| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sign Change & Interval Methods |
| Type | Sign Change with Function Evaluation |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard C3 techniques: (a) sign change demonstration by direct substitution, (b) algebraic rearrangement of a derivative equation, and (c) calculator-based iteration. All parts are routine applications with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y_{2.1} = -0.224\), \(y_{2.2} = (+)0.546\) | M1 | Sub both \(x=2.1\) and \(x=2.2\), achieve at least one correct to 1 sf |
| Change of sign \(\Rightarrow Q\) lies between | A1 | Both values correct to 1 sf, with reason and minimal conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dy}{dx} = -2x\sin\left(\frac{1}{2}x^2\right) + 3x^2 - 3\) | M1 A1 | Differentiating to get \(...\sin\left(\frac{1}{2}x^2\right) + 3x^2 - 3\) where \(...\) is a constant or linear function in \(x\) |
| \(-2x\sin\left(\frac{1}{2}x^2\right) + 3x^2 - 3 = 0 \Rightarrow x = \sqrt{1 + \frac{2}{3}x\sin\left(\frac{1}{2}x^2\right)}\) | M1 A1* | Sets \(\frac{dy}{dx} = 0\), makes \(x\) of \(3x^2\) the subject. Watch for missing \(x\)'s in formula |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x_1 = \sqrt{1 + \frac{2}{3} \times 1.3\sin\left(\frac{1}{2} \times 1.3^2\right)}\) | M1 | Substitute \(x = 1.3\) into iterative formula to find at least \(x_1\) |
| \(x_1 =\) awrt \(1.284\), \(x_2 =\) awrt \(1.276\) | A1 | Both correct to 3 dp. Subscripts not important |
# Question 6(a) - Change of Sign
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y_{2.1} = -0.224$, $y_{2.2} = (+)0.546$ | M1 | Sub both $x=2.1$ and $x=2.2$, achieve at least one correct to 1 sf |
| Change of sign $\Rightarrow Q$ lies between | A1 | Both values correct to 1 sf, with reason and minimal conclusion |
# Question 6(b) - Differentiation and Iteration Formula
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = -2x\sin\left(\frac{1}{2}x^2\right) + 3x^2 - 3$ | M1 A1 | Differentiating to get $...\sin\left(\frac{1}{2}x^2\right) + 3x^2 - 3$ where $...$ is a constant or linear function in $x$ |
| $-2x\sin\left(\frac{1}{2}x^2\right) + 3x^2 - 3 = 0 \Rightarrow x = \sqrt{1 + \frac{2}{3}x\sin\left(\frac{1}{2}x^2\right)}$ | M1 A1* | Sets $\frac{dy}{dx} = 0$, makes $x$ of $3x^2$ the subject. Watch for missing $x$'s in formula |
# Question 6(c) - Iteration
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x_1 = \sqrt{1 + \frac{2}{3} \times 1.3\sin\left(\frac{1}{2} \times 1.3^2\right)}$ | M1 | Substitute $x = 1.3$ into iterative formula to find at least $x_1$ |
| $x_1 =$ awrt $1.284$, $x_2 =$ awrt $1.276$ | A1 | Both correct to 3 dp. Subscripts not important |
---6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{16c69ee4-255e-4d77-955a-92e1eb2f7d3e-09_458_1164_239_383}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows a sketch of part of the curve with equation
$$y = 2 \cos \left( \frac { 1 } { 2 } x ^ { 2 } \right) + x ^ { 3 } - 3 x - 2$$
The curve crosses the $x$-axis at the point $Q$ and has a minimum turning point at $R$.
\begin{enumerate}[label=(\alph*)]
\item Show that the $x$ coordinate of $Q$ lies between 2.1 and 2.2
\item Show that the $x$ coordinate of $R$ is a solution of the equation
$$x = \sqrt { 1 + \frac { 2 } { 3 } x \sin \left( \frac { 1 } { 2 } x ^ { 2 } \right) }$$
Using the iterative formula
$$x _ { n + 1 } = \sqrt { 1 + \frac { 2 } { 3 } x _ { n } \sin \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } \right) } , \quad x _ { 0 } = 1.3$$
\item find the values of $x _ { 1 }$ and $x _ { 2 }$ to 3 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2014 Q6 [8]}}