| Exam Board | AQA |
|---|---|
| Module | Paper 2 (Paper 2) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Newton-Raphson convergence failure |
| Difficulty | Standard +0.3 This is a standard Newton-Raphson question with routine tasks: (a) uses the intermediate value theorem with simple substitution, (b) applies the Newton-Raphson formula mechanically with straightforward differentiation, and (c) asks for a textbook failure case (horizontal tangent when f'(-1)=0). All parts follow predictable patterns with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.09a Sign change methods: locate roots1.09d Newton-Raphson method |
| Answer | Marks |
|---|---|
| 4(a) | Starts an argument by showing |
| Answer | Marks | Guidance |
|---|---|---|
| explicit. | AO2.1 | R1 |
| Answer | Marks | Guidance |
|---|---|---|
| to continuous function | AO2.4 | E1 |
| (b) | Uses Newton–Raphson, must |
| Answer | Marks | Guidance |
|---|---|---|
| PI by correct substitution | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| (accept ‘their’ f(2) from part (a)) | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 9 9 | AO1.1b | A1 |
| (c) | Explains why the method fails |
| Answer | Marks | Guidance |
|---|---|---|
| eg division by zero | not possible | |
| gradient zero | method fails | AO2.4 |
| Answer | Marks | Guidance |
|---|---|---|
| Total | 6 | |
| Q | Marking Instructions | AO |
Question 4:
--- 4(a) ---
4(a) | Starts an argument by showing
that f(2)0 and f(1) 0
Both attempted and at least one
evaluated correctly
f must be clearly defined or
substitution of values must be
explicit. | AO2.1 | R1 | f(x) x3 3x1
f(2)(2)3 3(2)110
f(1)(1)3 3(1)130
Change of sign and f(x) is
continuous so a root must lie
between x = –2 and x = –1
Explains reasoning fully to
complete the argument
Evaluations above need to be of
opposite sign and ‘change of
sign’ OE seen and reference to
x-values –2 & –1 and reference
to continuous function | AO2.4 | E1
(b) | Uses Newton–Raphson, must
have f '(x) correct
PI by correct substitution | AO1.1a | M1 | (x )3 3x 1
x x n n
n1 n 3(x )2 3
n
1
x 2
2 3(2)2 3
17
9
Substitutes x 2 into ‘their’
1
Newton–Raphson formula
(accept ‘their’ f(2) from part (a)) | AO1.1a | M1
Obtains correct value for x
2
17 8
or 1 or1.89or better
9 9 | AO1.1b | A1
(c) | Explains why the method fails
when x 1
1
This must include a substitution of
x 1 and an explanation of
1
what goes wrong
eg division by zero | not possible
gradient zero | method fails | AO2.4 | E1 | x 1
1
3 3
x 2 2
2 3(1)2 3 0
causes division by zero (expression
undefined)
ALT
f(1)0, function has zero
gradient at this point, method will
fail
Total | 6
Q | Marking Instructions | AO | Marks | Typical SolutionThe equation $x^3 - 3x + 1 = 0$ has three real roots.
\begin{enumerate}[label=(\alph*)]
\item Show that one of the roots lies between $-2$ and $-1$
[2 marks]
\item Taking $x_1 = -2$ as the first approximation to one of the roots, use the Newton-Raphson method to find $x_2$, the second approximation.
[3 marks]
\item Explain why the Newton-Raphson method fails in the case when the first approximation is $x_1 = -1$
[1 mark]
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 2 Q4 [6]}}