| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | Specimen |
| Marks | 11 |
| Topic | Harmonic Form |
| Type | Range of rational function with harmonic denominator |
| Difficulty | Challenging +1.2 This requires expressing the denominator in harmonic form R cos(θ - α) to find its range, then inverting for the fraction's extrema. While it involves multiple techniques (harmonic form, finding R and α, solving trigonometric equations), these are standard Pre-U/Further Maths procedures with no novel insight required. The multi-step nature and need for careful algebraic manipulation elevate it above average difficulty. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
Attempt expression of $\cos\theta + \sqrt{2}\sin\theta$ in any of the forms $R\cos(\theta \mp \alpha)$ or $R\sin(\theta \pm \alpha)$ **M1**
Obtain e.g. $R\cos\alpha = 1$ **A1**
And $R\sin\alpha = \sqrt{2}$ **A1**
Solve to obtain $R = \sqrt{3}$ **A1**
And e.g. $\alpha = 54.7°$ or $0.955$ rad **A1**
Attempt to link at least one critical value with a value of $\theta$ **M1**
State that $\sqrt{3}$ corresponds to $\theta = 54.7°$ or $0.955$ rad **A1**
State that $-\sqrt{3}$ corresponds to $\theta = 234.7°$ or $4.097$ rad **A1**
Identify maximum as $\dfrac{1}{2-R}$ and/or minimum as $\dfrac{1}{2+R}$ **M1**
State maximum as $\dfrac{1}{2-\sqrt{3}}$, o.e., and $234.7°$ o.e. **A1**
State minimum as $\dfrac{1}{2+\sqrt{3}}$, o.e., and $54.7°$ o.e. **A1**
**Total: 11 marks**12 Calculate the maximum and minimum values of $\frac { 1 } { 2 + \cos \theta + \sqrt { 2 } \sin \theta }$ and the smallest positive values of $\theta$ for which they occur.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q12 [11]}}