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DSE Mathematics: Compulsory Practice

DSE Mathematics — Compulsory Part Practice

18 MCQ practice problems on the compulsory part of the HKDSE Mathematics curriculum. Select an answer, submit, and review the explanation. Questions cover number systems, quadratics, functions, coordinate geometry, trigonometry, and sequences.

Number Systems and Surds

Practice Problem Tips

  • Rationalise denominators: multiply numerator and denominator by the conjugate. For 1/(√3 - 1), multiply by (√3 + 1)/(√3 + 1) to get (√3 + 1)/2. The conjugate of a + b√c is a - b√c.
  • Check domain before solving: for logarithmic equations, ensure all arguments are positive. For square root equations, ensure the radicand is non-negative. Extraneous solutions often appear when squaring both sides.
  • Verify solutions: substitute answers back into the original equation, especially when squaring both sides (which can introduce extraneous solutions). This takes 30 seconds but saves losing marks.
  • Simplify surds fully: √12 = 2√3, not √12. Always extract perfect square factors. √50 = 5√2, √72 = 6√2.
  • Use indices rules: a^m × a^n = a^(m+n), (a^m)^n = a^(mn), a^(-n) = 1/a^n. These appear frequently in algebra and number questions.

Approach Strategy

  • For quadratic equations, first check if factorisation is possible before using the formula. Factorisation is faster and less error-prone when it works.
  • For simultaneous equations, decide whether substitution or elimination is more efficient based on the structure. If one equation has a variable with coefficient 1, substitution is in most cases easier.
  • For trigonometry, sketch the angle in the correct quadrant to determine the sign of the trig ratio. Remember: ASTC (All, Sine, Tangent, Cosine) tells you which ratios are positive in each quadrant.
  • For inequalities, remember to reverse the inequality sign when multiplying or dividing by a negative number. This is the most common source of errors in inequality questions.
  • For functions, identify the type first (linear, quadratic, exponential, logarithmic) and then apply the appropriate technique. Don’t try to solve a logarithmic equation using quadratic methods.

Intuition

Approach each problem by first classifying it: is this an equation to solve, a function to analyse, or a geometric relationship? Then select the appropriate technique from your toolkit. The DSE compulsory maths exam tests breadth of knowledge across many topics. A useful strategy is to scan all questions first, identify the ones you can solve quickly, and return to harder ones later. Time management is as important as mathematical skill.

Worked Examples

Example 1: Surd Simplification

Problem: Simplify 35−1\frac{3}{\sqrt{5} - 1}.

Solution: Step 1: Multiply by the conjugate: 35−1×5+15+1\frac{3}{\sqrt{5} - 1} \times \frac{\sqrt{5} + 1}{\sqrt{5} + 1}

Step 2: Numerator: 3(5+1)=35+33(\sqrt{5} + 1) = 3\sqrt{5} + 3

Step 3: Denominator: (5)2−12=5−1=4(\sqrt{5})^2 - 1^2 = 5 - 1 = 4

Step 4: Answer: 35+34\frac{3\sqrt{5} + 3}{4}

Key insight: Rationalising the denominator removes surds from the bottom. The conjugate of a+bca + b\sqrt{c} is a−bca - b\sqrt{c}, and (a+bc)(a−bc)=a2−b2c(a + b\sqrt{c})(a - b\sqrt{c}) = a^2 - b^2c.

Example 2: Quadratic Word Problem

Problem: A rectangular garden has length (2x+3)(2x + 3) m and width xx m. If the area is 44 m², find xx.

Solution: Step 1: Set up the equation: x(2x+3)=44x(2x + 3) = 44

Step 2: Expand: 2x2+3x=442x^2 + 3x = 44

Step 3: Rearrange: 2x2+3x−44=02x^2 + 3x - 44 = 0

Step 4: Factorise: (2x+11)(x−4)=0(2x + 11)(x - 4) = 0

Step 5: Solve: x=−112x = -\frac{11}{2} (rejected, width cannot be negative) or x=4x = 4

Step 6: Answer: x=4x = 4 m (width), length =2(4)+3=11= 2(4) + 3 = 11 m

Key insight: Always reject solutions that don’t make physical sense (negative lengths, negative time, etc.).

Example 3: Arithmetic Sequence

Problem: The 3rd term of an arithmetic sequence is 11 and the 7th term is 23. Find the first term and common difference.

Solution: Step 1: Set up equations using an=a1+(n−1)da_n = a_1 + (n-1)d:

  • 3rd term: a1+2d=11a_1 + 2d = 11 … (1)
  • 7th term: a1+6d=23a_1 + 6d = 23 … (2)

Step 2: Subtract (1) from (2): 4d=12⇒d=34d = 12 \Rightarrow d = 3

Step 3: Substitute into (1): a1+2(3)=11⇒a1=5a_1 + 2(3) = 11 \Rightarrow a_1 = 5

Step 4: Answer: a1=5a_1 = 5, d=3d = 3

Key insight: For arithmetic sequences, the difference between terms is constant. Subtracting equations eliminates a1a_1 and gives dd directly.

Common Mistakes

  • Incorrect sign when completing the square: (x + 3)² = x² + 6x + 9, not x² + 6x + 3. The constant term is (b/2)². When completing the square for x² + 6x, you add (6/2)² = 9.
  • Forgetting ± in square roots: √(x²) = |x|, not just x. If x² = 9, then x = ±3. This applies whenever you take the square root of both sides of an equation.
  • Confusing arithmetic and geometric sequences: arithmetic has common difference d, geometric has common ratio r. Mixing formulas gives wrong nth terms. Arithmetic: aₙ = a₁ + (n-1)d. Geometric: aₙ = a₁r^(n-1).
  • Errors in logarithm calculations: logₐ(x) = n means a^n = x. Students sometimes write a × n = x instead. The logarithm is the exponent, not a multiplier.
  • Misreading coordinate geometry questions: when asked for the equation of a line through two points, first find the gradient (m = (y₂-y₁)/(x₂-x₁)), then use point-slope form: y - y₁ = m(x - x₁).

Cross-References

  • Functions: Functions are core; composite and inverse functions are frequently tested in the compulsory part.
  • Quadratics: Quadratics are tested in equations, inequalities, coordinate geometry, and word problems.
  • Trigonometry: Trigonometry is covered extensively, including identities and equations in the compulsory syllabus.
  • Physics Kinematics: Kinematics problems directly apply quadratic equations and trigonometric ratios from compulsory maths.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.