Engineering Mathematics carries 50 questions (200 marks) in the BCECE LE examination. Along with Engineering Mechanics, it forms the core of your merit ranking. Because mathematical problems follow defined theorems and algebraic methods, this section provides the highest scoring potential when formulas and steps are thoroughly memorized.
1. Matrices & Determinants — Formula Cheat Sheet
Matrices and Determinants consistently yield 8 to 10 direct questions in BCECE LE. Key theorems and formulas you must memorize:
Properties of Determinants
- $|AB| = |A| \cdot |B|$ for any two square matrices $A$ and $B$ of the same order.
- $|A^T| = |A|$ (The determinant of a transpose equals the determinant of the original matrix).
- $|k \cdot A| = k^n |A|$ where $n$ is the order of matrix $A$ and $k$ is a scalar constant.
- $|A^{-1}| = 1 / |A|$, provided $|A| \neq 0$.
- If $A$ is an orthogonal matrix ($A A^T = I$), then $|A| = \pm 1$.
Eigenvalues & Eigenvectors Properties
Characteristic Equation: $\det(A - \lambda I) = 0$
Property 1: Sum of Eigenvalues = Trace of Matrix (Sum of main diagonal elements):
$$\sum_{i=1}^n \lambda_i = \text{Trace}(A) = a_{11} + a_{22} + \dots + a_{nn}$$
Property 2: Product of Eigenvalues = Determinant of Matrix:
$$\prod_{i=1}^n \lambda_i = |A|$$
Property 3: If $\lambda$ is an eigenvalue of $A$, then $\lambda^k$ is an eigenvalue of $A^k$, and $k\lambda$ is an eigenvalue of $kA$.
Property 4: The eigenvalues of a triangular or diagonal matrix are simply its principal diagonal elements.
Cayley-Hamilton Theorem
Every square matrix satisfies its own characteristic equation. If $\lambda^2 - 5\lambda + 6 = 0$ is the characteristic equation of matrix $A$, then:
$$A^2 - 5A + 6I = 0 \implies A^{-1} = \frac{1}{6}(5I - A)$$Use this theorem to calculate inverse matrices and higher matrix powers ($A^4, A^5$) without laborious manual multiplications.
2. Differential Calculus — Key Formulas & Theorems
Leibnitz's Theorem for n-th Derivative of a Product
If $u$ and $v$ are functions of $x$ possessing derivatives of $n$-th order, then:
$$(uv)_n = u_n v + n u_{n-1} v_1 + \frac{n(n-1)}{2} u_{n-2} v_2 + \dots + u v_n$$Exam Tip: Choose $v$ as the polynomial function (e.g., $x^2$ or $x$) whose derivatives terminate after 1 or 2 steps!
Euler's Theorem on Homogeneous Functions
If $u = f(x, y)$ is a homogeneous function of degree $n$ in $x$ and $y$, then:
$$x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n \cdot u$$3. Differential Equations — Solution Patterns
First-Order Linear Differential Equation
Standard form: $\frac{dy}{dx} + P(x) y = Q(x)$
- Integrating Factor (I.F.): $\text{I.F.} = e^{\int P(x) dx}$
- General Solution: $y \cdot (\text{I.F.}) = \int Q(x) \cdot (\text{I.F.}) dx + C$
Exact Differential Equations
An equation $M(x,y) dx + N(x,y) dy = 0$ is exact if and only if:
$$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$4. Solved Practice Questions (BCECE LE Standard)
Example 1: Eigenvalues
Problem: Find the sum and product of eigenvalues for the matrix $A = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$.
Solution:
Sum of eigenvalues = $\text{Trace}(A) = 3 + 4 = 7$.
Product of eigenvalues = $|A| = (3 \times 4) - (2 \times 1) = 12 - 2 = 10$.
Direct Answer without solving characteristic quadratic: Sum = 7, Product = 10.