| Exam Board | AQA |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Fixed Point Iteration |
| Type | Sketch graphs to show root existence |
| Difficulty | Standard +0.3 This is a standard A-level numerical methods question covering routine techniques: sketching graphs to show existence of roots, change of sign verification, algebraic rearrangement, and iteration with cobweb diagrams. All parts follow textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.09a Sign change methods: locate roots1.09b Sign change methods: understand failure cases1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams |
| Answer | Marks |
|---|---|
| 7(a) | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Correct asymptotes | 1.2 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| to first asymptote | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| axis. | 1.1b | A1 |
| Answer | Marks |
|---|---|
| 7(b) | Rearranges to the form |
| Answer | Marks | Guidance |
|---|---|---|
| interval [0.4,0.6] | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| be defined. | 2.1 | R1 |
| Answer | Marks |
|---|---|
| 7(c) | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| x cos2x cos2x | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| given answer | 2.1 | R1 |
| Answer | Marks |
|---|---|
| (d)(i) | Obtains any one correct value to |
| Answer | Marks | Guidance |
|---|---|---|
| labels. | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| CAO | 1.1b | A1 |
| Answer | Marks |
|---|---|
| (d)(ii) | Draws correct cobweb diagram |
| Answer | Marks | Guidance |
|---|---|---|
| 0.4 | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| not on y=x | 1.1b | A1 |
| Total | 11 | |
| Q | Marking instructions | AO |
Question 7:
--- 7(a) ---
7(a) | 1
Sketches graph of y =
x
Must not cross axes
Correct asymptotes | 1.2 | B1
Sketches graph of y =seckxup
to first asymptote | 1.1a | M1
Draws fully correct graphs in first
quadrant, intersecting at one
point with sec 2x up to asymptote
5π
at x= . Ignore fourth
4
quadrant/negative y.
Condone missing labels on y-
axis. | 1.1b | A1
--- 7(b) ---
7(b) | Rearranges to the form
f(x)=0
and evaluates f(x)at 0.4 and 0.6
Can evaluate at two values either
side of the root 0.515 in the
interval [0.4,0.6] | 1.1a | M1 | 1 1
=sec2x⇒ −sec2x=0
x x
1
f ( x )= −sec2x
x
( )=1.06..>0
f 0.4
( )=−1.09..<0
f 0.6
Hence the solution lies
between 0.4 and 0.6
Completes rigorous argument
with any reference to change of
sign. Must see evidence of
correct evaluation accepting
values correct to 1 sf.
If function notation used it must
be defined. | 2.1 | R1
--- 7(c) ---
7(c) | 1
Uses sec2x= to obtain a
cos2x
correct equation in cos2xeg,
1 1 x
= or 1=
x cos2x cos2x | 1.1a | M1 | 1
=sec2x
x
1 1
=
x cos2x
x=cos2x
2x=cos−1x
1
x= cos−1x
2
Completes rearrangement
cos−1x=2x
Must see before
given answer | 2.1 | R1
--- 7
(d)(i) ---
7
(d)(i) | Obtains any one correct value to
at least 3 decimal places, ignoring
labels. | 1.1a | M1 | x =0.5796
2
x =0.4763
3
x =0.5372
4
Obtains x ,x and x correct
2 3 4
to 4 decimal places
If no labels only accept the three
correct answers in the correct
order with no extras seen beyond
x
4
CAO | 1.1b | A1
--- 7
(d)(ii) ---
7
(d)(ii) | Draws correct cobweb diagram
Condone missing vertical line x =
0.4 | 1.1a | M1 | See diagram below
Shows positions
ofx ,x and x with clear
2 3 4
indication of positioning on x-axis
not on y=x | 1.1b | A1
Total | 11
Q | Marking instructions | AO | Mark | Typical solution\begin{enumerate}[label=(\alph*)]
\item By sketching the graphs of $y = \frac{1}{x}$ and $y = \sec 2x$ on the axes below, show that the equation
$$\frac{1}{x} = \sec 2x$$
has exactly one solution for $x > 0$ [3 marks]
\includegraphics{figure_7a}
\item By considering a suitable change of sign, show that the solution to the equation lies between 0.4 and 0.6 [2 marks]
\item Show that the equation can be rearranged to give
$$x = \frac{1}{2}\cos^{-1}x$$ [2 marks]
\item
\begin{enumerate}[label=(\roman*)]
\item Use the iterative formula
$$x_{n+1} = \frac{1}{2}\cos^{-1}x_n$$
with $x_1 = 0.4$, to find $x_2$, $x_3$ and $x_4$, giving your answers to four decimal places. [2 marks]
\item On the graph below, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of $x_2$, $x_3$ and $x_4$. [2 marks]
\includegraphics{figure_7d}
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 1 2019 Q7 [11]}}