| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Find stationary points of hyperbolic curves |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring differentiation of hyperbolic functions (quotient rule with sinh x), algebraic manipulation to reach the given form, application of Newton-Raphson method with hyperbolic derivatives (requiring sech²x), and analysis of error convergence. While the techniques are standard for FP2, the combination of hyperbolic calculus, numerical methods, and error estimation across three parts makes this moderately challenging, though still within expected Further Maths scope. |
| Spec | 1.09d Newton-Raphson method4.07b Hyperbolic graphs: sketch and properties |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt to diff. using product/quotient | M1 | |
| Attempt to solve \(\frac{dy}{dx} = 0\) | M1 | |
| Rewrite as A.G. | A1 | Clearly gain A.G. |
| (ii) Diff. to \(f'(x) = 1 \pm 2\sec^2 x\) | B1 | Or \(\pm 2\sec^2 x - 1\) |
| Use correct form of N-R with their expressions from correct \(f(x)\) | M1 | |
| Attempt N-R with \(x_1 = 2\) from previous | M1 | To get an \(x_2\) |
| Get \(x_2 = 1.9162(2)\) (3 s.f. min.) | A1 | |
| Get \(x_3 = 1.9150(1)\) (3 s.f. min.) | A1 | cao |
| (iii) Work out \(e_1\) and \(e_2\) (may be implied) | B1 | \(-0.083(8), -0.0012\) (allow \(\pm\) if both of same sign); \(e_1\) from \(0.083\) to \(0.085\) |
| Use \(e_2 \approx ke_1^2\) and \(e_3 \approx ke_2^2\) | M1 | |
| Get \(e_3 = e_2^2/e_1^2 = -0.0000002\)(or 3) | A1 | \(\pm\) if same sign as B1 |
| SC | B1 only for \(x_4 - x_3\) | |
| Use \(C^2 - S^2 = 1\) for \(C = \pm\sqrt{(1+x^2)}\) | M1 | |
| Use/state cosh \(y + \sinh y = e^y\) | A1 | |
| Justify one solution only | B1 |
**(i)** Attempt to diff. using product/quotient | M1 |
Attempt to solve $\frac{dy}{dx} = 0$ | M1 |
Rewrite as A.G. | A1 | Clearly gain A.G.
**(ii)** Diff. to $f'(x) = 1 \pm 2\sec^2 x$ | B1 | Or $\pm 2\sec^2 x - 1$
Use correct form of N-R with their expressions from correct $f(x)$ | M1 |
Attempt N-R with $x_1 = 2$ from previous | M1 | To get an $x_2$
Get $x_2 = 1.9162(2)$ (3 s.f. min.) | A1 |
Get $x_3 = 1.9150(1)$ (3 s.f. min.) | A1 | cao
**(iii)** Work out $e_1$ and $e_2$ (may be implied) | B1 | $-0.083(8), -0.0012$ (allow $\pm$ if both of same sign); $e_1$ from $0.083$ to $0.085$
Use $e_2 \approx ke_1^2$ and $e_3 \approx ke_2^2$ | M1 |
Get $e_3 = e_2^2/e_1^2 = -0.0000002$(or 3) | A1 | $\pm$ if same sign as B1
| SC | B1 only for $x_4 - x_3$ |
| | Use $C^2 - S^2 = 1$ for $C = \pm\sqrt{(1+x^2)}$ | M1
| | Use/state cosh $y + \sinh y = e^y$ | A1
| | Justify one solution only | B18 The curve with equation $y = \frac { \sinh x } { x ^ { 2 } }$, for $x > 0$, has one turning point.\\
(i) Show that the $x$-coordinate of the turning point satisfies the equation $x - 2 \tanh x = 0$.\\
(ii) Use the Newton-Raphson method, with a first approximation $x _ { 1 } = 2$, to find the next two approximations, $x _ { 2 }$ and $x _ { 3 }$, to the positive root of $x - 2 \tanh x = 0$.\\
(iii) By considering the approximate errors in $x _ { 1 }$ and $x _ { 2 }$, estimate the error in $x _ { 3 }$. (You are not expected to evaluate $x _ { 4 }$.)
\hfill \mbox{\textit{OCR FP2 2006 Q8 [11]}}