Section A (36 marks) with extension (revision) questions added to each. Since Section A has 45 minutes of exam time, this should take an hour (or less).
1 Express 12 cos q + 5 sin q in the form R sin (q + a). R>0 and 0<a<π/2
Hence solve 12 cos q + 5 sin q = 11 -π < q < π
2 Expand (8 – x) 2/3 to four terms. For what range of values is this valid?
What is the first move when expanding (a + bx)n ? How would you rather express the value a if it has only one prime factor, p? What would you expect a to equal in questions at this level?
3 Solve cosec² ø = 9 for 0 < ø ≤ 2π
Extend your answer to a general solution, i.e. any ø.
4 Solve 3 cos 2x = 1 – sin x for 0º < x < 360º
How would your answer be implied to change if ø had been used instead of x?
What are the other versions of the double angle formula you used?
(PF2+ extension: use hyperbolic functions instead of the circular ones)
5 The curve y = (1 – 2 e –3x) 1/2 is to be rotated around the x-axis. Find the volume enclosed by rotating the line segment between x=0 and x=1 through 360º about the x-axis. Give an exact answer.
Explain how your initial stages of this question would change if the same line segment was to be rotated around the other axis. What happens to the limits?
6 A curve has Cartesian equation x - y² + 48 = 0
Verify that x = t², y = 2t are parametric equations for the curve.
Show that dy/dx = t-1. State an asymptote, identify any stationary values and sketch the curve for the positive quadrant.
Check dy/dx = t-1 against the implicit differential of x - y² + 48 = 0
