| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2021 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find derivative of quotient |
| Difficulty | Moderate -0.8 This is a straightforward differentiation question testing basic rules. Part (a) requires only the chain rule for exponentials (standard recall), and part (b) is a direct application of the quotient rule with simple polynomial terms. Both are routine exercises with no problem-solving element, making this easier than average. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(-4e^{-4x}\) oe | B2 | B1 for \(e^{-4x}\) seen as part of answer, not in denominator |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| By quotient rule: \(\frac{(x+1)\times 2x - x^2 \times 1}{(x+1)^2}\) | B1, B1 | B1: correct denominator & 1 correct term in numerator, any form. ISW |
| Alternative (chain rule): \(-(x+1)^{-2}\times x^2 + (x+1)^{-1}\times 2x\) | B1, B1 | B1 for either term correct. ISW |
| \(\left(= \frac{x(x+2)}{(x+1)^2}\text{ or }\frac{x^2+2x}{x^2+2x+1}\right)\) |
## Question 1(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $-4e^{-4x}$ oe | **B2** | B1 for $e^{-4x}$ seen as part of answer, not in denominator |
---
## Question 1(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| By quotient rule: $\frac{(x+1)\times 2x - x^2 \times 1}{(x+1)^2}$ | **B1, B1** | B1: correct denominator & 1 correct term in numerator, any form. ISW |
| **Alternative** (chain rule): $-(x+1)^{-2}\times x^2 + (x+1)^{-1}\times 2x$ | **B1, B1** | B1 for either term correct. ISW |
| $\left(= \frac{x(x+2)}{(x+1)^2}\text{ or }\frac{x^2+2x}{x^2+2x+1}\right)$ | | |
---1 Differentiate the following with respect to $x$.
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { e } ^ { - 4 x }$
\item $\frac { x ^ { 2 } } { x + 1 }$
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2021 Q1 [4]}}