OCR Further Pure Core 2 (Further Pure Core 2) 2024 June

1

1. Use the method of differences to show that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \left( \frac { 1 } { \mathrm { r } } - \frac { 1 } { \mathrm { r } + 1 } \right) = 1 - \frac { 1 } { \mathrm { n } + 1 }\).
2. Hence determine the following sums.
   1. \(\quad \sum _ { r = 1 } ^ { 99 } \frac { 1 } { r } - \frac { 1 } { r + 1 }\)
   2. \(\quad \sum _ { r = 100 } ^ { \infty } \frac { 1 } { r } - \frac { 1 } { r + 1 }\)

2 In this question you must show detailed reasoning.

1. Solve the equation \(x ^ { 2 } - 6 x + 58 = 0\). Give your solutions in the form \(a + b\) i where \(a\) and \(b\) are real numbers.
2. Determine, in exact form, \(\arg ( - 10 + ( 5 \sqrt { 12 } ) \mathrm { i } ) ^ { 5 }\).

3 Matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by \(\mathbf { A } = \left( \begin{array} { r r } 4 & - 3
- 2 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 3 & - 5
0 & 1 \end{array} \right)\).

1. Find 2A - 4B.
2. Write down the matrix \(\mathbf { C }\) such that \(\mathbf { A C } = 2 \mathbf { A }\).
3. Find the value of \(\operatorname { det } \mathbf { A }\).
4. In this question you must show detailed reasoning. Use \(\mathbf { A } ^ { - 1 }\) to solve the equations \(4 \mathrm { x } - 3 \mathrm { y } = 7\) and \(- 2 \mathrm { x } + 2 \mathrm { y } = 9\).

4 In this question you must show detailed reasoning.
The series \(S\) is defined as being the sum of the squares of all positive odd integers from \(1 ^ { 2 }\) to \(779 ^ { 2 }\). Determine the value of \(S\).

5 Vectors, \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\), are given by \(\mathbf { a } = \mathbf { i } + ( 1 - p ) \mathbf { j } + ( p + 2 ) \mathbf { k } , \mathbf { b } = 2 \mathbf { i } + \mathbf { j } + \mathbf { k }\) and \(\mathbf { c } = \mathbf { i } + 14 \mathbf { j } + ( p - 3 ) \mathbf { k }\) where \(p\) is a constant. You are given that \(\mathbf { a } \times \mathbf { b }\) is perpendicular to \(\mathbf { c }\). Determine the possible values of \(p\).

6 In polar coordinates, the equation of a curve, \(C\), is \(r = 6 \sin ( 2 \theta ) \sinh \left( \frac { 1 } { 3 } \theta \right)\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
The pole of the polar coordinate system corresponds to the origin of the cartesian system and the initial line corresponds to the positive \(x\)-axis.

1. Explain how you can tell that \(C\) comprises a single loop in the first quadrant, passing through the pole. The incomplete table below shows values of \(r\) for various values of \(\theta\).

   \(\theta\)	0	\(\frac { 1 } { 12 } \pi\)	\(\frac { 1 } { 6 } \pi\)	\(\frac { 1 } { 4 } \pi\)	\(\frac { 1 } { 3 } \pi\)	\(\frac { 5 } { 12 } \pi\)	\(\frac { 1 } { 2 } \pi\)
   \(r\)	0	0.262			1.851

2. Use the copy of the table and the polar coordinate system diagram given in the Printed Answer Booklet to complete the table and sketch \(C\). The point on \(C\) which is furthest away from the pole is denoted by \(A\) and the value of \(\theta\) at \(A\) is denoted by \(\phi\).
3. Show that \(\phi\) satisfies the equation \(\phi = \frac { 3 } { 2 } \ln \left( \frac { 6 - \tan 2 \phi } { 6 + \tan 2 \phi } \right)\)
4. You are given that the relevant solution of the equation given in part (c) is \(\phi = 1.0207\) correct to 5 significant figures. Find the distance from \(A\) to the pole. Give your answer correct to \(\mathbf { 3 }\) significant figures.

7

1. Express \(17 \cosh x - 15 \sinh x\) in the form \(\mathrm { e } ^ { - \mathrm { x } } \left( \mathrm { ae } ^ { \mathrm { bx } } + \mathrm { c } \right)\) where \(a , b\) and \(c\) are integers to be determined. A function is defined by \(\mathrm { f } ( x ) = \frac { 1 } { \sqrt { 17 \cosh x - 15 \sinh x } }\). The region bounded by the curve \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), the \(x\)-axis, the \(y\)-axis and the line \(x = \ln 3\) is rotated by \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution \(S\). \section*{(b) In this question you must show detailed reasoning.} Use a suitable substitution, together with known results from the formula book, to show that the volume of \(S\) is given by \(k \pi \tan ^ { - 1 } q\) where \(k\) and \(q\) are rational numbers to be determined.

9 In this question, the argument of a complex number is defined as being in the range \([ 0,2 \pi )\).
You are given that \(\omega _ { k }\), where \(k = 0,1,2 , \ldots , n - 1\), are the \(n n ^ { \text {th } }\) roots of unity for some integer \(n , n \geqslant 3\), and that these are given in order of increasing argument (so that \(\omega _ { 0 } = 1\) ).

1. With the help of a diagram explain why \(\omega _ { k } = \left( \omega _ { 1 } \right) ^ { k }\) for \(k = 2 , \ldots , n - 1\).
2. Using the identity given in part (a), show that \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } \omega _ { \mathrm { k } } = 0\).
3. Show that if \(z\) is a complex number then \(z + z ^ { * } = 2 \operatorname { Re } ( z )\).
4. Using the results from parts (b) and (c) show that \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } \operatorname { Re } \left( \omega _ { \mathrm { k } } \right) = 0\).
5. With the help of a diagram explain why \(\operatorname { Re } \left( \omega _ { \mathrm { k } } \right) = \operatorname { Re } \left( \omega _ { \mathrm { n } - \mathrm { k } } \right)\) for \(k = 1,2 , \ldots , n - 1\). You should now consider the case where \(n = 5\).
6. 1. Use parts (d) and (e) to deduce that \(\cos \frac { 4 \pi } { 5 } = \mathrm { a } + \mathrm { b } \cos \frac { 2 \pi } { 5 }\), for some rational constants \(a\) and \(b\).
   2. Hence determine the exact value of \(\cos \frac { 2 \pi } { 5 }\).