Question 3 - A-Level Maths

OCR Further Pure Core 1 2021 June — Question 3 6 marks

Exam Board	OCR
Module	Further Pure Core 1 (Further Pure Core 1)
Year	2021
Session	June
Marks	6
Topic	Second order differential equations
Type	Particular solution with initial conditions
Difficulty	Standard +0.8 This is a multi-part Further Maths question combining 3D geometry (reflection in a plane), matrix determinants, simultaneous equations with parameters, and second-order differential equations with particular solutions. While each component uses standard techniques, the combination of topics, the parameter work in parts (b)-(c), and the differential equation requiring both complementary function and particular integral make this moderately challenging for Further Maths students.
Spec	4.03i Determinant: area scale factor and orientation 4.03r Solve simultaneous equations: using inverse matrix 4.04j Shortest distance: between a point and a plane

3 The equation of a plane is $4 x + 2 y + z = 7$.
The point $A$ has coordinates $( 9,6,1 )$ and the point $B$ is the reflection of $A$ in the plane.
Find the coordinates of the point $B$. You are given the matrix $\mathbf { A }$ where $\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 0 \\ 0 & a & 2 \\ 4 & 5 & 1 \end{array} \right)$.

Find, in terms of $a$, the determinant of $\mathbf { A }$, simplifying your answer.
Hence find the values of $a$ for which $\mathbf { A }$ is singular. You are given the following equations which are to be solved simultaneously. $$\begin{aligned} a x + 2 y & = 6 \\ a y + 2 z & = 8 \\ 4 x + 5 y + z & = 16 \end{aligned}$$
For each of the values of $a$ found in part (b) determine whether the equations have
A particle is suspended in a resistive medium from one end of a light spring. The other end of the spring is attached to a point which is made to oscillate in a vertical line. The displacement of the particle may be modelled by the differential equation $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 10 \sin t$ where $x$ is the displacement of the particle below the equilibrium position at time $t$.
When $t = 0$ the particle is stationary and its displacement is 2 .
1. Find the particular solution of the differential equation.
2. Write down an approximate equation for the displacement when $t$ is large.

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3 The equation of a plane is $4 x + 2 y + z = 7$.\\
The point $A$ has coordinates $( 9,6,1 )$ and the point $B$ is the reflection of $A$ in the plane.\\
Find the coordinates of the point $B$.

You are given the matrix $\mathbf { A }$ where $\mathbf { A } = \left( \begin{array} { l l l } a & 2 & 0 \\ 0 & a & 2 \\ 4 & 5 & 1 \end{array} \right)$.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $a$, the determinant of $\mathbf { A }$, simplifying your answer.
\item Hence find the values of $a$ for which $\mathbf { A }$ is singular.

You are given the following equations which are to be solved simultaneously.

$$\begin{aligned}
a x + 2 y & = 6 \\
a y + 2 z & = 8 \\
4 x + 5 y + z & = 16
\end{aligned}$$
\item For each of the values of $a$ found in part (b) determine whether the equations have

\begin{itemize}
  \item a unique solution, which should be found, or
  \item an infinite set of solutions or
  \item no solution.
\end{itemize}

A particle is suspended in a resistive medium from one end of a light spring. The other end of the spring is attached to a point which is made to oscillate in a vertical line.

The displacement of the particle may be modelled by the differential equation\\
$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 10 \sin t$\\
where $x$ is the displacement of the particle below the equilibrium position at time $t$.\\
When $t = 0$ the particle is stationary and its displacement is 2 .\\
(a) Find the particular solution of the differential equation.\\
(b) Write down an approximate equation for the displacement when $t$ is large.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q3 [6]}}

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