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Chapter 5 Practice Paper: Exponential and Logarithmic Functions — question paper

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QQ/S3AM/CH5

Secondary 3 · Additional Mathematics (G3)

Chapter 5 Practice Paper

Exponential and Logarithmic Functions

Duration: 40 minutes
Total: 30 marks
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Class:
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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 5 (A6 Exponential and logarithmic functions).

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. Give non-exact numerical answers correct to 3 significant figures unless told otherwise. Throughout the paper, lg x means log₁₀ x and ln x means logₑ x.

  1. 1

    This question is about the laws of logarithms.

    1. (a)

      Evaluate lg 2 + lg 50, showing the law you use.

      [2]
    2. (b)

      Solve the equation lg x = 3.

      x =[2]
    3. (c)

      State the value of ln e.

      ln e =[1]
  2. 2

    Answer the whole of this question about exponential and logarithmic equations.

    1. (a)

      Find x if 5^x = 40. Give your answer to 3 significant figures.

      [3]
    2. (b)

      It is given that log₃ 7 = k. Express log₇ 27 in terms of k.

      [3]
  3. 3

    Solve the equation log₂(3x − 1) = 4.

    1. (a)

      Express the equation without logarithms.

      [2]
    2. (b)

      Hence solve the equation, leaving your answer as a fraction.

      x =[2]
    3. (c)

      Check that your answer is valid for the original equation.

      [2]
  4. 4

    Use the substitution y = 3^x to solve 3^(2x) − 10(3^x) + 9 = 0.

    1. (a)

      Show that, with the substitution y = 3^x, the equation becomes y² − 10y + 9 = 0.

      [2]
    2. (b)

      Find the two possible values of y.

      y =[2]
    3. (c)

      Hence find both values of x, checking each against the original equation.

      [3]
  5. 5

    A physics experiment tracks a decaying sample. Its mass, M grams, t years after the experiment begins, is modelled by M = 80 × (0.9)^t.

    1. (a)

      Find the mass of the sample after 10 years, correct to 3 significant figures.

      [3]
    2. (b)

      Find the number of years the sample takes to halve in mass, correct to 3 significant figures.

      [3]

— End of paper —