| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard C3 question on inverse trig functions and iteration. Part (i) requires simple substitution into the domain of arcsin. Part (ii)(a) is routine sign-change verification. Part (ii)(b) involves rearranging to iterative form (straightforward algebra) and applying iteration with a calculator—all standard textbook techniques with no novel insight required. Slightly easier than average due to the guided structure. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.09a Sign change methods: locate roots1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or clearly imply \(a=\frac{1}{2}\) | B1 | \(a=\frac{5}{2}\) and \(b=\frac{1}{2}\) earn B0 B0. \(\sin(-\frac{1}{2}\pi)+\frac{3}{2}\) and \(\sin(\frac{1}{2}\pi)+\frac{3}{2}\) earn B0 B0 |
| State or clearly imply \(b=\frac{5}{2}\) | B1 | (Implied by, for example, just \(\frac{1}{2}\) and \(\frac{5}{2}\) stated in that order) |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Carry out relevant calculations using radians | M1 | Involving \(8\sin^{-1}(x-\frac{3}{2})\) or \(8\sin^{-1}(x-\frac{3}{2})-x\) or equiv; needs two explicit calculations. May carry out calculations in, for example, \(\frac{3}{2}+\sin(\frac{1}{8}x)-x\) |
| Obtain \(1.6\) and \(2.4\) or \(-0.1\) and \(0.6\) | A1 | Or equivs |
| Conclude with reference to \(1.6<1.7\) but \(2.4>1.8\), or to sign change | A1 | Or equiv |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply \(p=\frac{3}{2}\) and \(q=\frac{1}{8}\) | B1 | Implied by presence in iterative formula. Answer only can earn no more than the first B1 for values of \(p\) and \(q\); working in degrees can earn no more than the first B1 (for \(p\) and \(q\)) and M1 |
| Obtain correct first iterate | B1 | Having started with value \(x_1\) such that \(1.7\leq x_1\leq1.8\); given to at least 4 s.f. |
| Carry out iteration process | M1 | Obtaining at least three iterates in all; having started with any non-negative value; implied by an apparently converging sequence of plausible values; all values to at least 4 s.f. |
| Obtain at least three correct iterates | A1 | Allowing recovery after error |
| Conclude with clear statement that root is \(1.712\) | A1 | Final answer required to exactly 4 significant figures |
| [5] |
# Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or clearly imply $a=\frac{1}{2}$ | B1 | $a=\frac{5}{2}$ and $b=\frac{1}{2}$ earn B0 B0. $\sin(-\frac{1}{2}\pi)+\frac{3}{2}$ and $\sin(\frac{1}{2}\pi)+\frac{3}{2}$ earn B0 B0 |
| State or clearly imply $b=\frac{5}{2}$ | B1 | (Implied by, for example, just $\frac{1}{2}$ and $\frac{5}{2}$ stated in that order) |
| **[2]** | | |
---
# Question 6(ii)(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out relevant calculations using radians | M1 | Involving $8\sin^{-1}(x-\frac{3}{2})$ or $8\sin^{-1}(x-\frac{3}{2})-x$ or equiv; needs two explicit calculations. May carry out calculations in, for example, $\frac{3}{2}+\sin(\frac{1}{8}x)-x$ |
| Obtain $1.6$ and $2.4$ or $-0.1$ and $0.6$ | A1 | Or equivs |
| Conclude with reference to $1.6<1.7$ but $2.4>1.8$, or to sign change | A1 | Or equiv |
| **[3]** | | |
---
# Question 6(ii)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $p=\frac{3}{2}$ and $q=\frac{1}{8}$ | B1 | Implied by presence in iterative formula. Answer only can earn no more than the first B1 for values of $p$ and $q$; working in degrees can earn no more than the first B1 (for $p$ and $q$) and M1 |
| Obtain correct first iterate | B1 | Having started with value $x_1$ such that $1.7\leq x_1\leq1.8$; given to at least 4 s.f. |
| Carry out iteration process | M1 | Obtaining at least three iterates in all; having started with any non-negative value; implied by an apparently converging sequence of plausible values; all values to at least 4 s.f. |
| Obtain at least three correct iterates | A1 | Allowing recovery after error |
| Conclude with clear statement that root is $1.712$ | A1 | Final answer required to exactly 4 significant figures |
| **[5]** | | |
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{00a4be37-c095-4d9c-a1cd-d03b8ab1d411-3_553_579_274_726}
The diagram shows the curve $y = 8 \sin ^ { - 1 } \left( x - \frac { 3 } { 2 } \right)$. The end-points $A$ and $B$ of the curve have coordinates ( $a , - 4 \pi$ ) and ( $b , 4 \pi$ ) respectively.\\
(i) State the values of $a$ and $b$.\\
(ii) It is required to find the root of the equation $8 \sin ^ { - 1 } \left( x - \frac { 3 } { 2 } \right) = x$.
\begin{enumerate}[label=(\alph*)]
\item Show by calculation that the root lies between 1.7 and 1.8.
\item In order to find the root, the iterative formula
$$x _ { n + 1 } = p + \sin \left( q x _ { n } \right) ,$$
with a suitable starting value, is to be used. Determine the values of the constants $p$ and $q$ and hence find the root correct to 4 significant figures. Show the result of each step of the iteration process.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2015 Q6 [10]}}