Mathematics: Core Skills and Exam Traps
What you must understand
Grade 12 mathematics in Afghanistan draws on algebra, functions, trigonometry, sequences and series, coordinate geometry, and basic probability and statistics. Be fluent with: factoring and the quadratic formula; simplifying rational and radical expressions; the laws of exponents and logarithms; the behaviour of linear, quadratic, exponential, and trigonometric functions; the sine and cosine rules; arithmetic and geometric progressions; the equation of a line and circle; and the meaning of mean, median, mode, and simple probability.
Traps that appear in questions
- Sign errors when expanding brackets or moving terms across an equals sign.
- Domain restrictions: a denominator cannot be zero, and the argument of a logarithm must be positive. Questions often ask for the value that makes an expression undefined.
- Degrees versus radians in trigonometry; check which unit the question uses.
- Geometric series: the sum to infinity exists only when |r| < 1.
- Distractors built from a common mistake, such as forgetting to square when applying the cosine rule.
Worked example
Solve x² − 5x + 6 = 0. Factor as (x − 2)(x − 3) = 0, so x = 2 or x = 3. Check by substitution: 4 − 10 + 6 = 0 and 9 − 15 + 6 = 0. Always substitute back; it catches most arithmetic slips.
Rule of thumb
Write every step on its own line, keep units attached, and finish by checking the answer against the original equation or the wording of the problem.