Extensions added to each question. This is roughly Section A standard, which is to take 45 minutes in exam conditions, so this should take about an hour at this time of year.
This sheet is a little harder than 2006.C4.1.
1 Express 8 cos ø + 15 sin ø in the form R cos (ø + ∂) where R > 0 and
|∂| < π/2.Hence write the range of ß for which there are solutions to
8 cos ø + 15 sin ø = ß-π < ø < π
2 Expand (4 – x) 3/2 to four terms. For what range of values is this valid? Use your answer to estimate 3.993/2 and comment on the precision achieved in using four terms instead of three.
What is precision? Your formula book gives some words that have precise meanings in their use at A-level. You could compare your answer with the calculator of with the size of the next term.
3 Solve cot² ø = 36 for 0 < ø < 2π
Extend your answer to a general solution, i.e. any ø.
4 Solve 3 cos² 2x + sin² x – 1 = 0 for 0º < x < 360º
Express your answers in the form cos x = p/q where p & q are integers or surds before transforming to degrees.
5 i) The manufacturers of a particular brand of soap powder are concerned that the number, n, of people buying their product at any time t has remained constant for some months. They launch a major advertising programme which results in the number of customers increasing at a rate proportional to the square root of n. express the progress of sales as a differential equation (a) before and (b) after advertising.
ii)If N0 is the number of customers at the start of the advertising campaign show that, for some constant, µ, the number of customers is of the form
n = N0 + µt √N0 + µ² t² / 4
Criticise the model. How would a multiplier of e–kt improve the solution?
6 The parametric equations of a curve are x = et and y = sin t. Find the first and second differentials of y with respect to x (written y’ and y”) as functions of t.
Hence show that x² y” + x y’ + y = 0.
