| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2012 |
| Session | June |
| Marks | 15 |
| Topic | Trig Proofs |
| Type | Prove trigonometric identity |
| Difficulty | Challenging +1.2 This is a multi-part question combining differentiation, range-finding with trigonometry, parametric curves, and algebraic manipulation. Part (i) is routine differentiation. Part (ii) requires finding stationary points and evaluating endpoints—standard but requires care with trigonometric values. Parts (iii)-(iv) involve algebraic manipulation of trigonometric identities and geometric interpretation, which is moderately challenging. Part (v) is standard parametric differentiation. Overall, this requires competent technique across several topics but no particularly novel insights, placing it somewhat above average difficulty. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07n Stationary points: find maxima, minima using derivatives1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{df}{dt} = 2\cos t - 2\sin 2t = 2\cos t - 4\sin\cos t = 2\cos t(1-2\sin t)\) | M1, A1 [2] | \(k\cos t + m\sin 2t\); AG |
| (ii) Find local maxima/minima: \(f'(x) = 0\) implies \(x = \frac{1}{4}\pi, \frac{5}{6}\pi, \frac{7}{6}\pi, \frac{11}{6}\pi\) | M1, A1, A1 | Any four; All eight |
| Values of \(f\): \(1, -3, 1.5, 1.5\) | ||
| Values of \(f\) at endpoints: \(1, 1\) | B1 | |
| Hence range is \([-3, 1.5]\) | A1 [5] | |
| (iii) Substitute for \(x\) and \(y\), and multiply out: \((2\cos t + \sin 2t)^2 + (2\sin t + \cos 2t)^2\) | M1 | Including cross-terms |
| \(= 4\cos^2 t + 4\cos t\sin 2t + \sin^2 2t + 4\sin^2 t + 4\sin t\cos 2t + \cos^2 2t\) | ||
| \(= 4(\cos^2 t + \sin^2 t) + (\cos^2 2t + \sin^2 2t) + 4(\cos t\sin 2t + \sin t\cos 2t)\) | DM1 | Pythagorean identity OR addition formula |
| \(= 5 + 4\sin(t + 2t) = 5 + 4\sin 3t\) | A1 [3] | AG |
| (iv) \(x^2 + y^2 = r^2\) is a circle centre the origin; \(5 + 4\sin 3t \in [1, 9]\) | B0, 1, 2 | Either statement B1; Both and conclusion B2 |
| so \(C\) lies between and on circles of radius 1 and 3 | ||
| (v) \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2\cos t - 2\sin 2t}{2\cos t - 4\sin\cos t} = \frac{-2\sin t + 2\cos 2t}{-2\sin t + 2\cos 2t}\) | M1 | \(\frac{a\cos t + b\sin 2t}{c\sin t + d\cos 2t}\) |
| at \(t = 0\), \(\frac{dy}{dx} = \frac{2-0}{-0+2} = 1\) | A1, A1 [3] | [15] |
**(i)** $\frac{df}{dt} = 2\cos t - 2\sin 2t = 2\cos t - 4\sin\cos t = 2\cos t(1-2\sin t)$ | M1, A1 [2] | $k\cos t + m\sin 2t$; AG
**(ii)** Find local maxima/minima: $f'(x) = 0$ implies $x = \frac{1}{4}\pi, \frac{5}{6}\pi, \frac{7}{6}\pi, \frac{11}{6}\pi$ | M1, A1, A1 | Any four; All eight
Values of $f$: $1, -3, 1.5, 1.5$ | |
Values of $f$ at endpoints: $1, 1$ | B1 |
Hence range is $[-3, 1.5]$ | A1 [5] |
**(iii)** Substitute for $x$ and $y$, and multiply out: $(2\cos t + \sin 2t)^2 + (2\sin t + \cos 2t)^2$ | M1 | Including cross-terms
$= 4\cos^2 t + 4\cos t\sin 2t + \sin^2 2t + 4\sin^2 t + 4\sin t\cos 2t + \cos^2 2t$ | |
$= 4(\cos^2 t + \sin^2 t) + (\cos^2 2t + \sin^2 2t) + 4(\cos t\sin 2t + \sin t\cos 2t)$ | DM1 | Pythagorean identity OR addition formula
$= 5 + 4\sin(t + 2t) = 5 + 4\sin 3t$ | A1 [3] | AG
**(iv)** $x^2 + y^2 = r^2$ is a circle centre the origin; $5 + 4\sin 3t \in [1, 9]$ | B0, 1, 2 | Either statement B1; Both and conclusion B2
so $C$ lies between and on circles of radius 1 and 3 | |
**(v)** $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{2\cos t - 2\sin 2t}{2\cos t - 4\sin\cos t} = \frac{-2\sin t + 2\cos 2t}{-2\sin t + 2\cos 2t}$ | M1 | $\frac{a\cos t + b\sin 2t}{c\sin t + d\cos 2t}$
at $t = 0$, $\frac{dy}{dx} = \frac{2-0}{-0+2} = 1$ | A1, A1 [3] | [15]The function f is defined by $f : t \mapsto 2 \sin t + \cos 2t$ for $0 \leqslant t < 2\pi$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{df}{dt} = 2 \cos t(1 - 2 \sin t)$. [2]
\item Determine the range of f. [5]
\end{enumerate}
A curve $C$ is given parametrically by $x = 2 \cos t + \sin 2t$, $y = f(t)$ for $0 \leqslant t < 2\pi$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Show that $x^2 + y^2 = 5 + 4 \sin 3t$. [3]
\item Deduce that $C$ lies between two circles centred at the origin, and touches both. [2]
\item Find the gradient of the tangent to $C$ at the point at which $t = 0$. [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2012 Q11 [15]}}