Mathematics Majorship Reviewer 📐

The Math majorship in the LET tests your ability to solve complex problems and your understanding of the "why" behind the formulas. This module focuses on the logical foundations and shortcuts needed for the exam.

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1. Advanced Algebra

Polynomials & Functions

• Remainder Theorem: If a polynomial $f(x)$ is divided by $(x - c)$, the remainder is $f(c)$.
• Factor Theorem: $(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$.
• Quadratic Discriminant ($D = b^2 - 4ac$):
  • $D > 0$: Two real roots.
  • $D = 0$: One real root (repeated).
  • $D < 0$: Two complex roots.
• Logarithms: Remember the change of base formula: $\log_b a = \frac{\log_c a}{\log_c b}$.

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2. Geometry: Beyond the Basics

Circle Theorems

• Inscribed Angle: The measure of an inscribed angle is half the measure of its intercepted arc.
• Tangent-Radius: A tangent to a circle is perpendicular to the radius at the point of tangency.
• Power of a Point: If two chords $AB$ and $CD$ intersect at $P$, then $AP \cdot PB = CP \cdot PD$.

Analytic Geometry

• Distance Formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$
• Slope-Intercept Form: $y = mx + b$
• General Form of a Circle: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center.

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3. Trigonometry: The Unit Circle

Visual Breakdown: The Unit Circle 🎡

(cos θ, sin θ) θ

sin²θ + cos²θ = 1

Essential Identities

• Pythagorean: $\sin^2\theta + \cos^2\theta = 1$
• Sum/Difference: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
• Double Angle: $\sin 2\theta = 2 \sin \theta \cos \theta$

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4. Introduction to Calculus

• Limits: A limit exists if the left-hand limit equals the right-hand limit.
• Derivatives (The Slope):
  • Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$
  • Product Rule: $(uv)' = u'v + uv'$
• Integrals (The Area): The integral is the "anti-derivative."
  • $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

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💡 LET Strategy: Math Major Tips

1. Look for Symmetries: Many geometry and trig problems can be solved faster by recognizing symmetry rather than doing full calculations.
2. Substitution is Valid: In complex algebra problems, if the options are numbers, plug them into the equation to see which one works!
3. Draw it Out: Never solve a word problem or a geometry problem without a quick sketch. It prevents "logic slips."
4. The "Longest Option" pattern in Math:
   • In Math, the longest option is often NOT the correct one. Usually, answers are concise numbers or short expressions.
   • Strategy: If you see one very long, complicated-looking option among three short ones, it might be a "scare tactic" distractor. Solve the problem first!

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🔗 Related Topics

• GenEd: Mathematics - Reviewing the fundamentals.
• Science (Majorship) - Applying calculus and algebra to physics.

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Practice Quiz

Crunch the numbers and master the logic: Try the Math Majorship Quiz.