CSCA maths: a revision checklist

Use this checklist to find the CSCA maths topics that need another look. Try a few tasks and note whether the difficulty is recalling a formula, choosing a method or reading the question.

The grouping follows the official 2025 syllabus. Compare it with the syllabus available for your new sitting. Practising a formula in a problem makes it easier to see when it applies.

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| < a and |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) and cos(α ± β);
• double-angle formulas.

For real logarithms, require a > 0, a ≠ 1 and positive arguments: x > 0, y > 0 in the formulas above. Check these conditions before transforming an expression.

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) when q ≠ 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/n for 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

1. Day 1: diagnostic test, 48 questions in 60 minutes.
2. Day 2: inequalities and functions.
3. Day 3: sequences and derivatives.
4. Day 4: analytic geometry.
5. Day 5: vectors, complex numbers and solid geometry.
6. Day 6: probability and statistics.
7. 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.

Choosing the first revision topic

Start with an area that caused repeated difficulty in the diagnostic. For inequalities, for example, solve (x - 2)(x - 3) < 0: mark the roots, check the signs and obtain 2 < x < 3. Then try a rational inequality where zeros of the denominator must also be excluded.

Return to the task without notes after reviewing it. Add timing once the method is clear. If you still need to look up the formula, keep that topic in the plan before adding much more new material.