| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find gradient at point |
| Difficulty | Moderate -0.8 This is a straightforward differentiation exercise testing quotient rule and chain rule at a specific point. Both parts are routine applications of standard techniques with no problem-solving required—students simply apply learned rules and substitute x=2. Easier than average for C3. |
| Spec | 1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Either Attempt use of quotient rule | M1 | allow numerator wrong way round but needs minus sign in numerator and both terms in numerator involving \(x\); for M1 condone minor errors such as absence of square in denominator, absence of brackets, … |
| Obtain \(\frac{3(2x+1) - 6x}{(2x+1)^2}\) or equiv | A1 | give A0 if necessary brackets absent unless subsequent calculation indicates their 'presence' |
| Substitute 2 to obtain \(\frac{3}{25}\) or 0.12 | A1 | or simplified equiv but A0 for final \(\frac{3}{25}\) |
| Or Attempt use of product rule for \(3x(2x+1)^{-1}\) | M1 | allow sign error; condone no use of chain rule |
| Obtain \(3(2x+1)^{-1} - 6x(2x+1)^{-2}\) or equiv | A1 | |
| Substitute 2 to obtain \(\frac{3}{25}\) or 0.12 | A1 | or simplified equiv |
| Answer | Marks | Guidance |
|---|---|---|
| Differentiate to obtain form \(kx(4x^2 + 9)^n\) | M1 | any non-zero constants \(k\) and \(n\) (including 1 or \(\frac{1}{2}\) for \(n\)) |
| Obtain \(4x(4x^2 + 9)^{-\frac{1}{2}}\) | A1 | or (unsimplified) equiv |
| Substitute 2 to obtain \(\frac{8}{5}\) or 1.6 | A1 | or simplified equiv but A0 for final \(\frac{8}{5}\) |
## (i)
**Either** Attempt use of quotient rule | M1 | allow numerator wrong way round but needs minus sign in numerator and both terms in numerator involving $x$; for M1 condone minor errors such as absence of square in denominator, absence of brackets, …
Obtain $\frac{3(2x+1) - 6x}{(2x+1)^2}$ or equiv | A1 | give A0 if necessary brackets absent unless subsequent calculation indicates their 'presence'
Substitute 2 to obtain $\frac{3}{25}$ or 0.12 | A1 | or simplified equiv but A0 for final $\frac{3}{25}$
**Or** Attempt use of product rule for $3x(2x+1)^{-1}$ | M1 | allow sign error; condone no use of chain rule
Obtain $3(2x+1)^{-1} - 6x(2x+1)^{-2}$ or equiv | A1 |
Substitute 2 to obtain $\frac{3}{25}$ or 0.12 | A1 | or simplified equiv
**Total: [3]**
## (ii)
Differentiate to obtain form $kx(4x^2 + 9)^n$ | M1 | any non-zero constants $k$ and $n$ (including 1 or $\frac{1}{2}$ for $n$)
Obtain $4x(4x^2 + 9)^{-\frac{1}{2}}$ | A1 | or (unsimplified) equiv
Substitute 2 to obtain $\frac{8}{5}$ or 1.6 | A1 | or simplified equiv but A0 for final $\frac{8}{5}$
**Total: [3]**
---For each of the following curves, find the gradient at the point with $x$-coordinate 2.
\begin{enumerate}[label=(\roman*)]
\item $y = \frac{3x}{2x + 1}$ [3]
\item $y = \sqrt{4x^2 + 9}$ [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2013 Q1 [6]}}