Mathematics
"UPCAT Math? Algebra and Geometry ang puhunan dyan besh. Ito ang foundation ng lahat ng STEM courses sa UP! Let's master the essentials para confident ka sa exam!"
1. Algebra Fundamentals
Algebra is the backbone of UPCAT Math! Approximately 40-50% of the math questions involve algebraic concepts. Let's break down the essentials.
Key Algebraic Operations
| Operation | Formula/Method | Example |
|---|---|---|
| FOIL Method | (a+b)(c+d) = ac + ad + bc + bd | (x+2)(x+3) = x² + 5x + 6 |
| Difference of Squares | a² - b² = (a+b)(a-b) | x² - 9 = (x+3)(x-3) |
| Perfect Square Trinomial | a² + 2ab + b² = (a+b)² | x² + 6x + 9 = (x+3)² |
| Sum of Cubes | a³ + b³ = (a+b)(a²-ab+b²) | x³ + 8 = (x+2)(x²-2x+4) |
| Difference of Cubes | a³ - b³ = (a-b)(a²+ab+b²) | x³ - 27 = (x-3)(x²+3x+9) |
UPCAT Tip:
Memorize factoring patterns! They appear in almost every UPCAT Math section. The quadratic formula x = (-b ± √(b²-4ac))/2a is your lifesaver when factoring fails.
2. Linear & Quadratic Equations
Equation solving is essential! Know how to handle both single-variable and systems of equations.
Linear Equations
- • Standard form: ax + b = c
- • Slope-intercept: y = mx + b
- • Point-slope: y - y₁ = m(x - x₁)
- • Slope formula: m = (y₂-y₁)/(x₂-x₁)
Quadratic Equations
- • Standard form: ax² + bx + c = 0
- • Quadratic formula for solutions
- • Discriminant: b² - 4ac
- • Vertex: x = -b/2a
Discriminant Guide (b² - 4ac):
- • D > 0: Two distinct real roots
- • D = 0: One repeated real root (perfect square)
- • D < 0: No real roots (imaginary)
3. Geometry Essentials
Geometry comprises about 25-30% of UPCAT Math. Focus on formulas and properties!
Essential Geometry Formulas
| Shape | Area | Perimeter/Circumference |
|---|---|---|
| Rectangle | A = l × w | P = 2l + 2w |
| Triangle | A = ½bh | P = a + b + c |
| Circle | A = πr² | C = 2πr = πd |
| Trapezoid | A = ½(b₁+b₂)h | P = sum of all sides |
| Parallelogram | A = bh | P = 2(a + b) |
3D Solids - Volume
- • Cube: V = s³
- • Rectangular Prism: V = lwh
- • Cylinder: V = πr²h
- • Cone: V = ⅓πr²h
- • Sphere: V = (4/3)πr³
3D Solids - Surface Area
- • Cube: SA = 6s²
- • Rectangular Prism: SA = 2(lw+lh+wh)
- • Cylinder: SA = 2πr² + 2πrh
- • Sphere: SA = 4πr²
4. Trigonometry Basics
Basic trig appears in UPCAT! Know SOH-CAH-TOA and special angles.
SOH-CAH-TOA
SOH
sin θ = Opposite/Hypotenuse
CAH
cos θ = Adjacent/Hypotenuse
TOA
tan θ = Opposite/Adjacent
Special Angles to Memorize:
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | √3/3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
5. Word Problems & Applications
UPCAT loves word problems! Here are the common types:
Age Problems
Setup: Let x = present age. Past = x - n years, Future = x + n years
Example: "Maria is 3x older than Juan. In 5 years, she'll be 2x older..."
Rate Problems (Distance = Rate × Time)
D = RT, so R = D/T and T = D/R
Example: "Train A leaves at 50 km/hr, Train B leaves 2 hours later at 70 km/hr..."
Work Problems
Combined rate: 1/T = 1/T₁ + 1/T₂
Example: "Juan can paint a room in 6 hours, Maria in 4 hours. Together?"
Mixture Problems
Concentration × Volume = Amount of solute
6. Number Theory & Sequences
These concepts appear frequently in UPCAT!
Arithmetic Sequence
- • Common difference: d = a₂ - a₁
- • nth term: aₙ = a₁ + (n-1)d
- • Sum: Sₙ = n/2 (a₁ + aₙ)
- • or Sₙ = n/2 [2a₁ + (n-1)d]
Geometric Sequence
- • Common ratio: r = a₂/a₁
- • nth term: aₙ = a₁ × r^(n-1)
- • Sum (finite): Sₙ = a₁(1-rⁿ)/(1-r)
- • Sum (infinite, |r|<1): S = a₁/(1-r)
Number Properties to Remember:
- • Even ± Even = Even | Odd ± Odd = Even | Even ± Odd = Odd
- • Even × Any = Even | Odd × Odd = Odd
- • Divisibility by 3: Sum of digits divisible by 3
- • Divisibility by 9: Sum of digits divisible by 9
- • Perfect squares end in: 0, 1, 4, 5, 6, or 9
7. Practice Questions
1. Factor completely: x² - 5x - 14
Show Answer
(x - 7)(x + 2). Find two numbers that multiply to -14 and add to -5: -7 and +2.
2. A circle has an area of 49π cm². What is its circumference?
Show Answer
14π cm. Area = πr² = 49π, so r² = 49, r = 7. Circumference = 2πr = 2π(7) = 14π cm.
3. In an arithmetic sequence, a₁ = 5 and d = 3. Find a₁₀.
Show Answer
32. Using aₙ = a₁ + (n-1)d: a₁₀ = 5 + (10-1)(3) = 5 + 27 = 32.
4. Solve: 2x² - 5x - 3 = 0
Show Answer
x = 3 or x = -1/2. Factor: (2x + 1)(x - 3) = 0, or use quadratic formula.
UPCAT Math Exam Tips
- • Time management is crucial - approximately 1-2 minutes per question
- • Skip difficult questions and return to them later
- • Use process of elimination for multiple choice
- • Double-check calculations, especially signs (+ and -)
- • Memorize formulas - they're not provided!
- • Practice mental math for faster computation
- • Bring an analog watch - electronic devices are prohibited
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