Edexcel FP2 (Further Pure Mathematics 2) Specimen

1. (i) Use the Euclidean algorithm to find the highest common factor of 602 and 161.

Show each step of the algorithm.
(ii) The digits which can be used in a security code are the numbers \(1,2,3,4,5,6,7,8\) and 9. Originally the code used consisted of two distinct odd digits, followed by three distinct even digits. To enable more codes to be generated, a new system is devised. This uses two distinct even digits, followed by any three other distinct digits. No digits are repeated. Find the increase in the number of possible codes which results from using the new system.

1. A transformation from the \(z\)-plane to the \(w\)-plane is given by

$$w = z ^ { 2 }$$

1. Show that the line with equation \(\operatorname { Im } ( z ) = 1\) in the \(z\)-plane is mapped to a parabola in the \(w\)-plane, giving an equation for this parabola.
2. Sketch the parabola on an Argand diagram.

1. The matrix \(\mathbf { M }\) is given by

$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 0
1 & 2 & 0
- 1 & 0 & 4 \end{array} \right)$$

1. Show that 4 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
2. For each of the eigenvalues find a corresponding eigenvector.
3. Find a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P }\) is a diagonal matrix.

1. 1. A group \(G\) contains distinct elements \(a , b\) and \(e\) where \(e\) is the identity element and the group operation is multiplication.

Given \(a ^ { 2 } b = b a\), prove \(a b \neq b a\)
(ii) The set \(H = \{ 1,2,4,7,8,11,13,14 \}\) forms a group under the operation of multiplication modulo 15

1. Find the order of each element of \(H\).
2. Find three subgroups of \(H\) each of order 4, and describe each of these subgroups. The elements of another group \(J\) are the matrices \(\left( \begin{array} { c c } \cos \left( \frac { k \pi } { 4 } \right) & \sin \left( \frac { k \pi } { 4 } \right)
   - \sin \left( \frac { k \pi } { 4 } \right) & \cos \left( \frac { k \pi } { 4 } \right) \end{array} \right)\)
   where \(k = 1,2,3,4,5,6,7,8\) and the group operation is matrix multiplication.
3. Determine whether \(H\) and \(J\) are isomorphic, giving a reason for your answer.

5. \begin{figure}[h] \end{figure} An engineering student makes a miniature arch as part of the design for a piece of coursework. The cross-section of this arch is modelled by the curve with equation $$y = A - \frac { 1 } { 2 } \cosh 2 x , \quad - \ln a \leqslant x \leqslant \ln a$$ where \(a > 1\) and \(A\) is a positive constant. The curve begins and ends on the \(x\)-axis, as shown in Figure 1.

1. Show that the length of this curve is \(k \left( a ^ { 2 } - \frac { 1 } { a ^ { 2 } } \right)\), stating the value of the constant \(k\). The length of the curved cross-section of the miniature arch is required to be 2 m long.
2. Find the height of the arch, according to this model, giving your answer to 2 significant figures.
3. Find also the width of the base of the arch giving your answer to 2 significant figures.
4. Give the equation of another curve that could be used as a suitable model for the cross-section of an arch, with approximately the same height and width as you found using the first model.
   (You do not need to consider the arc length of your curve)

1. A curve has equation

$$| z + 6 | = 2 | z - 6 | \quad z \in \mathbb { C }$$

1. Show that the curve is a circle with equation \(x ^ { 2 } + y ^ { 2 } - 20 x + 36 = 0\)
2. Sketch the curve on an Argand diagram. The line \(l\) has equation \(a z ^ { * } + a ^ { * } z = 0\), where \(a \in \mathbb { C }\) and \(z \in \mathbb { C }\)
   Given that the line \(l\) is a tangent to the curve and that \(\arg a = \theta\)
3. find the possible values of \(\tan \theta\)

7. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$

1. Prove that, for \(n \geqslant 2\), $$n I _ { n } = ( n - 1 ) I _ { n - 2 }$$
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1c262813-4160-4eda-9a36-e4ba38182c8a-22_588_1018_630_520} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} A designer is asked to produce a poster to completely cover the curved surface area of a solid cylinder which has diameter 1 m and height 0.7 m . He uses a large sheet of paper with height 0.7 m and width of \(\pi \mathrm { m }\).
   Figure 2 shows the first stage of the design, where the poster is divided into two sections by a curve. The curve is given by the equation $$y = \sin ^ { 2 } ( 4 x ) - \sin ^ { 10 } ( 4 x )$$ relative to axes taken along the bottom and left hand edge of the paper.
   The region of the poster below the curve is shaded and the region above the curve remains unshaded, as shown in Figure 2. Find the exact area of the poster which is shaded.

1. A staircase has \(n\) steps. A tourist moves from the bottom (step zero) to the top (step \(n\) ). At each move up the staircase she can go up either one step or two steps, and her overall climb up the staircase is a combination of such moves.

If \(u _ { n }\) is the number of ways that the tourist can climb up a staircase with \(n\) steps,

1. explain why \(u _ { n }\) satisfies the recurrence relation $$u _ { n } = u _ { n - 1 } + u _ { n - 2 } , \text { with } u _ { 1 } = 1 \text { and } u _ { 2 } = 2$$
2. Find the number of ways in which she can climb up a staircase when there are eight steps. A staircase at a certain tourist attraction has 400 steps.
3. Show that the number of ways in which she could climb up to the top of this staircase is given by $$\frac { 1 } { \sqrt { 5 } } \left[ \left( \frac { 1 + \sqrt { 5 } } { 2 } \right) ^ { 401 } - \left( \frac { 1 - \sqrt { 5 } } { 2 } \right) ^ { 401 } \right]$$