Question 2 - A-Level Maths

OCR Further Pure Core 2 2021 June — Question 2 6 marks

Exam Board	OCR
Module	Further Pure Core 2 (Further Pure Core 2)
Year	2021
Session	June
Marks	6
Topic	Hyperbolic functions
Type	Solve using substitution u = cosh x or u = sinh x
Difficulty	Standard +0.8 This is a Further Maths hyperbolic functions question requiring use of the identity cosh²x = 1 + sinh²x to convert to a quadratic in sinh x, then solving and using the inverse hyperbolic function formula to express answers in logarithmic form. While it involves multiple steps and careful algebraic manipulation, it follows a standard substitution technique taught in FP2/Further Pure, making it moderately challenging but not requiring novel insight.
Spec	1.05l Double angle formulae: and compound angle formulae 4.02n Euler's formula: e^(itheta) = cos(theta) + isin(theta)4.03f Linear transformations 3D: reflections and rotations about axes 4.03h Determinant 2x2: calculation 4.03n Inverse 2x2 matrix 4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials 4.07c Hyperbolic identity: cosh^2(x) - sinh^2(x) = 1 4.07d Differentiate/integrate: hyperbolic functions

2 In this question you must show detailed reasoning.
Solve the equation $2 \cosh ^ { 2 } x + 5 \sinh x - 5 = 0$ giving each answer in the form $\ln ( p + q \sqrt { r } )$ where $p$ and $q$ are rational numbers, and $r$ is an integer, whose values are to be determined. You are given that the matrix $\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { 2 a - a ^ { 2 } } { 3 } & 0 \\ 0 & 0 & 1 \end{array} \right)$, where $a$ is a positive constant, represents the transformation R which is a reflection in 3-D.

State the plane of reflection of $R$.
Determine the value of $a$.
With reference to R explain why $\mathbf { A } ^ { 2 } = \mathbf { I }$, the $3 \times 3$ identity matrix.
1. By using Euler's formula show that $\cosh ( \mathrm { iz } ) = \cos z$.
2. Hence, find, in logarithmic form, a root of the equation $\cos z = 2$. [You may assume that $\cos z = 2$ has complex roots.] A swing door is a door to a room which is closed when in equilibrium but which can be pushed open from either side and which can swing both ways, into or out of the room, and through the equilibrium position. The door is sprung so that when displaced from the equilibrium position it will swing back towards it. The extent to which the door is open at any time, $t$ seconds, is measured by the angle at the hinge, $\theta$, which the plane of the door makes with the plane of the equilibrium position. See the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-03_317_954_497_255} In an initial model of the motion of a certain swing door it is suggested that $\theta$ satisfies the following differential equation. $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 25 \theta = 0$$
3. 1. Write down the general solution to (\textit{).
  2. With reference to the behaviour of your solution in part (a)(i) explain briefly why the model using (}) is unlikely to be realistic. In an improved model of the motion of the door an extra term is introduced to the differential equation so that it becomes $$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm {~d} t } + 25 \theta = 0$$ where $\lambda$ is a positive constant.
4. In the case where $\lambda = 16$ the door is held open at an angle of 0.9 radians and then released from rest at time $t = 0$.
  1. Find, in a real form, the general solution of ( $\dagger$ ).
  2. Find the particular solution of ( $\dagger$ ).
  3. With reference to the behaviour of your solution found in part (b)(ii) explain briefly how the extra term in ( $\dagger$ ) improves the model.
  4. Find the value of $\lambda$ for which the door is critically damped.

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2 In this question you must show detailed reasoning.\\
Solve the equation $2 \cosh ^ { 2 } x + 5 \sinh x - 5 = 0$ giving each answer in the form $\ln ( p + q \sqrt { r } )$ where $p$ and $q$ are rational numbers, and $r$ is an integer, whose values are to be determined.

You are given that the matrix $\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { 2 a - a ^ { 2 } } { 3 } & 0 \\ 0 & 0 & 1 \end{array} \right)$, where $a$ is a positive constant, represents the transformation R which is a reflection in 3-D.
\begin{enumerate}[label=(\alph*)]
\item State the plane of reflection of $R$.
\item Determine the value of $a$.
\item With reference to R explain why $\mathbf { A } ^ { 2 } = \mathbf { I }$, the $3 \times 3$ identity matrix.\\
(a) By using Euler's formula show that $\cosh ( \mathrm { iz } ) = \cos z$.\\
(b) Hence, find, in logarithmic form, a root of the equation $\cos z = 2$. [You may assume that $\cos z = 2$ has complex roots.]

A swing door is a door to a room which is closed when in equilibrium but which can be pushed open from either side and which can swing both ways, into or out of the room, and through the equilibrium position. The door is sprung so that when displaced from the equilibrium position it will swing back towards it.

The extent to which the door is open at any time, $t$ seconds, is measured by the angle at the hinge, $\theta$, which the plane of the door makes with the plane of the equilibrium position. See the diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{20816f61-154d-4491-9d2d-4c62687bf81e-03_317_954_497_255}

In an initial model of the motion of a certain swing door it is suggested that $\theta$ satisfies the following differential equation.

$$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 25 \theta = 0$$

(a) (i) Write down the general solution to (\textit{).\\
(ii) With reference to the behaviour of your solution in part (a)(i) explain briefly why the model using (}) is unlikely to be realistic.

In an improved model of the motion of the door an extra term is introduced to the differential equation so that it becomes

$$4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm {~d} t } + 25 \theta = 0$$

where $\lambda$ is a positive constant.\\
(b) In the case where $\lambda = 16$ the door is held open at an angle of 0.9 radians and then released from rest at time $t = 0$.
\begin{enumerate}[label=(\roman*)]
\item Find, in a real form, the general solution of ( $\dagger$ ).
\item Find the particular solution of ( $\dagger$ ).
\item With reference to the behaviour of your solution found in part (b)(ii) explain briefly how the extra term in ( $\dagger$ ) improves the model.\\
(c) Find the value of $\lambda$ for which the door is critically damped.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q2 [6]}}

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