C Module Tips

Below are some suggestions for students who are sitting papers for modules C1 – C4 in GCE Mathematics:

Sketches

  • A sketch should indicate the overall shape of the entire curve. It should not be an accurate graph of a section of the curve.
  • You are not expected to sketch curves on graph paper.
  • To sketch a line, 2 points need to be labelled.
  • To sketch a quadratic, you should have the correct basic shape as well as the points where the graph crosses the axes clearly labelled.
  • Turning points do not have to be labelled (unless they have been asked for in a previous part).
  • Sketches are often asked for in part (i) of a question. Sketches are intended to help you check that your answers to later parts look sensible.

    Example: Paper C1 May 2005, Question 7

    Given the equations x + y = 12 and y = x2 –5x

    (i) Sketch the graphs.

    (ii) Find the points of intersection.

    The sketches made in (i) should confirm that the solutions to (ii) are sensible.

Trigonometric Graphs

  • For sine and cosine graphs:
    • show the basic wave shape;
    • show the turning points; and
    • label the points where the graphs cross the axes.
  • For a tangent graph:
    • show the basic shape;
    • label the points where it crosses the axes; and
    • indicate and label the asymptotes.

Show Full Development

  • You must show the development of your answer.
  • If you use a graphic calculator, you must write down sufficient steps of the solution to let the examiner see that you know how to solve the problem.
  • When solving a quadratic inequality, you must show how you selected the valid region.

'Show That'

  • ‘Show that’ means the same as ‘find’. The result is given so that if you cannot do the first part of the question, you can still attempt the later parts.

    Example: Paper C2 May 2005, Question 2

    The value of a car depreciates by 20% each year. The original cost of the car was £14000.

    • Show that the value, V, of the car after n years is given by
      V       = 14000 × 0.8n
    You need to find the expression for V as if it had not been given.

Transformations

  • Remember to transform the whole graph.

    Example: Paper C1 January 2005, Question 2

    A sketch of the curve y = f(x) is provided; the curve cuts the y-axis at A(0, 3) and has a minimum turning point at (2, −1).

    (iii) Sketch the transformation from y = f(x) to y =3f(x).

    Many candidates who knew the transformation was a stretch parallel to the y -axis and showed the image of A at (0, 9) left the minimum turning point at (2, −1) instead of (2, −3).

Turning Points

  • If you are asked to find the turning point on a curve, you must positively identify the sort of turning point it is. You can use the sign of the second derivative at the turning point, or consider the sign of the first derivative just to either side of the turning point.

    Example: Paper C1 January 2005, Question 5

    (ii) Show that

    (iii) Find the value of x that gives the minimum value of A.

    It is not enough to find the value of x that gives the turning point. You must also verify that the turning point is a minimum.

Common Errors

  • Sequences: ‘converging to infinity’ is not acceptable (‘Divergent’ is what is meant).
  • Solving quadratic equations by factorising : remember to rearrange the expression into the form f(x) = 0 before factorising f(x).
  • Solving cubic equations : remember to include the first factor you find.
  • Solving trigonometric equations : factorise instead of dividing through by a common factor.

    Example:
    Solve the equation: 2sinxcosx = sinx for 0 ≤ x <2π
    Rearrange to give 2sinxcosx – sinx = 0
    Factorise to get sinx(2cosx – 1) = 0
    Therefore sinx = 0 or 2cosx – 1 = 0
    (Dividing through by sinx leaves 2cosx = 1, so sinx = 0 is missed)