| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Newton-Raphson method |
| Type | Find stationary point coordinate |
| Difficulty | Standard +0.3 This is a structured, multi-part question that guides students through a standard Newton-Raphson application. Part (a) requires routine differentiation using product rule, part (b) is algebraic manipulation following a template, and parts (c)-(d) are calculator work. While it involves multiple steps, each step is straightforward with no novel insight required, making it slightly easier than average. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.09d Newton-Raphson method |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2x\ln x + \dfrac{x^2 - 2}{x}\) | M1 | Attempt differentiation using product rule; may expand first to give \(2x\ln x + \frac{x^2}{x} - \frac{2}{x}\) (allow middle term as just \(x\)) |
| \(2x\ln x + \dfrac{x^2-2}{x} = 0\) leading to \(2x^2\ln x + x^2 - 2 = 0\) A.G. | A1 | Equate to 0 and obtain given answer; must be equated to 0 before clearing fractions; must be equation ie \(\ldots = 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(f'(x) = 4x\ln x + 2x^2 \cdot \frac{1}{x} + 2x\) | B1 | Correct derivative seen; allow simplified middle term of \(2x\) |
| \(x_{n+1} = x_n - \dfrac{2x_n^2\ln x_n + x_n^2 - 2}{4x_n\ln x_n + 2x_n^2 \cdot \frac{1}{x_n} + 2x_n}\) | M1 | Use correct Newton-Raphson formula with numerator correct and their derivative in denominator; allow fractional term without subscripts; SC condone use of N-R on \((x^2-2)\ln x\) |
| \(x_{n+1} = \dfrac{x_n(4x_n\ln x_n + 4x_n) - (2x_n^2\ln x_n + x_n^2 - 2)}{4x_n\ln x_n + 4x_n}\) | M1 | Attempt rearrangement into single fraction with brackets expanded; allow without subscripts; N-R not necessarily correct but must be recognisable attempt; SC rearrange their N-R on \((x^2-2)\ln x\) |
| \(x_{n+1} = \dfrac{2x_n^2\ln x_n + 3x_n^2 + 2}{4x_n(\ln x_n + 1)}\) A.G. | A1 | Obtain given answer with no errors seen; subscripts needed on RHS at least one step before AG; LHS needs \(x_{n+1}\) seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x_2 = 1.25,\ x_3 = 1.2075\) | B1 | Condone 1.21 or better for \(x_3\); \(x_3 = 1.207515437\ldots\) |
# Question 7:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2x\ln x + \dfrac{x^2 - 2}{x}$ | M1 | Attempt differentiation using product rule; may expand first to give $2x\ln x + \frac{x^2}{x} - \frac{2}{x}$ (allow middle term as just $x$) |
| $2x\ln x + \dfrac{x^2-2}{x} = 0$ leading to $2x^2\ln x + x^2 - 2 = 0$ **A.G.** | A1 | Equate to 0 and obtain given answer; must be equated to 0 **before** clearing fractions; must be equation ie $\ldots = 0$ |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $f'(x) = 4x\ln x + 2x^2 \cdot \frac{1}{x} + 2x$ | B1 | Correct derivative seen; allow simplified middle term of $2x$ |
| $x_{n+1} = x_n - \dfrac{2x_n^2\ln x_n + x_n^2 - 2}{4x_n\ln x_n + 2x_n^2 \cdot \frac{1}{x_n} + 2x_n}$ | M1 | Use correct Newton-Raphson formula with numerator correct and their derivative in denominator; allow fractional term without subscripts; **SC** condone use of N-R on $(x^2-2)\ln x$ |
| $x_{n+1} = \dfrac{x_n(4x_n\ln x_n + 4x_n) - (2x_n^2\ln x_n + x_n^2 - 2)}{4x_n\ln x_n + 4x_n}$ | M1 | Attempt rearrangement into single fraction with brackets expanded; allow without subscripts; N-R not necessarily correct but must be recognisable attempt; **SC** rearrange their N-R on $(x^2-2)\ln x$ |
| $x_{n+1} = \dfrac{2x_n^2\ln x_n + 3x_n^2 + 2}{4x_n(\ln x_n + 1)}$ **A.G.** | A1 | Obtain given answer with no errors seen; subscripts needed on RHS at least one step before AG; LHS needs $x_{n+1}$ seen |
## Part (c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x_2 = 1.25,\ x_3 = 1.2075$ | B1 | Condone 1.21 or better for $x_3$; $x_3 = 1.207515437\ldots$ |7 The curve $y = \left( x ^ { 2 } - 2 \right) \ln x$ has one stationary point which is close to $x = 1$.
\begin{enumerate}[label=(\alph*)]
\item Show that the $x$-coordinate of this stationary point satisfies the equation $2 x ^ { 2 } \ln x + x ^ { 2 } - 2 = 0$.
\item Show that the Newton-Raphson iterative formula for finding the root of the equation in part (a) can be written in the form $x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } \ln x _ { n } + 3 x _ { n } ^ { 2 } + 2 } { 4 x _ { n } \left( \ln x _ { n } + 1 \right) }$.
\item Apply the Newton-Raphson formula with initial value $x _ { 1 } = 1$ to find $x _ { 2 }$ and $x _ { 3 }$.
\item Find the coordinates of this stationary point, giving each coordinate correct to $\mathbf { 3 }$ decimal places.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2021 Q7 [9]}}