OCR FP3 (Further Pure Mathematics 3)

1. By writing \(z\) in the form \(re^{i\theta}\), show that \(zz^* = |z|^2\). [1]
2. Given that \(zz^* = 9\), describe the locus of \(z\). [2]

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]

1. A cyclic multiplicative group \(G\) has order 12. The identity element of \(G\) is \(e\) and another element is \(r\), with order 12.
   1. Write down, in terms of \(e\) and \(r\), the elements of the subgroup of \(G\) which is of order 4. [2]
   2. Explain briefly why there is no proper subgroup of \(G\) in which two of the elements are \(e\) and \(r\). [1]
2. A group \(H\) has order \(mnp\), where \(m, n\) and \(p\) are prime. State the possible orders of proper subgroups of \(H\). [2]

In this question \(G\) is a group of order \(n\), where \(3 \leqslant n < 8\).

1. In each case, write down the smallest possible value of \(n\):
   1. if \(G\) is cyclic, [1]
   2. if \(G\) has a proper subgroup of order 3, [1]
   3. if \(G\) has at least two elements of order 2. [1]
2. Another group has the same order as \(G\), but is not isomorphic to \(G\). Write down the possible value(s) of \(n\). [2]

Find the cube roots of \(\frac{1}{2}\sqrt{3} + \frac{1}{2}i\), giving your answers in the form \(\cos \theta + i \sin \theta\), where \(0 \leqslant \theta < 2\pi\). [4]

A line \(l\) has equation \(\mathbf{r} = 3\mathbf{i} + \mathbf{j} - 2\mathbf{k} + t(\mathbf{i} + 4\mathbf{j} + 2\mathbf{k})\) and a plane \(\Pi\) has equation \(8x - 7y + 10z = 7\). Determine whether \(l\) lies in \(\Pi\), is parallel to \(\Pi\) without intersecting it, or intersects \(\Pi\) at one point. [5]

Find the general solution of the differential equation $$\frac{d^2y}{dx^2} - 8\frac{dy}{dx} + 16y = 4x.$$ [7]

Find the acute angle between the line with equation \(\mathbf{r} = 2\mathbf{i} + 3\mathbf{k} + t(\mathbf{i} + 4\mathbf{j} - \mathbf{k})\) and the plane with equation \(\mathbf{r} = 2\mathbf{i} + 3\mathbf{k} + \lambda(\mathbf{i} + 3\mathbf{j} + 2\mathbf{k}) + \mu(\mathbf{i} + 2\mathbf{j} - \mathbf{k})\). [7]

1. Express \(\frac{\sqrt{3} + i}{\sqrt{3} - i}\) in the form \(re^{i\theta}\), where \(r > 0\) and \(0 \leqslant \theta < 2\pi\). [3]
2. Hence find the smallest positive value of \(n\) for which \(\left(\frac{\sqrt{3} + i}{\sqrt{3} - i}\right)^n\) is real and positive. [2]

It is given that the set of complex numbers of the form \(re^{i\theta}\) for \(-\pi < \theta \leqslant \pi\) and \(r > 0\), under multiplication, forms a group.

1. Write down the inverse of \(5e^{3\pi i}\). [1]
2. Prove the closure property for the group. [2]
3. \(Z\) denotes the element \(e^{i\gamma}\), where \(\frac{1}{2}\pi < \gamma < \pi\). Express \(Z^2\) in the form \(e^{i\theta}\), where \(-\pi < \theta \leqslant 0\). [2]

Find the general solution of the differential equation $$\frac{d^2y}{dx^2} - c\frac{dy}{dx} + 8y = e^{3x}.$$ [6]

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 \(0 \leqslant \lambda \leqslant 1\). [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]

1. Use the substitution \(z = x + y\) to show that the differential equation $$\frac{dy}{dx} = \frac{x + y + 3}{x + y - 1} \qquad (A)$$ may be written in the form \(\frac{dz}{dx} = \frac{2(z + 1)}{z - 1}\). [3]
2. Hence find the general solution of the differential equation (A). [4]

Two skew lines have equations $$\frac{x}{2} = \frac{y + 3}{1} = \frac{z - 6}{3} \quad \text{and} \quad \frac{x - 5}{3} = \frac{y + 1}{1} = \frac{z - 7}{5}.$$

1. Find the direction of the common perpendicular to the lines. [2]
2. Find the shortest distance between the lines. [4]

A line \(l\) has equation \(\frac{x - 6}{-4} = \frac{y + 7}{8} = \frac{z + 10}{7}\) and a plane \(p\) has equation \(3x - 4y - 2z = 8\).

1. Find the point of intersection of \(l\) and \(p\). [3]
2. Find the equation of the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(ax + by + cz = d\). [5]

Elements of the set \(\{p, q, r, s, t\}\) are combined according to the operation table shown below.

	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
\(p\)	\(t\)	\(s\)	\(p\)	\(r\)	\(q\)
\(q\)	\(s\)	\(p\)	\(q\)	\(t\)	\(r\)
\(r\)	\(p\)	\(q\)	\(r\)	\(s\)	\(t\)
\(s\)	\(r\)	\(t\)	\(s\)	\(q\)	\(p\)
\(t\)	\(q\)	\(r\)	\(t\)	\(p\)	\(s\)

1. Verify that \(q(st) = (qs)t\). [2]
2. Assuming that the associative property holds for all elements, prove that the set \(\{p, q, r, s, t\}\), with the operation table shown, forms a group \(G\). [4]
3. A multiplicative group \(H\) is isomorphic to the group \(G\). The identity element of \(H\) is \(e\) and another element is \(d\). Write down the elements of \(H\) in terms of \(e\) and \(d\). [2]

The integrals \(C\) and \(S\) are defined by $$C = \int_0^{2\pi} e^{3x} \cos 3x \, dx \quad \text{and} \quad S = \int_0^{2\pi} e^{3x} \sin 3x \, dx.$$ By considering \(C + iS\) as a single integral, show that $$C = \frac{1}{13}(2 + 3e^\pi),$$ 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. By expressing \(\cos \theta\) in terms of \(e^{i\theta}\) and \(e^{-i\theta}\), show that $$\cos^5 \theta \equiv \frac{1}{16}(\cos 5\theta + 5\cos 3\theta + 10\cos \theta).$$ [5]
2. Hence solve the equation \(\cos 5\theta + 5\cos 3\theta + 9\cos \theta = 0\) for \(0 \leqslant \theta \leqslant \pi\). [4]

Find the general solution of the differential equation $$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 5y = 65 \sin 2x.$$ [9]

The differential equation $$\frac{dy}{dx} + \frac{1}{1 - x^2} y = (1 - x)^{\frac{1}{2}}, \quad \text{where } |x| < 1,$$ can be solved by the integrating factor method.

1. Use an appropriate result given in the List of Formulae (MF1) to show that the integrating factor can be written as \(\left(\frac{1 + x}{1 - x}\right)^{\frac{1}{2}}\). [2]
2. Hence find the solution of the differential equation for which \(y = 2\) when \(x = 0\), giving your answer in the form \(y = f(x)\). [6]

1. Use de Moivre's theorem to prove that $$\cos 6\theta = 32\cos^6 \theta - 48\cos^4 \theta + 18\cos^2 \theta - 1.$$ [4]
2. Hence find the largest positive root of the equation $$64x^6 - 96x^4 + 36x^2 - 3 = 0,$$ giving your answer in trigonometrical form. [4]

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 \(x = \frac{1}{4}\pi\).
2. Find the solution of the differential equation in this case. [2]
3. Write down a function to which \(y\) approximates when \(x\) is large and positive. [1]

Two lines have equations $$\frac{x-k}{2} = \frac{y+1}{-5} = \frac{z-1}{-3} \quad \text{and} \quad \frac{x-k}{1} = \frac{y+4}{-4} = \frac{z}{-2},$$ where \(k\) is a constant.

1. Show that, for all values of \(k\), the lines intersect, and find their point of intersection in terms of \(k\). [6]
2. For the case \(k = 1\), find the equation of the plane in which the lines lie, giving your answer in the form \(ax + by + cz = d\). [4]

The variables \(x\) and \(y\) are related by the differential equation $$x^3 \frac{dy}{dx} = xy + x + 1. \qquad (A)$$

1. Use the substitution \(y = u - \frac{1}{x}\), where \(u\) is a function of \(x\), to show that the differential equation may be written as $$x^2 \frac{du}{dx} = u.$$ [4]
2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y = f(x)\). [5]

The variables \(x\) and \(y\) satisfy the differential equation $$\frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 9y = e^{3x}.$$

1. Find the complementary function. [3]
2. Explain briefly why there is no particular integral of either of the forms \(y = ke^{3x}\) or \(y = kxe^{3x}\). [1]
3. Given that there is a particular integral of the form \(y = kx^2e^{3x}\), find the value of \(k\). [5]

Lines \(l_1\) and \(l_2\) have equations $$\frac{x-3}{2} = \frac{y-4}{-1} = \frac{z+1}{1} \quad \text{and} \quad \frac{x-5}{4} = \frac{y-1}{3} = \frac{z-1}{2}$$ respectively.

1. Find the equation of the plane \(\Pi_1\) which contains \(l_1\) and is parallel to \(l_2\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [5]
2. Find the equation of the plane \(\Pi_2\) which contains \(l_2\) and is parallel to \(l_1\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [2]
3. Find the distance between the planes \(\Pi_1\) and \(\Pi_2\). [2]
4. State the relationship between the answer to part (iii) and the lines \(l_1\) and \(l_2\). [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]

The operation \(\circ\) on real numbers is defined by \(a \circ b = a|b|\).

1. Show that \(\circ\) is not commutative. [2]
2. Prove that \(\circ\) is associative. [4]
3. Determine whether the set of real numbers, under the operation \(\circ\), forms a group. [4]

\includegraphics{figure_6} The cuboid \(OABCDEFG\) shown in the diagram has \(\overrightarrow{OA} = 4\mathbf{i}, \overrightarrow{OC} = 2\mathbf{j}, \overrightarrow{OD} = 3\mathbf{k}\), and \(M\) is the mid-point of \(GF\).

1. Find the equation of the plane \(ACGE\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4]
2. The plane \(OEFC\) has equation \(\mathbf{r} \cdot (3\mathbf{i} - 4\mathbf{k}) = 0\). Find the acute angle between the planes \(OEFC\) and \(ACGE\). [4]
3. The line \(AM\) meets the plane \(OEFC\) at the point \(W\). Find the ratio \(AW : WM\). [5]

The plane \(\Pi_1\) has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + \mu \begin{pmatrix} -5 \\ -2 \end{pmatrix}\).

1. Express the equation of \(\Pi_1\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4] The plane \(\Pi_2\) has equation \(\mathbf{r} \cdot \begin{pmatrix} 7 \\ 1 \\ -3 \end{pmatrix} = 21\).
2. Find an equation of the line of intersection of \(\Pi_1\) and \(\Pi_2\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [5]

1. Show that \((z - e^{i\theta})(z - e^{-i\theta}) \equiv z^2 - (2\cos \theta)z + 1\). [1]
2. Write down the seven roots of the equation \(z^7 = 1\) in the form \(e^{i\theta}\) and show their positions in an Argand diagram. [4]
3. Hence express \(z^7 - 1\) as the product of one real linear factor and three real quadratic factors. [5]

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 \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\), or otherwise, find the other solution of the equation \(\sin 6\theta = \sin 2\theta\) in the interval \(0 < \frac{1}{2}\pi\). [2]
2. Use de Moivre's theorem to prove that $$\sin 6\theta \equiv \sin 2\theta(16\cos^4 \theta - 16\cos^2 \theta + 3).$$ [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]

The roots of the equation \(z^3 - 1 = 0\) are denoted by \(1, \omega\) and \(\omega^2\).

1. Sketch an Argand diagram to show these roots. [1]
2. Show that \(1 + \omega + \omega^2 = 0\). [2]
3. Hence evaluate
   1. \((2 + \omega)(2 + \omega^2)\), [2]
   2. \(\frac{1}{2 + \omega} + \frac{1}{2 + \omega^2}\). [2]
4. Hence find a cubic equation, with integer coefficients, which has roots \(2, \frac{1}{2 + \omega}\) and \(\frac{1}{2 + \omega^2}\). [4]

1. The operation \(*\) is defined by \(x * y = x + y - a\), where \(x\) and \(y\) are real numbers and \(a\) is a real constant.
   1. Prove that the set of real numbers, together with the operation \(*\), forms a group. [6]
   2. State, with a reason, whether the group is commutative. [1]
   3. Prove that there are no elements of order 2. [2]
2. The operation \(\circ\) is defined by \(x \circ y = x + y - 5\), where \(x\) and \(y\) are positive real numbers. By giving a numerical example in each case, show that two of the basic group properties are not necessarily satisfied. [4]

1. Find the general solution of the differential equation $$\frac{dy}{dx} + y\tan x = \cos^3 x,$$ expressing \(y\) in terms of \(x\) in your answer. [8]
2. Find the particular solution for which \(y = 2\) when \(x = \pi\). [2]

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

1. Write down the identity element for each of groups \(A, B, C\) and \(D\). [2]
2. Determine in each case whether the groups \begin{align} &A \text{ and } B,
   &B \text{ and } C,
   &A \text{ and } C \end{align} 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]

1. Find the complementary function of the differential equation $$\frac{d^2y}{dx^2} + y = \cosec x.$$ [2]
2. It is given that \(y = p(\ln \sin x) \sin x + qx \cos x\), where \(p\) and \(q\) are constants, is a particular integral of this differential equation.
   1. Show that \(p - 2(p + q) \sin^2 x \equiv 1\). [6]
   2. Deduce the values of \(p\) and \(q\). [2]
3. Write down the general solution of the differential equation. State the set of values of \(x\), in the interval \(0 \leqslant x \leqslant 2\pi\), for which the solution is valid, justifying your answer. [3]

1. By expressing \(\sin \theta\) in terms of \(e^{i\theta}\) and \(e^{-i\theta}\), show that $$\sin^6 \theta \equiv \frac{1}{32}(\cos 6\theta - 6\cos 4\theta + 15\cos 2\theta - 10).$$ [5]
2. Replace \(\theta\) by \(\left(\frac{1}{2}\pi - \theta\right)\) in the identity in part (i) to obtain a similar identity for \(\cos^6 \theta\). [3]
3. Hence find the exact value of \(\int_0^{2\pi} \left(\sin^6 \theta - \cos^6 \theta\right) d\theta\). [4]

The set \(S\) consists of the numbers \(3^n\), where \(n \in \mathbb{Z}\). (\(\mathbb{Z}\) denotes the set of integers \(\{0, \pm 1, \pm 2, \ldots\}\).)

1. Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.) [6]
2. Determine whether or not each of the following subsets of \(S\), under multiplication, forms a subgroup of \(G\), justifying your answers.
   1. The numbers \(3^{2n}\), where \(n \in \mathbb{Z}\). [2]
   2. The numbers \(3^n\), where \(n \in \mathbb{Z}\) and \(n \geqslant 0\). [2]
   3. The numbers \(3^{(±n^2)}\), where \(n \in \mathbb{Z}\). [2]