| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find gradient at point |
| Difficulty | Moderate -0.3 This is a straightforward application of product rule (part a) and quotient rule (part b) with no algebraic complications. Both parts are standard textbook exercises requiring direct application of differentiation rules with minimal simplification, making it slightly easier than average for A-level. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt use of product rule | M1 | Involving \(\ldots + \ldots\) |
| Obtain \(2x(x+1)^6 \ldots\) | A1 | |
| Obtain \(\ldots + 6x^2(x+1)^5\) | A1 | 3 Or equivs; ignore subsequent attempt at simplification |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt use of quotient rule | M1 | Or, with adjustment, product rule; allow \(u/v\) confusion |
| Obtain \(\dfrac{(x^2-3)2x-(x^2+3)2x}{(x^2-3)^2}\) | A1 | Or equiv |
| Obtain \(-3\) | A1 | 3 From correct derivative only |
## Question 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt use of product rule | M1 | Involving $\ldots + \ldots$ |
| Obtain $2x(x+1)^6 \ldots$ | A1 | |
| Obtain $\ldots + 6x^2(x+1)^5$ | A1 | **3** Or equivs; ignore subsequent attempt at simplification |
## Question 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt use of quotient rule | M1 | Or, with adjustment, product rule; allow $u/v$ confusion |
| Obtain $\dfrac{(x^2-3)2x-(x^2+3)2x}{(x^2-3)^2}$ | A1 | Or equiv |
| Obtain $-3$ | A1 | **3** From correct derivative only |
---3
\begin{enumerate}[label=(\alph*)]
\item Differentiate $x ^ { 2 } ( x + 1 ) ^ { 6 }$ with respect to $x$.
\item Find the gradient of the curve $y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } - 3 }$ at the point where $x = 1$.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2006 Q3 [6]}}