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Chapter 4 Practice Paper: Differentiation and Applications — question paper

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QQ/S4AM/CH4

Secondary 4 · Additional Mathematics (G3)

Chapter 4 Practice Paper

Differentiation and Applications

Duration: 40 minutes
Total: 30 marks
Name:
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 4 (C1 Differentiation: rules, stationary points, applications).

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.

  1. 1

    Differentiate each of the following with respect to x.

    1. (a)

      4x³ − 6/x².

      [2]
    2. (b)

      2√x − 5eˣ.

      [2]
    3. (c)

      ln 3x.

      d/dx =[1]
  2. 2

    The curve C has equation y = 3 − 2/x.

    1. (a)

      Find dy/dx.

      [2]
    2. (b)

      The point P lies on C with x-coordinate 1. Write down the coordinates of P and the gradient of the tangent to C at P.

      [2]
    3. (c)

      Hence write down the equation of the tangent at P, and give the equation of the normal at P in the form ax + by = c.

      [2]
  3. 3

    The curve y = 2x³ + 3x² − 36x + 5 has two stationary points.

    1. (a)

      Find dy/dx and show that the x-coordinates of the stationary points are 2 and −3.

      [3]
    2. (b)

      Find the coordinates of each stationary point and determine the nature of each, using the second derivative test.

      [4]
  4. 4

    A curve has equation y = x²√(3x + 1). Find dy/dx in the form x(px + q)/(2√(3x + 1)), stating the values of the constants p and q.

    1. Apply the product and chain rules, and simplify to the required form.

      [6]
  5. 5

    A rectangle has a perimeter of 36 cm. Its sides are x cm and (18 − x) cm.

    1. (a)

      Show that the area A cm² of the rectangle is given by A = 18x − x².

      [1]
    2. (b)

      Find the value of x that gives the maximum area, justifying that it is a maximum.

      [3]
    3. (c)

      State the maximum area and the dimensions of the rectangle.

      area =[2]

— End of paper —