OCR FP3 (Further Pure Mathematics 3) 2008 January

1. A group \(G\) of order 6 has the combination table shown below.

   	\(e\)	\(a\)	\(b\)	\(p\)	\(q\)	\(r\)
   \(e\)	\(e\)	\(a\)	\(b\)	\(p\)	\(q\)	\(r\)
   \(a\)	\(a\)	\(b\)	\(e\)	\(r\)	\(p\)	\(q\)
   \(b\)	\(b\)	\(e\)	\(a\)	\(q\)	\(r\)	\(p\)
   \(p\)	\(p\)	\(q\)	\(r\)	\(e\)	\(a\)	\(b\)
   \(q\)	\(q\)	\(r\)	\(p\)	\(b\)	\(e\)	\(a\)
   \(r\)	\(r\)	\(p\)	\(q\)	\(a\)	\(b\)	\(e\)

   1. State, with a reason, whether or not \(G\) is commutative. [1]
   2. State the number of subgroups of \(G\) which are of order 2. [1]
   3. List the elements of the subgroup of \(G\) which is of order 3. [1]
2. A multiplicative group \(H\) of order 6 has elements \(e, c, c^2, c^3, c^4, c^5\), where \(e\) is the identity. Write down the order of each of the elements \(c^3, c^4\) and \(c^5\). [3]

Two fixed points, \(A\) and \(B\), have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) relative to the origin \(O\), and a variable point \(P\) has position vector \(\mathbf{r}\).

1. Give a geometrical description of the locus of \(P\) when \(\mathbf{r}\) satisfies the equation \(\mathbf{r} = \lambda\mathbf{a}\), where \(0 \leq \lambda \leq 1\). [2]
2. Given that \(P\) is a point on the line \(AB\), use a property of the vector product to explain why \((\mathbf{r} - \mathbf{a}) \times (\mathbf{r} - \mathbf{b}) = \mathbf{0}\). [2]
3. Give a geometrical description of the locus of \(P\) when \(\mathbf{r}\) satisfies the equation \(\mathbf{r} \times (\mathbf{a} - \mathbf{b}) = \mathbf{0}\). [3]

The integrals \(C\) and \(S\) are defined by $$C = \int_0^{3\pi} e^{3x} \cos 3x \, dx \quad \text{and} \quad S = \int_0^{3\pi} e^{3x} \sin 3x \, dx.$$ By considering \(C + iS\) as a single integral, show that $$C = -\frac{1}{3}(2 + 3e^{\pi}),$$ and obtain a similar expression for \(S\). [8] (You may assume that the standard result for \(\int e^{kx} dx\) remains true when \(k\) is a complex constant, so that \(\int e^{(a+ib)x} dx = \frac{1}{a+ib}e^{(a+ib)x}\).)

1. Find the general solution of the differential equation $$\frac{dy}{dx} - \frac{y}{x} = \sin 2x,$$ expressing \(y\) in terms of \(x\) in your answer. [6]

In a particular case, it is given that \(y = \frac{2}{\pi}\) when \(x = \frac{1}{4}\pi\).

1. Find the solution of the differential equation in this case. [2]
2. Write down a function to which \(y\) approximates when \(x\) is large and positive. [1]

A tetrahedron \(ABCD\) is such that \(AB\) is perpendicular to the base \(BCD\). The coordinates of the points \(A\), \(C\) and \(D\) are \((-1, -7, 2)\), \((5, 0, 3)\) and \((-1, 3, 3)\) respectively, and the equation of the plane \(BCD\) is \(x + 2y - 2z = -1\).

1. Find, in either order, the coordinates of \(B\) and the length of \(AB\). [5]
2. Find the acute angle between the planes \(ACD\) and \(BCD\). [6]

1. 1. Verify, without using a calculator, that \(\theta = \frac{1}{8}\pi\) is a solution of the equation \(\sin 6\theta = \sin 2\theta\). [1]
   2. By sketching the graphs of \(y = \sin 6\theta\) and \(y = \sin 2\theta\) for \(0 < \theta < \frac{1}{2}\pi\), or otherwise, find the other solution of the equation \(\sin 6\theta = \sin 2\theta\) in the interval \(0 < \theta < \frac{1}{2}\pi\). [2]
2. Use de Moivre's theorem to prove that $$\sin 6\theta = \sin 2\theta (16 \cos^4 \theta - 16 \cos^2 \theta + 3).$$ [5]
3. Hence show that one of the solutions obtained in part (i) satisfies \(\cos^2 \theta = \frac{1}{4}(2 - \sqrt{2})\), and justify which solution it is. [3]

Groups \(A\), \(B\), \(C\) and \(D\) are defined as follows: \(A\): the set of numbers \(\{2, 4, 6, 8\}\) under multiplication modulo 10, \(B\): the set of numbers \(\{1, 5, 7, 11\}\) under multiplication modulo 12, \(C\): the set of numbers \(\{2^0, 2^1, 2^2, 2^3\}\) under multiplication modulo 15, \(D\): the set of numbers \(\left\{\frac{1+2m}{1+2n}, \text{ where } m \text{ and } n \text{ are integers}\right\}\) under multiplication.

1. Write down the identity element for each of groups \(A\), \(B\), \(C\) and \(D\). [2]
2. Determine in each case whether the groups
   \(A\) and \(B\), \(B\) and \(C\), \(A\) and \(C\)
   are isomorphic or non-isomorphic. Give sufficient reasons for your answers. [5]
3. Prove the closure property for group \(D\). [4]
4. Elements of the set \(\left\{\frac{1+2m}{1+2n}, \text{ where } m \text{ and } n \text{ are integers}\right\}\) are combined under addition. State which of the four basic group properties are not satisfied. (Justification is not required.) [2]