| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2024 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Proof by contradiction |
| Difficulty | Standard +0.8 This requires proof by contradiction (a less routine proof technique at A-level), finding dy/dx = 2 + 3x² + (-sin x), then showing this can never equal zero by recognizing 2 + 3x² ≥ 2 while -1 ≤ -sin x ≤ 1, so their sum is always positive. The conceptual insight about combining inequalities elevates this above standard differentiation questions, though the calculus itself is straightforward. |
| Spec | 1.01d Proof by contradiction1.07i Differentiate x^n: for rational n and sums1.07k Differentiate trig: sin(kx), cos(kx), tan(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Assume (there is a stationary point so) \(\frac{dy}{dx} = 0\) for some value of \(x\) | B1 | Clear suitable assumption referencing \(\frac{dy}{dx}=0\); stating "for all \(x\)" is B0 |
| \(\frac{dy}{dx} = 2 + 3x^2 - \sin x\) | M1 | Attempts derivative achieving \(2 + 3x^2 \pm \sin x\) |
| \(\frac{dy}{dx} = 0 \Rightarrow \sin x = 3x^2 + 2\ \ldots 2\) | dM1 | Makes progress with correct method leading to contradiction; must reference both \(3x^2 \geq 0\) and range of \(\sin x\) |
| This is a contradiction as \(\ | \sin x\ | \leq 1\) for all \(x\), hence the assumption is false and so the function has no stationary points |
# Question 8:
| Working/Answer | Mark | Guidance |
|---|---|---|
| Assume (there is a stationary point so) $\frac{dy}{dx} = 0$ for some value of $x$ | B1 | Clear suitable assumption referencing $\frac{dy}{dx}=0$; stating "for all $x$" is B0 |
| $\frac{dy}{dx} = 2 + 3x^2 - \sin x$ | M1 | Attempts derivative achieving $2 + 3x^2 \pm \sin x$ |
| $\frac{dy}{dx} = 0 \Rightarrow \sin x = 3x^2 + 2\ \ldots 2$ | dM1 | Makes progress with correct method leading to contradiction; must reference both $3x^2 \geq 0$ and range of $\sin x$ |
| This is a contradiction as $\|\sin x\| \leq 1$ for all $x$, hence the assumption is false and so the function has **no stationary points** | A1cso | Fully correct proof; contradiction clear; conclusion drawn; condone strict inequalities |\begin{enumerate}
\item Use proof by contradiction to prove that the curve with equation
\end{enumerate}
$$y = 2 x + x ^ { 3 } + \cos x$$
has no stationary points.
\hfill \mbox{\textit{Edexcel P4 2024 Q8 [4]}}