Mathematics: Algebra, Functions, and Practical Calculation
Mathematics tests algebra, functions and graphs, logarithms and exponents, trigonometry, geometry, probability, statistics, and practical calculation such as drug doses and dilutions. The exam favors multi-step problems where one algebraic move unlocks the rest.
Core rules and concepts
- Solve equations by performing the same operation on both sides; check for extraneous roots after squaring or multiplying by an expression that can be zero.
- Exponent rules: a^m × a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n. To solve a^x = a^y, equate exponents when the base is the same and positive.
- Logarithms: log(ab) = log a + log b; log(a/b) = log a − log b; log(a^n) = n log a. The argument of a logarithm must be positive.
- In right triangles, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; check whether the angle is in degrees or radians.
- Probability of independent events multiplies; for mutually exclusive events, add. Percentage change = (new − old)/old × 100%, not new/old × 100%.
- A dose stated as mg/kg/day must be multiplied by body mass and then divided by the number of doses per day.
Worked example
Solve 2^(x+1) = 8^(x−1). Rewrite 8 as 2³, so the right side is (2³)^(x−1) = 2^(3x−3). The equation becomes 2^(x+1) = 2^(3x−3). Since the base 2 is positive and not 1, equate exponents: x + 1 = 3x − 3. Then 1 + 3 = 3x − x, so 4 = 2x and x = 2. Substitution confirms 2³ = 8 and 8¹ = 8. The key move is expressing both sides in the same base; without that, guessing may work here but will fail on harder items.
Traps
Do not cancel terms across a sum, such as (x+1)/x = 1. Do not ignore the domain of a log or the possibility of a zero denominator. In statistics, distinguish mean from median when outliers are present, and do not confuse percentage points with percent change.