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