OCR FP3 (Further Pure Mathematics 3) 2012 June

1 The plane \(p\) has equation \(\mathbf { r } . ( \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } ) = 4\) and the line \(l _ { 1 }\) has equation \(\mathbf { r } = 2 \mathbf { j } - \mathbf { k } + t ( 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\). The line \(l _ { 2 }\) is parallel to \(p\) and perpendicular to \(l _ { 1 }\), and passes through the point with position vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\). Find the equation of \(l _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).

2

1. Solve the equation \(z ^ { 4 } = 2 ( 1 + \mathrm { i } \sqrt { 3 } )\), giving the roots exactly in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
2. Sketch an Argand diagram to show the lines from the origin to the point representing \(2 ( 1 + i \sqrt { 3 } )\) and from the origin to the points which represent the roots of the equation in part (i).

3 Find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \cot x = 2 x$$ for which \(y = 2\) when \(x = \frac { 1 } { 6 } \pi\). Give your answer in the form \(y = \mathrm { f } ( x )\).

4 The elements \(a , b , c , d\) are combined according to the operation table below, to form a group \(G\) of order 4. Group \(G\) is isomorphic either to the multiplicative group \(H = \left\{ e , r , r ^ { 2 } , r ^ { 3 } \right\}\) or to the multiplicative group \(K = \{ e , p , q , p q \}\). It is given that \(r ^ { 4 } = e\) in group \(H\) and that \(p ^ { 2 } = q ^ { 2 } = e\) in group \(K\), where \(e\) denotes the identity in each group.

1. Write down the operation tables for \(H\) and \(K\).
2. State the identity element of \(G\).
3. Demonstrate the isomorphism between \(G\) and either \(H\) or \(K\) by listing how the elements of \(G\) correspond to the elements of the other group. If the correspondence can be shown in more than one way, list the alternative correspondence(s).

5

1. By expressing \(\sin \theta\) and \(\cos \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), prove that $$\sin ^ { 3 } \theta \cos ^ { 2 } \theta \equiv - \frac { 1 } { 16 } ( \sin 5 \theta - \sin 3 \theta - 2 \sin \theta )$$
2. Hence show that all the roots of the equation $$\sin 5 \theta = \sin 3 \theta + 2 \sin \theta$$ are of the form \(\theta = \frac { n \pi } { k }\), where \(n\) is any integer and \(k\) is to be determined.

6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 12 \mathrm { e } ^ { 2 x }$$

1. Find the general solution of the differential equation.
2. It is given that the curve which represents a particular solution of the differential equation has gradient 6 when \(x = 0\), and approximates to \(y = \mathrm { e } ^ { 2 x }\) when \(x\) is large and positive. Find the equation of the curve.

7 With respect to the origin \(O\), the position vectors of the points \(U , V\) and \(W\) are \(\mathbf { u } , \mathbf { v }\) and \(\mathbf { w }\) respectively. The mid-points of the sides \(V W , W U\) and \(U V\) of the triangle \(U V W\) are \(M , N\) and \(P\) respectively.

1. Show that \(\overrightarrow { U M } = \frac { 1 } { 2 } ( \mathbf { v } + \mathbf { w } - 2 \mathbf { u } )\).
2. Verify that the point \(G\) with position vector \(\frac { 1 } { 3 } ( \mathbf { u } + \mathbf { v } + \mathbf { w } )\) lies on \(U M\), and deduce that the lines \(U M , V N\) and \(W P\) intersect at \(G\).
3. Write down, in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), an equation of the line through \(G\) which is perpendicular to the plane \(U V W\). (It is not necessary to simplify the expression for \(\mathbf { b }\).)
4. It is now given that \(\mathbf { u } = \left( \begin{array} { l } 1
   0
   0 \end{array} \right) , \mathbf { v } = \left( \begin{array} { l } 0
   1
   0 \end{array} \right)\) and \(\mathbf { w } = \left( \begin{array} { l } 0
   0
   1 \end{array} \right)\). Find the perpendicular distance from \(O\) to the plane \(U V W\).

8 The set \(M\) of matrices \(\left( \begin{array} { l l } a & b
c & d \end{array} \right)\), where \(a , b , c\) and \(d\) are real and \(a d - b c = 1\), forms a group \(( M , \times )\) under matrix multiplication. \(R\) denotes the set of all matrices \(\left( \begin{array} { c c } \cos \theta & - \sin \theta
\sin \theta & \cos \theta \end{array} \right)\).

1. Prove that ( \(R , \times\) ) is a subgroup of ( \(M , \times\) ).
2. By considering geometrical transformations in the \(x - y\) plane, find a subgroup of \(( R , \times )\) of order 6 . Give the elements of this subgroup in exact numerical form. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}