx dy
= ∫₁³ y² [x³/3]₁² dy
= ∫₁³ y² * 7/3 dy
= 7/3 * 26/3 = 182/9

Q1(g) Prove curl r = 0

r = xi + yj + zk
∇ × r = 0
All second partial derivatives vanish.

SECTION B (7 × 3 = 21)

Q2(a) Normal form of matrix

Row and column operations reduce A to rank 3 normal form:
PAQ = [[1 0 0 0],
       [0 1 0 0],
       [0 0 1 0]]

Q2(b) nth derivative of arctan(x/a)

dⁿ/dxⁿ [tan⁻¹(x/a)] = (-1)^(n-1)(n-1)! a / (a²+x²)^(n/2) * sin(nθ)

Q2(c) Largest parallelepiped in ellipsoid

V_max = 8abc / (3√3)

Q2(d) Dirichlet’s theorem

∭ xyz dV = abc / 8

Q2(e) Show f(r)r is irrotational

∇ × (f(r)r) = 0

Suggested Reading:

Integration by Partial Fraction Integration by Parts Example Q2 Integration of e^(ax) sin(bx) Click here for more links

SECTION C (7 × 1 = 7 each)

Q3(a) Eigenvalues and eigenvectors

Eigenvalues: λ = 1, 3, 3
Corresponding eigenvectors from (A - λI)x = 0

Q3(b) Nature of solution

  • Unique if K ≠ 1
  • Infinite solutions if K = 1

Q4(a) Curve tracing

Symmetric about x-axis, intercepts at origin and x = 3a.

Q4(b) Laplacian proof

∇²u = f''(r) + (1/r) f'(r)

Q5(a) Jacobian

∂(x,y,z)/∂(u,v,w) = 1 / (2(x²+y²+z²))

Q5(b) Maxima and minima

Maximum = 3 at (π/2, π/2)

Q6(a) Area between curves

Area = a²(π - 2)

Q6(b) Change order and evaluate

= 16 a⁴ / 3

Q7(a)

∇² rⁿ = n(n+1) r^(n-2)

Q7(b) Stokes theorem verified

LHS = RHS = ab(a+b)

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