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

1 Plane \(\Pi\) has equation \(3 x - y + 2 z = 33\). Line \(l\) has the following vector equation. $$l : \quad \mathbf { r } = \left( \begin{array} { l } 1
0
5 \end{array} \right) + \lambda \left( \begin{array} { l }

3 \end{array} \right)$$

1. Find the acute angle between \(\Pi\) and \(l\).
2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\).
3. \(S\) is the point \(( 4,5 , - 5 )\). Find the shortest distance from \(S\) to \(\Pi\). 2 The complex number \(2 + \mathrm { i }\) is denoted by \(z\).
4. Show that \(z ^ { 2 } = 3 + 4 \mathrm { i }\).
5. Plot the following on the Argand diagram in the Printed Answer Booklet.
   • \(z\)
   • \(z ^ { 2 }\)
   • State the relationship between \(\left| z ^ { 2 } \right|\) and \(| z |\).
   • State the relationship between \(\arg \left( z ^ { 2 } \right)\) and \(\arg ( z )\).
   3 In this question you must show detailed reasoning. Use the formula \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to evaluate \(121 ^ { 2 } + 122 ^ { 2 } + 123 ^ { 2 } + \ldots + 300 ^ { 2 }\).

5 \end{array} \right) + \lambda \left( \begin{array} { l } 2
2
3 \end{array} \right)$$

1. Find the acute angle between \(\Pi\) and \(l\).
2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\).
3. \(S\) is the point \(( 4,5 , - 5 )\). Find the shortest distance from \(S\) to \(\Pi\). 2 The complex number \(2 + \mathrm { i }\) is denoted by \(z\).
4. Show that \(z ^ { 2 } = 3 + 4 \mathrm { i }\).
5. Plot the following on the Argand diagram in the Printed Answer Booklet.
   • \(z\)
   • \(z ^ { 2 }\)
   • State the relationship between \(\left| z ^ { 2 } \right|\) and \(| z |\).
   • State the relationship between \(\arg \left( z ^ { 2 } \right)\) and \(\arg ( z )\).
   3 In this question you must show detailed reasoning. Use the formula \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) to evaluate \(121 ^ { 2 } + 122 ^ { 2 } + 123 ^ { 2 } + \ldots + 300 ^ { 2 }\). 4 You are given that the cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + x + 4 = 0\) has three roots, \(\alpha , \beta\) and \(\gamma\).
   By making a suitable substitution to obtain a related cubic equation, determine the value of \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\). 5 In this question you must show detailed reasoning.
   An ant starts from a fixed point \(O\) and walks in a straight line for 1.5 s . Its velocity, \(v \mathrm {~cm} \mathrm {~s} ^ { - 1 }\), can be modelled by \(v = \frac { 1 } { \sqrt { 9 - t ^ { 2 } } }\). By finding the mean value of \(v\) in \(0 \leqslant t \leqslant 1.5\), deduce the average velocity of the ant.

6 In this question you must show detailed reasoning.

1. Find the coordinates of all stationary points on the graph of \(y = 6 \sinh ^ { 2 } x - 13 \cosh x\), giving your answers in an exact, simplified form.
2. By finding the second derivative, classify the stationary points found in part (i).

7 In the following set of simultaneous equations, \(a\) and \(b\) are constants. $$\begin{aligned} 3 x + 2 y - z & = 5
2 x - 4 y + 7 z & = 60
a x + 20 y - 25 z & = b \end{aligned}$$

1. In the case where \(a = 10\), solve the simultaneous equations, giving your solution in terms of \(b\).
2. Determine the value of \(a\) for which there is no unique solution for \(x , y\) and \(z\).
3. (a) Find the values of \(\alpha\) and \(\beta\) for which \(\alpha ( 2 y - z ) + \beta ( - 4 y + 7 z ) = 20 y - 25 z\) for any \(y\) and \(z\).
   (b) Hence, for the case where there is no unique solution for \(x , y\) and \(z\), determine the value of \(b\) for which there is an infinite number of solutions.
   (c) When \(a\) takes the value in part (ii) and \(b\) takes the value in part (iii)(b) describe the geometrical arrangement of the planes represented by the three equations.

8 In this question you must show detailed reasoning.
Show that \(\int _ { 0 } ^ { 2 } \frac { 2 x ^ { 2 } + 3 x - 1 } { x ^ { 3 } - 3 x ^ { 2 } + 4 x - 12 } \mathrm {~d} x = \frac { 3 } { 8 } \pi - \ln 9\).

9 In this question you must show detailed reasoning.

1. Show that \(\mathrm { e } ^ { \mathrm { i } \theta } - \mathrm { e } ^ { - \mathrm { i } \theta } = 2 \mathrm { i } \sin \theta\).
2. Hence, show that \(\frac { 2 } { \mathrm { e } ^ { 2 \mathrm { i } \theta } - 1 } = - ( 1 + \mathrm { i } \cot \theta )\).
3. Two series, \(C\) and \(S\), are defined as follows. $$\begin{aligned} & C = 2 + 2 \cos \frac { \pi } { 10 } + 2 \cos \frac { \pi } { 5 } + 2 \cos \frac { 3 \pi } { 10 } + 2 \cos \frac { 2 \pi } { 5 }
   & S = 2 \sin \frac { \pi } { 10 } + 2 \sin \frac { \pi } { 5 } + 2 \sin \frac { 3 \pi } { 10 } + 2 \sin \frac { 2 \pi } { 5 } \end{aligned}$$ By considering \(C + \mathrm { i } S\), find a simplified expression for \(C\) in terms of only integers and \(\cot \frac { \pi } { 20 }\).
4. Verify that \(S = C - 2\) and, by considering the series in their original form, explain why this is so. \section*{END OF QUESTION PAPER}