| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2008 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Solve using double angle formulas |
| Difficulty | Standard +0.8 This is a Further Maths question requiring manipulation of hyperbolic identities using exponential definitions, analysis of equation solvability conditions, and solving a cubic-like equation in sinh x. Part (i) is routine algebraic manipulation, but parts (ii)-(iii) require insight into when the cubic equation has real solutions and solving for x in logarithmic form, which goes beyond standard techniques. |
| Spec | 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use correct exponential for \(\sinh x\). Attempt to expand cube of this. Correct cubic. Clearly replace in terms of \(\sinh\) | B1, M1, A1, B1 | Must be 4 terms. (Allow RHS \(\to\) LHS or RHS = LHS separately) |
| (ii) Replace and factorise. Attempt to solve for \(\sinh^2 x\). Get \(k>3\) | M1, M1, A1 | Or state \(\sinh x \neq 0\) (= \(\frac{1}{4}(k-3)\)) or for \(k\) and use \(\sinh^2 x>0\). Not \(\geq\) |
| (iii) Get \(x = \sinh^{-1}c\). Replace in ln equivalent. Repeat for negative root | M1, A1, A1 | \((c = \pm\frac{1}{2})\); allow \(\sinh x = c\). As \(\ln(2q+\sqrt[3]{1}t_1)\); their \(x\). May be given as neg. of first answer (no need for \(x=0\) implied). SR: Use of exponential definitions. Express as cubic in \(e^{2x} = u\) M1. Factorise to \((u-1)(u^2-3u+1)=0\) A1. Solve for \(x=0, \frac{1}{2}\ln(\frac{3}{2} \pm \frac{\sqrt{5}}{2})\) A1 |
**(i)** Use correct exponential for $\sinh x$. Attempt to expand cube of this. Correct cubic. Clearly replace in terms of $\sinh$ | B1, M1, A1, B1 | Must be 4 terms. (Allow RHS $\to$ LHS or RHS = LHS separately)
**(ii)** Replace and factorise. Attempt to solve for $\sinh^2 x$. Get $k>3$ | M1, M1, A1 | Or state $\sinh x \neq 0$ (= $\frac{1}{4}(k-3)$) or for $k$ and use $\sinh^2 x>0$. Not $\geq$
**(iii)** Get $x = \sinh^{-1}c$. Replace in ln equivalent. Repeat for negative root | M1, A1, A1 | $(c = \pm\frac{1}{2})$; allow $\sinh x = c$. As $\ln(2q+\sqrt[3]{1}t_1)$; their $x$. May be given as neg. of first answer (no need for $x=0$ implied). SR: Use of exponential definitions. Express as cubic in $e^{2x} = u$ M1. Factorise to $(u-1)(u^2-3u+1)=0$ A1. Solve for $x=0, \frac{1}{2}\ln(\frac{3}{2} \pm \frac{\sqrt{5}}{2})$ A18 (i) By using the definition of $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$, show that
$$\sinh ^ { 3 } x = \frac { 1 } { 4 } \sinh 3 x - \frac { 3 } { 4 } \sinh x$$
(ii) Find the range of values of the constant $k$ for which the equation
$$\sinh 3 x = k \sinh x$$
has real solutions other than $x = 0$.\\
(iii) Given that $k = 4$, solve the equation in part (ii), giving the non-zero answers in logarithmic form.
\hfill \mbox{\textit{OCR FP2 2008 Q8 [10]}}