OCR FP3 (Further Pure Mathematics 3) 2011 June

A line \(l\) has equation \(\frac{x-1}{5} = \frac{y-6}{6} = \frac{z+3}{-7}\) and a plane \(p\) has equation \(x + 2y - z = 40\).

1. Find the acute angle between \(l\) and \(p\). [4]
2. Find the perpendicular distance from the point \((1, 6, -3)\) to \(p\). [2]

It is given that \(z = e^{i\theta}\), where \(0 < \theta < 2\pi\), and \(w = \frac{1+z}{1-z}\).

1. Prove that \(w = i \cot \frac{1}{2}\theta\). [3]
2. Sketch separate Argand diagrams to show the locus of \(z\) and the locus of \(w\). You should show the direction in which each locus is described when \(\theta\) increases in the interval \(0 < \theta < 2\pi\). [3]

The variables \(x\) and \(y\) satisfy the differential equation $$\frac{dy}{dx} + 4y = 5 \cos 3x.$$

1. Find the complementary function. [2]
2. Hence, or otherwise, find the general solution. [7]
3. Find the approximate range of values of \(y\) when \(x\) is large and positive. [2]

A group \(G\), of order 8, is generated by the elements \(a\), \(b\), \(c\). \(G\) has the properties $$a^2 = b^2 = c^2 = e, \quad ab = ba, \quad bc = cb, \quad ca = ac,$$ where \(e\) is the identity.

1. Using these properties and basic group properties as necessary, prove that \(abc = cba\). [2]

The operation table for \(G\) is shown below.

1. List all the subgroups of order 2. [2]
2. List five subgroups of order 4. [3]
3. Determine whether all the subgroups of \(G\) which are of order 4 are isomorphic. [2]

The substitution \(y = u^k\), where \(k\) is an integer, is to be used to solve the differential equation $$x \frac{dy}{dx} + 3y = x^2 y^2 \qquad (A)$$ by changing it into an equation (B) in the variables \(u\) and \(x\).

1. Show that equation (B) may be written in the form $$\frac{du}{dx} + \frac{3}{kx} u = \frac{1}{k} x u^{k+1}.$$ [4]
2. Write down the value of \(k\) for which the integrating factor method may be used to solve equation (B). [1]
3. Using this value of \(k\), solve equation (B) and hence find the general solution of equation (A), giving your answer in the form \(y = f(x)\). [4]

1. The set of polynomials \(\{ax + b\}\), where \(a, b \in \mathbb{R}\), is denoted by \(P\). Assuming that the associativity property holds, prove that \(P\), under addition, is a group. [4]
2. The set of polynomials \(\{ax + b\}\), where \(a, b \in \{0, 1, 2\}\), is denoted by \(Q\). It is given that \(Q\), under addition modulo 3, is a group, denoted by \((Q, +(\text{mod}3))\).
   1. State the order of the group. [1]
   2. Write down the inverse of the element \(2x + 1\). [1]
   3. \(q(x) = ax + b\) is any element of \(Q\) other than the identity. Find the order of \(q(x)\) and hence determine whether \((Q, +(\text{mod}3))\) is a cyclic group. [4]

(In this question, the notation \(\Delta ABC\) denotes the area of the triangle \(ABC\).) The points \(P\), \(Q\) and \(R\) have position vectors \(p\mathbf{i}\), \(q\mathbf{j}\) and \(r\mathbf{k}\) respectively, relative to the origin \(O\), where \(p\), \(q\) and \(r\) are positive. The points \(O\), \(P\), \(Q\) and \(R\) are joined to form a tetrahedron.

1. Draw a sketch of the tetrahedron and write down the values of \(\Delta OPQ\), \(\Delta OQR\) and \(\Delta ORP\). [3]
2. Use the definition of the vector product to show that \(\frac{1}{2}|\overrightarrow{RP} \times \overrightarrow{RQ}| = \Delta PQR\). [1]
3. Show that \((\Delta OPQ)^2 + (\Delta OQR)^2 + (\Delta ORP)^2 = (\Delta PQR)^2\). [6]

1. Use de Moivre's theorem to express \(\cos 4\theta\) as a polynomial in \(\cos \theta\). [4]
2. Hence prove that \(\cos 4\theta \cos 2\theta \equiv 16 \cos^6 \theta - 24 \cos^4 \theta + 10 \cos^2 \theta - 1\). [1]
3. Use part (ii) to show that the only roots of the equation \(\cos 4\theta \cos 2\theta = 1\) are \(\theta = n\pi\), where \(n\) is an integer. [3]
4. Show that \(\cos 4\theta \cos 2\theta = -1\) only when \(\cos \theta = 0\). [3]