OCR Further Pure Core 2 (Further Pure Core 2) 2018 December

1

1. The Argand diagram below shows the two points which represent two complex numbers, \(z _ { 1 }\) and \(z _ { 2 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{e792797e-6d20-4fc3-9733-43db10f764d7-2_371_689_429_360} On the copy of the diagram in the Printed Answer Booklet
   • draw an appropriate shape to illustrate the geometrical effect of adding \(z _ { 1 }\) and \(z _ { 2 }\),
   • indicate with a cross \(( \times )\) the location of the point representing the complex number \(z _ { 1 } + z _ { 2 }\).
   • You are given that \(\arg z _ { 3 } = \frac { 1 } { 4 } \pi\) and \(\arg z _ { 4 } = \frac { 3 } { 8 } \pi\).
   In each part, sketch and label the points representing the numbers \(z _ { 3 } , z _ { 4 }\) and \(z _ { 3 } z _ { 4 }\) on the diagram in the Printed Answer Booklet. You should join each point to the origin with a straight line.
   (i) \(\left| z _ { 3 } \right| = 1.5\) and \(\left| z _ { 4 } \right| = 1.2\)
   (ii) \(\left| z _ { 3 } \right| = 0.7\) and \(\left| z _ { 4 } \right| = 0.5\)

2 In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the \(x\)-axis. \(\mathbf { A }\) is the matrix which represents S .

1. Write down \(\mathbf { A }\).
2. By considering the transformation represented by \(\mathbf { A } ^ { - 1 }\), determine the matrix \(\mathbf { A } ^ { - 1 }\). Matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { c c } 0 & - 1
   - 1 & 0 \end{array} \right)\). T is the transformation represented by \(\mathbf { B }\).
3. Describe T.
4. Determine the matrix which represents the transformation S followed by T .
5. Demonstrate, by direct calculation, that \(( \mathbf { B A } ) ^ { - 1 } = \mathbf { A } ^ { - 1 } \mathbf { B } ^ { - 1 }\).

3 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.

4 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.

1. State the plane of reflection of R .
2. Determine the value of \(a\).
3. With reference to R explain why \(\mathbf { A } ^ { 2 } = \mathbf { I }\), the \(3 \times 3\) identity matrix.

5

1. Find the shortest distance between the point ( \(- 6,4\) ) and the line \(y = - 0.75 x + 7\). Two lines, \(l _ { 1 }\) and \(l _ { 2 }\), are given by
   \(l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 4
   3
   - 2 \end{array} \right) + \lambda \left( \begin{array} { c } 2
   1
   - 4 \end{array} \right)\) and \(l _ { 2 } : \mathbf { r } = \left( \begin{array} { c } 11
   - 1
   5 \end{array} \right) + \mu \left( \begin{array} { c } 3
   - 1
   1 \end{array} \right)\).
2. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
3. Hence determine the geometrical arrangement of \(l _ { 1 }\) and \(l _ { 2 }\).

6 Three matrices, \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\), are given by \(\mathbf { A } = \left( \begin{array} { c c } 1 & 2
a & - 1 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c } 2 & - 1
4 & 1 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { c c } 5 & 0
- 2 & 2 \end{array} \right)\) where \(a\) is a

1. Using \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) in that order demonstrate explicitly the associativity property of matrix multiplication.
2. Use \(\mathbf { A }\) and \(\mathbf { C }\) to disprove by counterexample the proposition 'Matrix multiplication is commutative'. For a certain value of \(a , \mathbf { A } \binom { x } { y } = 3 \binom { x } { y }\).
3. Find
   • \(y\) in terms of \(x\),
   • the value of \(a\).
     \(7 C\) is the locus of numbers, \(z\), for which \(\operatorname { Im } \left( \frac { z + 7 \mathrm { i } } { z - 24 } \right) = \frac { 1 } { 4 }\).
     By writing \(z = x + \mathrm { i } y\) give a complete description of the shape of \(C\) on an Argand diagram.

8
\includegraphics[max width=\textwidth, alt={}, center]{e792797e-6d20-4fc3-9733-43db10f764d7-4_677_1182_587_440} The figure shows part of the graph of \(y = ( x - 3 ) \sqrt { \ln x }\). The portion of the graph below the \(x\)-axis is rotated by \(2 \pi\) radians around the \(x\)-axis to form a solid of revolution, \(S\). Determine the exact volume of \(S\).

9

1. By using Euler's formula show that \(\cosh ( \mathrm { i } z ) = \cos z\).
2. Hence, find, in logarithmic form, a root of the equation \(\cos z = 2\). [You may assume that \(\cos z = 2\) has complex roots.]

10 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]{e792797e-6d20-4fc3-9733-43db10f764d7-5_367_1116_625_246} In an initial model of the motion of a certain swing door it is suggested that \(\theta\) satisfies the following differential equation.
\(4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + 25 \theta = 0\)

1. 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
      \(4 \frac { \mathrm {~d} ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } + \lambda \frac { \mathrm { d } \theta } { \mathrm { d } t } + 25 \theta = 0 \quad ( \dagger )\)
      where \(\lambda\) is a positive constant.
2. 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.
3. Find the value of \(\lambda\) for which the door is critically damped.