The danger in CSCA Mathematics is not only forgetting a difficult formula. It is spending three minutes reconstructing a formula you “almost remember” when 48 questions must fit into 60 minutes.
The official 2025 syllabus has four modules: sets and inequalities, functions, geometry and algebra, and probability and statistics. This is a revision checklist, not a copy of exam questions.
Exam format
- 60 minutes;
- 48 single-answer multiple-choice questions;
- a score from 0 to 100;
- Chinese or English, subject to the chosen university’s rules.
The average is 75 seconds per question. That does not mean every question should take exactly that long: fast questions buy time for difficult ones.
1. Sets and inequalities
Check that you can quickly:
- find unions, intersections and complements;
- work with intervals;
- solve quadratic inequalities;
- respect the domain of rational expressions;
- apply absolute-value properties.
Core tools:
- discriminant:
D = b² - 4ac; - roots:
x = (-b ± √D) / 2a; - the sign of a quadratic from its roots and the direction of the parabola;
- the geometric meaning of
|x| < aand|x| > a.
A common mistake is to solve the numerator of a rational inequality and forget the zeros of the denominator.
2. Functions
The syllabus includes domain, range, monotonicity, parity and the main types of functions.
Review:
- power functions;
- exponentials;
- logarithms;
- trigonometry;
- graph transformations;
- composition and inverse functions.
Minimum formula set:
a^(x+y) = a^x · a^y;log_a(xy) = log_a x + log_a y;log_a(x/y) = log_a x - log_a y;log_a x = ln x / ln a;sin²x + cos²x = 1;sin(α ± β)andcos(α ± β);- double-angle formulas.
Do not memorise an identity without its domain and sign conditions.
3. Sequences
The syllabus covers arithmetic and geometric sequences, including general terms and sums.
Arithmetic sequence
a_n = a_1 + (n - 1)d;S_n = n(a_1 + a_n)/2;S_n = n[2a_1 + (n - 1)d]/2.
Geometric sequence
a_n = a_1q^(n-1);S_n = a_1(1 - q^n)/(1 - q)whenq ≠ 1;- infinite sum
S = a_1/(1-q)when|q| < 1.
4. Derivatives and basic calculus
You need the definition, geometric meaning and simple applications of a derivative.
Review:
- the power rule;
- sums and products;
- tangent lines;
- increasing and decreasing intervals;
- extrema;
- basic optimisation.
Core facts:
(x^n)' = nx^(n-1);(uv)' = u'v + uv';- the tangent slope equals the derivative value;
- a critical point is not always an extremum, so check the sign change.
5. Analytic geometry
The syllabus includes lines, circles, ellipses, hyperbolas and parabolas.
Recognise:
y = kx + b;- distance between two points;
- the midpoint of a segment;
- the circle
(x-a)² + (y-b)² = r²; - standard conic forms;
- foci, vertices and asymptotes in basic problems.
Draw a quick sketch. A ten-second picture often prevents a sign error that memory will not catch.
6. Vectors, complex numbers and solid geometry
Review:
- vector addition and length;
- dot products;
- the angle between vectors;
- basic operations with
a + bi; - modulus of a complex number;
- three-dimensional coordinates;
- areas and volumes of common solids.
Formulas:
a·b = |a||b|cosθ;|a+bi| = √(a²+b²);- 3D distance:
√[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²].
7. Probability and statistics
The syllabus includes classical probability, mean, variance and basic normal distribution.
Check:
P(A) = m/nfor equally likely outcomes;- complement:
P(Ā) = 1 - P(A); - addition rules;
- independent events;
- mean;
- variance;
- the meaning of standard deviation;
- symmetry of a normal distribution.
Do not confuse independent and mutually exclusive events. They are different properties.
A seven-day revision plan
- Day 1: diagnostic test, 48 questions in 60 minutes.
- Day 2: inequalities and functions.
- Day 3: sequences and derivatives.
- Day 4: analytic geometry.
- Day 5: vectors, complex numbers and solid geometry.
- Day 6: probability and statistics.
- Day 7: full test and error-log review.
A formula is not revised when you reread it. It is revised when you apply it to three different timed problems.
Bottom line
CSCA Mathematics tests a broad school foundation under severe time pressure. A secret formula sheet cannot replace automatic recall.
The uncomfortable advice is to stop revising only your favourite topics. Scores are usually lost in the areas you prefer to avoid.
