| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Determine increasing/decreasing intervals |
| Difficulty | Standard +0.3 This is a straightforward quotient rule application followed by solving a rational inequality. Part (a) requires routine differentiation and algebraic simplification. Part (b) involves analyzing the sign of a simple quadratic expression, which is standard C3 material but slightly above average difficulty due to the inequality solving step. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y = \frac{4x}{(x^2+5)} \Rightarrow \left(\frac{dy}{dx}\right) = \frac{4(x^2+5) - 4x \times 2x}{(x^2+5)^2}\) | M1A1 | Attempt to use quotient rule \(\frac{vu' - uv'}{v^2}\) with \(u = 4x\), \(v = x^2 + 5\). If rule quoted it must be correct. May be implied by \(u=4x, u'=A, v=x^2+5, v'=Bx\). Alternatively uses product rule with \(u(1/v) = 4x\) and \(v(1/u) = (x^2+5)^{-1}\) |
| \(\Rightarrow \left(\frac{dy}{dx}\right) = \frac{20 - 4x^2}{(x^2+5)^2}\) | M1A1 | Simplifies to form \(f'(x) = \frac{A + Bx^2}{(x^2+5)^2}\). CAO. Accept \(\frac{4(5-x^2)}{(x^2+5)^2}\), \(\frac{4x^2-20}{(x^2+5)^2}\), or \(\frac{-4(x^2-5)}{x^4+10x^2+25}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{20-4x^2}{(x^2+5)^2} < 0 \Rightarrow x^2 > \frac{20}{4}\); critical values of \(\pm\sqrt{5}\) | M1 | Sets numerator \(= 0\), \(< 0\), \(> 0\) and proceeds to at least one value for \(x\). Cannot be scored from numerator such as 4 or \(20 + 4x^2\) |
| \(x < -\sqrt{5},\ x > \sqrt{5}\) or equivalent | dM1 | Achieves two critical values for numerator \(= 0\) and chooses outside region. Dependent on previous M. Condone \(x < -\sqrt{5}\), \(x > \sqrt{5}\) and expressions like \(-\sqrt{5} > x > \sqrt{5}\) |
| A1 | Allow \(x < -\sqrt{5}, x > \sqrt{5}\); \(\{x: -\infty < x < -\sqrt{5}\} \cup \{\sqrt{5} < x < \infty\}\); \( | x |
# Question 2:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = \frac{4x}{(x^2+5)} \Rightarrow \left(\frac{dy}{dx}\right) = \frac{4(x^2+5) - 4x \times 2x}{(x^2+5)^2}$ | M1A1 | Attempt to use quotient rule $\frac{vu' - uv'}{v^2}$ with $u = 4x$, $v = x^2 + 5$. If rule quoted it must be correct. May be implied by $u=4x, u'=A, v=x^2+5, v'=Bx$. Alternatively uses product rule with $u(1/v) = 4x$ and $v(1/u) = (x^2+5)^{-1}$ |
| $\Rightarrow \left(\frac{dy}{dx}\right) = \frac{20 - 4x^2}{(x^2+5)^2}$ | M1A1 | Simplifies to form $f'(x) = \frac{A + Bx^2}{(x^2+5)^2}$. CAO. Accept $\frac{4(5-x^2)}{(x^2+5)^2}$, $\frac{4x^2-20}{(x^2+5)^2}$, or $\frac{-4(x^2-5)}{x^4+10x^2+25}$ |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{20-4x^2}{(x^2+5)^2} < 0 \Rightarrow x^2 > \frac{20}{4}$; critical values of $\pm\sqrt{5}$ | M1 | Sets numerator $= 0$, $< 0$, $> 0$ and proceeds to at least one value for $x$. Cannot be scored from numerator such as 4 or $20 + 4x^2$ |
| $x < -\sqrt{5},\ x > \sqrt{5}$ or equivalent | dM1 | Achieves two critical values for numerator $= 0$ and chooses outside region. Dependent on previous M. Condone $x < -\sqrt{5}$, $x > \sqrt{5}$ and expressions like $-\sqrt{5} > x > \sqrt{5}$ |
| | A1 | Allow $x < -\sqrt{5}, x > \sqrt{5}$; $\{x: -\infty < x < -\sqrt{5}\} \cup \{\sqrt{5} < x < \infty\}$; $|x| > \sqrt{5}$. Do not allow $x < -\sqrt{5}$ and $x > \sqrt{5}$ |
---2.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, writing your answer as a single fraction in its simplest form.
\item Hence find the set of values of $x$ for which $\frac { \mathrm { d } y } { \mathrm {~d} x } < 0$\\
2.
$$y = \frac { 4 x } { x ^ { 2 } + 5 }$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2016 Q2 [7]}}