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Chapter 1 Practice Paper: Quadratic Functions, Equations and Inequalities — question paper

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QQ/S3AM/CH1

Secondary 3 · Additional Mathematics (G3)

Chapter 1 Practice Paper

Quadratic Functions, Equations and Inequalities

Duration: 40 minutes
Total: 30 marks
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Class:
Register No.:
Date:

READ THESE INSTRUCTIONS FIRST

  • Write your name, class and register number in the spaces at the top of this page.
  • Answer all questions.
  • Show all necessary working. Omission of essential working will result in loss of marks.
  • Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.
  • The number of marks is given in brackets [ ] at the end of each question or part question.
  • The total number of marks for each paper is shown on its cover.

Original practice material written for the MOE Additional Mathematics syllabus. It is not an official MOE or school examination paper. Chapter 1 (A1 Quadratic functions, A2 Equations and inequalities).

For Examiner’s Use
Section A/ 30
Total/ 30

Section A: Structured questions[30 marks]

Answer all questions in the spaces provided. Show all necessary working — omission of essential working will result in loss of marks.

  1. 1

    Answer the whole of this question on the quadratic expression 2x² + 8x − 7.

    1. (a)

      Express 2x² + 8x − 7 in the form a(x + p)² + q, stating the value of each constant.

      [3]
    2. (b)

      Hence state the coordinates of the minimum point of the curve y = 2x² + 8x − 7.

      minimum point =[1]
    3. (c)

      Solve the equation 2x² + 8x − 7 = 3, giving each answer exactly.

      x =[1]
  2. 2

    The curve y = x² + 4x + 7 and the line y = 2x + k, where k is a constant, may intersect.

    1. (a)

      Show that the x-coordinates of any point of intersection satisfy x² + 2x + (7 − k) = 0.

      [1]
    2. (b)

      Find the value of k for which the line is a tangent to the curve.

      k =[2]
    3. (c)

      Find the range of values of k for which the line intersects the curve at two distinct points.

      range of k =[2]
    4. (d)

      When k takes the value found in part (b), state the coordinates of the point of tangency.

      point of tangency =[1]
  3. 3

    Solve the pair of simultaneous equations

    Simultaneous equations for Question 3

    Equation 1Equation 2
    x − y = 2x² + y² = 20
    1. (a)

      Show that the equation reduces to x² − 2x − 8 = 0.

      [2]
    2. (b)

      Solve the equation x² − 2x − 8 = 0.

      x =[2]
    3. (c)

      Write down both points of intersection of the line and the curve.

      points =[2]
  4. 4

    This question is about the inequality 2x² − 7x + 3 ≤ 0.

    1. (a)

      Factorise 2x² − 7x + 3.

      factorisation =[2]
    2. (b)

      Find the solution set of 2x² − 7x + 3 ≤ 0, showing it on a number line.

      [3]
    3. (c)

      State the largest integer satisfying the inequality.

      largest integer =[1]
  5. 5

    A rectangular community garden bed is built against a wall. Fencing is placed along the other three sides only, and 12 m of fencing is available. The width of the bed is x metres and its length is (12 − 2x) metres.

    1. (a)

      Show that the area A m² of the bed can be written as A = −2(x − 3)² + 18.

      [2]
    2. (b)

      Using your answer to part (a), state the maximum possible area of the bed and the corresponding value of x.

      maximum area =[2]
    3. (c)

      Find the range of values of x for which the area of the bed is greater than 8 m². Give your answers in exact form.

      [3]

— End of paper —