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Mathematics - COLLEGE ENTRANCE Study Notes

01

Algebra: The Language of Math

Algebra is about finding the unknown. It is the foundation for almost every question in college entrance exams (UPCAT, USTET, DLSUCET).

Order of Operations (PEMDAS)

Always solve in this order: Parentheses, Exponents, Multiplication/Division (Left to Right), Addition/Subtraction (Left to Right).

🚀 PR Hack: Sign Rules

Multiplying two negatives? Result is POSITIVE.
Adding two negatives? Result is NEGATIVE (Stay in debt!).

02

Linear Equations & Inequalities

Solving for x. What you do to one side, you MUST do to the other.

3x + 5 = 17 \to 3x = 12 \to x = 4

⚠️ THE INEQUALITY TRAP

When you Multiply or Divide by a Negative Number, you must FLIP the inequality sign!

-2x < 10 → x > -5

03

Quadratic Equations (ax² + bx + c = 0)

Equations with a squared term. Use factoring or the Quadratic Formula.

x = -b \pm \sqrt(b² - 4ac) 2a

🚀 PR Hack: Vieta's Shortcuts

Sum of roots = -b/a
Product of roots = c/a
Often, you don't even need to solve for x!

04

Polynomials & Special Products

Mastering these patterns will save you minutes during the test.

Name	Pattern
Diff. of Two Squares	a² - b² = (a-b)(a+b)
Square of Sum	(a+b)² = a² + 2ab + b²
Square of Diff.	(a-b)² = a² - 2ab + b²

05

Geometry: Lines & Angles

1
Complementary: Sum is 90° (Right Angle).
2
Supplementary: Sum is 180° (Straight Line).
3
Vertical Angles: Always equal.

06

Triangles Mastery

a² + b² = c²

Common Pythagorean Triples

Save time by memorizing these: 3-4-5, 5-12-13, 8-15-17, and 7-24-25.

⚠️ THE TRIANGLE INEQUALITY

The sum of any two sides of a triangle MUST be greater than the third side.
a + b > c

07

Circles: Pi-Powered Formulas

Circumference

C = 2πr

Area

A = πr²

Pro Tip: Use π ≈ 3.14 or 22/7 depending on the choices.

08

Trigonometry: SOH-CAH-TOA

Basic relationships in right-angled triangles.

Function	Ratio
Sin	Opposite / Hypotenuse
Cos	Adjacent / Hypotenuse
Tan	Opposite / Adjacent

09

Functions & Coordinate Geometry

Slope (m) = y² - y¹ x² - x¹

Slope-Intercept Form

y = mx + b
m = slope, b = y-intercept.

10

Systems of Linear Equations

Solving for two variables (x and y) using two equations.

1
Substitution: Solve for one variable and plug into the other.
2
Elimination: Add or subtract equations to cancel a variable.

11

Word Problems Mastery

Work Shortcut

Together Time =

Product of Times

Sum of Times

Distance Formula

Distance = Rate × Time (D-R-T)

12

Statistics & Probability

Mean

Average

Median

Middle

Mode

Frequent

P(E) = Favorable Outcomes Total Possible

13

Exponents & Logs

Rule	Expression
Product Rule	xª × xº = xªº
Quotient Rule	xª / xº = xªº
Power Rule	(xª)º = xªº

14

Sequences & Series

Finding the nth term of a sequence.

Arithmetic: aº = a¹ + (n-1)d
Geometric: aº = a¹rº¹

15

Entrance Exam Speed Hacks

1
Back-solving: Plug choices into the equation instead of solving from scratch.
2
Elimination: Cross out obviously wrong choices (e.g., negative length).
3
Estimation: Round 4.98 to 5 to quickly find the ballpark answer.

Future Scholar 🎓

You've mastered the essentials for the college entrance math exams.

ACE IT!