CSCA Math Sample Questions: Four Original Problems with Solutions

These CSCA math sample questions show how to turn a short problem into a reliable calculation. Each question is original Passumi practice, written for this guide. They are not official sample questions, recalled exam items or leaked papers. The topics come from the published mathematics syllabus; four examples cannot represent its full coverage or difficulty distribution.

The official CSCA introduction lists Mathematics as 48 objective multiple-choice questions in 60 minutes, available in Chinese or English. Follow your intended university's language requirement. The English terminology below helps you recognize what a question asks before choosing a formula.

For a longer practice session, open CSCA Mathematics practice. First attempt these four problems without reading the solutions, then review both the arithmetic and the reason each method works.

Question 1: Solve a quadratic inequality

Original practice question. Which interval contains exactly the real numbers satisfying x^2 - 5x + 6 <= 0?

• A. x <= 2 or x >= 3
• B. 2 <= x <= 3
• C. 2 < x < 3
• D. x <= -3 or x >= -2

Answer: B.

Factor the quadratic: x^2 - 5x + 6 = (x - 2)(x - 3). The boundary values, also called roots, are 2 and 3. They divide the number line into three regions.

For x < 2, both factors are negative, so their product is positive. Between 2 and 3, the first factor is positive and the second negative, so the product is negative. For x > 3, both factors are positive. Only the middle region satisfies the inequality away from the boundaries.

At x = 2 or x = 3, the product equals zero. Because the question says less than or equal to zero, include both endpoints. In interval notation the solution is [2, 3].

Check with x = 2.5: 6.25 - 12.5 + 6 = -0.25, which satisfies the condition. Option C wrongly excludes the roots. Option A selects the positive regions instead.

English terms to know: quadratic, factor, root, endpoint, interval, non-positive. “Non-positive” means zero or negative; it does not mean strictly negative.

Question 2: Find a function's minimum value

Original practice question. For all real x, let f(x) = x^2 - 6x + 11. What is the minimum value of f(x)?

• A. 2
• B. 3
• C. 9
• D. 11

Answer: A.

Complete the square: x^2 - 6x + 11 = (x - 3)^2 + 2. The square of a real number cannot be negative, so the smallest possible squared term is zero. This happens at x = 3. Therefore f(3) = 2, and 2 is the minimum value.

A common error is answering 3 because that is the input where the minimum occurs. The question asks for the output, not the input. In graph language, the vertex is (3, 2), and the minimum is its vertical coordinate.

You can also check with a derivative: f'(x) = 2x - 6, which is zero at 3. The derivative is negative before 3 and positive after 3, confirming that the function decreases and then increases. Completing the square is shorter here and proves the minimum is global.

The domain matters. This problem allows every real number. If a different problem restricted x to an interval that excluded 3, you would need to inspect the permitted values, often including an endpoint.

English terms to know: domain, range, minimum value, vertex, increasing, decreasing. For this function, the domain is all real numbers and the range is y >= 2.

Question 3: Intersect a line and a circle

Original practice question. The circle x^2 + y^2 = 25 meets the line y = 3 at points A and B. What is the length of segment AB?

• A. 4
• B. 6
• C. 8
• D. 10

Answer: C.

Substitute the line equation into the circle equation: x^2 + 3^2 = 25. Therefore x^2 = 16, giving two solutions, x = -4 and x = 4. The intersection points are A = (-4, 3) and B = (4, 3).

Both points have the same vertical coordinate, so the distance is the horizontal difference: AB = 4 - (-4) = 8 coordinate units. The full distance formula gives the same result: sqrt((4 - (-4))^2 + (3 - 3)^2) = sqrt(64) = 8.

Geometrically, the circle has center (0, 0) and radius 5. The chord is 3 units from the center. A right triangle from the center to the chord midpoint has hypotenuse 5 and one leg 3, so half the chord is 4. Doubling that length gives 8.

Option A forgets to double the half-chord. Option D gives the diameter, which would apply to a line through the center. This line does not pass through the center.

English terms to know: intersection, center, radius, diameter, chord, midpoint. Keep “radius” and “diameter” distinct even when the arithmetic looks familiar.

Question 4: Calculate probability without replacement

Original practice question. A bag contains three red balls and two blue balls. Two balls are drawn at random without replacement. What is the probability that exactly one is red?

• A. 3/10
• B. 2/5
• C. 1/2
• D. 3/5

Answer: D.

There are two successful orders: red then blue, or blue then red. For red followed by blue, the probability is (3/5) × (2/4) = 3/10. After the first draw only four balls remain, which explains the second denominator.

For blue followed by red, the probability is (2/5) × (3/4) = 3/10. The two orders are mutually exclusive: one draw sequence cannot have both first colors. Add their probabilities to obtain 3/10 + 3/10 = 3/5.

An independent check uses unordered pairs. There are 5 × 4 / 2 = 10 possible pairs of distinct balls. Choosing one of three red balls and one of two blue balls gives 3 × 2 = 6 favorable pairs. Thus the probability is 6/10 = 3/5.

“Exactly one red” excludes two reds. “At least one red” would be a different event, including both one-red and two-red outcomes. Sampling with replacement would also change the calculation because the second draw would again have five possible balls.

English terms to know: without replacement, exactly, at least, favorable outcome, mutually exclusive. Underline these conditions before calculating.

Turn four solutions into a study plan

Record why you missed a question, even if you later recognized the answer. An endpoint error calls for inequality practice; confusing minimum input and output calls for function interpretation. A probability error may come from reading “exactly” incorrectly rather than from weak arithmetic.

The wider syllabus also includes sequences, trigonometry, logarithms, vectors, complex numbers, spatial geometry and statistics. Use the CSCA exam guide to confirm your subject plan, and the CSCA preparation hub to find other required papers. These four questions are a diagnostic starting point.

Passumi currently offers the first 25% of each available paper free. Continue with CSCA math practice, keeping a short error log and repeating missed methods with different numbers.

Sources and last checked

Last checked: October 15, 2026. All questions and worked solutions above are original educational examples.

• Official CSCA exam introduction: current language and paper-format information.
• Official CSCA preparation resources: syllabus and official sample downloads.
• GDUFS introduction and reproduced 2025-edition syllabuses: topic coverage; its older calendar is not used for upcoming dates.