| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find equation of tangent |
| Difficulty | Standard +0.3 Part (a) is a standard bookwork proof using the quotient rule on sin x/cos x. Part (b) requires product rule, substitution of x=π/4, finding the tangent equation, and algebraic manipulation. This is a routine multi-step C3 question with no novel insight required, slightly easier than average due to straightforward application of standard techniques. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07m Tangents and normals: gradient and equations1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{d}{dx}(\tan x) = \frac{d}{dx}\left(\frac{\sin x}{\cos x}\right)\) | M1 A1 | |
| \(= \frac{\cos x \times \cos x - \sin x \times (-\sin x)}{\cos^2 x}\) | M1 A1 | |
| \(= \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x\) | M1 A1 | |
| (b) \(\frac{dy}{dx} = 2x\tan x + 2x \times \sec^2 x = 2\tan x + 2x\sec^2 x\) | M1 A1 | |
| At \(x = \frac{\pi}{4}\), \(y = \frac{\pi}{2}\), grad \(= 2 + \pi\) | B1 | |
| \(y - \frac{\pi}{2} = (2 + \pi)\left(x - \frac{\pi}{4}\right)\) | M1 | |
| At \(P\), \(x = 0\): \(\therefore y = \frac{\pi}{2} - \frac{\pi}{4}(2 + \pi) = -\frac{1}{4}\pi^2\) | M1 A1 | (10 marks) |
**(a)** $\frac{d}{dx}(\tan x) = \frac{d}{dx}\left(\frac{\sin x}{\cos x}\right)$ | M1 A1 |
$= \frac{\cos x \times \cos x - \sin x \times (-\sin x)}{\cos^2 x}$ | M1 A1 |
$= \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x$ | M1 A1 |
**(b)** $\frac{dy}{dx} = 2x\tan x + 2x \times \sec^2 x = 2\tan x + 2x\sec^2 x$ | M1 A1 |
At $x = \frac{\pi}{4}$, $y = \frac{\pi}{2}$, grad $= 2 + \pi$ | B1 |
$y - \frac{\pi}{2} = (2 + \pi)\left(x - \frac{\pi}{4}\right)$ | M1 |
At $P$, $x = 0$: $\therefore y = \frac{\pi}{2} - \frac{\pi}{4}(2 + \pi) = -\frac{1}{4}\pi^2$ | M1 A1 | (10 marks)\begin{enumerate}
\item (a) Use the derivatives of $\sin x$ and $\cos x$ to prove that
\end{enumerate}
$$\frac { \mathrm { d } } { \mathrm {~d} x } ( \tan x ) = \sec ^ { 2 } x$$
The tangent to the curve $y = 2 x \tan x$ at the point where $x = \frac { \pi } { 4 }$ meets the $y$-axis at the point $P$.\\
(b) Find the $y$-coordinate of $P$ in the form $k \pi ^ { 2 }$ where $k$ is a rational constant.\\
\hfill \mbox{\textit{Edexcel C3 Q4 [10]}}