OCR Further Pure Core 2 (Further Pure Core 2) 2023 June

1

1. The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { l l l l } 1 & 0 & - 2 & 2
   4 & 2 & - 2 & 3 \end{array} \right)\).
   1. Write down the dimensions of \(\mathbf { P }\).
   2. Write down the transpose of \(\mathbf { P }\).
2. The matrices \(\mathbf { Q } , \mathbf { R }\) and \(\mathbf { S }\) are given by \(\mathbf { Q } = \left( \begin{array} { l l } 1 & 2 \end{array} \right) , \mathbf { R } = \left( \begin{array} { r r } 3 & - 4
   2 & 3 \end{array} \right)\) and \(\mathbf { S } = \left( \begin{array} { l l } 3 & - 2 \end{array} \right)\). Write down the sum of the two of these matrices which are conformable for addition.
3. The dimensions of matrix \(\mathbf { A }\) are 4 by 5. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are conformable for multiplication so that the matrix \(\mathbf { C } = \mathbf { B A }\) can be formed. The matrix \(\mathbf { C }\) has 6 rows.
   1. Write down the number of columns that \(\mathbf { C }\) has.
   2. Write down the dimensions of \(\mathbf { B }\).
   3. Explain whether the matrix \(\mathbf { A B }\) can be formed.
4. Find the value of \(c\) for which \(\left( \begin{array} { r r } - 2 & 3
   6 & 10 \end{array} \right) \left( \begin{array} { r r } c & 5
   10 & 13 \end{array} \right) = \left( \begin{array} { r r } c & 5
   10 & 13 \end{array} \right) \left( \begin{array} { r r } - 2 & 3
   6 & 10 \end{array} \right)\).

2 In this question you must show detailed reasoning.

1. Write the complex number \(- 24 + 7 \mathrm { i }\) in modulus-argument form.
2. Solve the simultaneous equations given below, giving your answers in cartesian form. $$\begin{aligned} i z + 3 w & = - 7 i
   - 6 z + 5 i w & = 3 + 13 i \end{aligned}$$

3

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } u } \left( \sinh ^ { - 1 } u \right) = \frac { 1 } { \sqrt { u ^ { 2 } + 1 } }\).
2. Find the equation of the normal to the graph of \(\mathrm { y } = \sinh ^ { - 1 } 2 \mathrm { x }\) at the point where \(x = \sqrt { 6 }\). Give your answer in the form \(\mathrm { y } = \mathrm { mx } + \mathrm { c }\) where \(m\) and \(c\) are given in exact, non-hyperbolic form.

4 In this question you must show detailed reasoning. The region \(R\) is bounded by the curve with equation \(\mathrm { y } = \frac { 1 } { \sqrt { 3 \mathrm { x } ^ { 2 } - 3 \mathrm { x } + 1 } }\), the \(x\)-axis and the lines with equations \(x = \frac { 1 } { 2 }\) and \(x = 1\) (see diagram). The units of the axes are cm .
\includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-3_778_1241_497_242} A pendant is to be made out of a precious metal. The shape of the pendant is modelled as the shape formed when \(R\) is rotated by \(2 \pi\) radians about the \(x\)-axis. Find the exact value of the volume of precious metal required to make the pendant, according to the model.

5 In this question you must show detailed reasoning.

1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, show that \(\sinh 2 x \equiv 2 \sinh x \cosh x\).
2. Solve the equation \(15 \sinh x + 16 \cosh x - 6 \sinh 2 x = 20\), giving all your answers in logarithmic form.

6 The equation of the plane \(\Pi\) is \(\mathbf { r } = \left( \begin{array} { r } - 1
2
1 \end{array} \right) + \lambda \left( \begin{array} { l } 4
4
3 \end{array} \right) + \mu \left( \begin{array} { r } - 2
3
1 \end{array} \right)\).

1. Find the acute angle between \(\Pi\) and the plane with equation \(\mathbf { r } . \left( \begin{array} { l } 2
   0
   3 \end{array} \right) = 4\). The point \(A\) has coordinates ( \(9 , - 7,20\) ).
   The point \(F\) is the point of intersection between \(\Pi\) and the perpendicular from \(A\) to \(\Pi\).
2. Determine the coordinates of \(F\).

7 In this question you must show detailed reasoning.

1. Show that $$\sum _ { r = 1 } ^ { n } \frac { 5 r + 6 } { r ^ { 3 } + r ^ { 2 } } = \frac { a } { n + 1 } + b + c \sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are integers whose values are to be determined. You are given that \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ^ { 2 } }\) exists and is equal to \(\frac { 1 } { 6 } \pi ^ { 2 }\).
2. Show that \(\sum _ { r = 1 } ^ { \infty } \frac { 5 r + 6 } { r ^ { 3 } + r ^ { 2 } }\) exists and is equal to \(( \pi - 1 ) ( \pi + 1 )\).

8 A surge in the current, \(I\) units, through an electrical component at a time, \(t\) seconds, is to be modelled. The surge starts when \(t = 0\) and there is initially no current through the component. When the current has surged for 1 second it is measured as being 5 units. While the surge is occurring, \(I\) is modelled by the following differential equation.
\(\left( 2 t - t ^ { 2 } \right) \frac { d l } { d t } = \left( 2 t - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } - 2 ( t - 1 ) l\)

1. By using an integrating factor show that, according to the model, while the surge is occurring, \(I\) is given by \(\mathrm { I } = \left( 2 \mathrm { t } - \mathrm { t } ^ { 2 } \right) \left( \sin ^ { - 1 } ( \mathrm { t } - 1 ) + 5 \right)\). The surge lasts until there is again no current through the component.
2. Determine the length of time that the surge lasts according to the model.
3. Determine, according to the model, the rate of increase of the current at the start of the surge. Give your answer in an exact form.

9 A function is defined by \(y = f ( t )\) where \(f ( t ) = \ln ( 1 + a t )\) and \(a\) is a constant.

1. By considering \(\frac { d y } { d t } , \frac { d ^ { 2 } y } { d t ^ { 2 } } , \frac { d ^ { 3 } y } { d t ^ { 3 } }\) and \(\frac { d ^ { 4 } y } { d t ^ { 4 } }\), make a conjecture for a general formula for \(\frac { d ^ { n } y } { d t ^ { n } }\) in terms of \(n\) and \(a\) for any integer \(n \geqslant 1\).
2. Use induction to prove the formula conjectured in part (a).
3. In the case where \(\mathrm { f } ( t ) = \ln ( 1 + 2 t )\), find the rate at which the \(6 ^ { \text {th } }\) derivative of \(\mathrm { f } ( t )\) is varying when \(t = \frac { 3 } { 2 }\).

10 In this question you must show detailed reasoning. A region, \(R\), of the floor of an art gallery is to be painted for the purposes of an art installation. A suitable polar coordinate system is set up on the floor of the gallery with units in metres and radians. \(R\) is modelled as being the region enclosed by two curves, \(C _ { 1 }\) and \(C _ { 2 }\). The polar equations of \(C _ { 1 }\) and \(C _ { 2 }\) are $$\begin{array} { l l } C _ { 1 } : r = 5 , & - \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi
C _ { 2 } : r = 3 \cosh \theta , & - \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi \end{array}$$ Both curves are shown in the diagram, with \(R\) indicated.
\includegraphics[max width=\textwidth, alt={}, center]{7b2bfb4e-524f-4d1c-ae98-075c7fb404f9-6_1481_821_836_251} The gallery must buy tins of paint to paint \(R\). Each tin of paint can cover an area of \(0.5 \mathrm {~m} ^ { 2 }\).
Determine the smallest number of tins of paint that the gallery must buy in order to be able to paint \(R\) completely.