6 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.
- Find the equation of the plane \(\Pi _ { 1 }\) which contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\), giving your answer in the form r.n \(= p\).
- Find the equation of the plane \(\Pi _ { 2 }\) which contains \(l _ { 2 }\) and is parallel to \(l _ { 1 }\), giving your answer in the form r.n \(= p\).
- Find the distance between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
- State the relationship between the answer to part (iii) and the lines \(l _ { 1 }\) and \(l _ { 2 }\).
- Show that \(\left( z - \mathrm { e } ^ { \mathrm { i } \phi } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \phi } \right) \equiv z ^ { 2 } - ( 2 \cos \phi ) z + 1\).
- Write down the seven roots of the equation \(z ^ { 7 } = 1\) in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\) and show their positions in an Argand diagram.
- Hence express \(z ^ { 7 } - 1\) as the product of one real linear factor and three real quadratic factors.
8
- Find the general solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \tan x = \cos ^ { 3 } x$$
expressing \(y\) in terms of \(x\) in your answer.
- Find the particular solution for which \(y = 2\) when \(x = \pi\).
9 The set \(S\) consists of the numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\). ( \(\mathbb { Z }\) denotes the set of integers \(\{ 0 , \pm 1 , \pm 2 , \ldots \}\).)
- Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.)
- Determine whether or not each of the following subsets of \(S\), under multiplication, forms a subgroup of \(G\), justifying your answers.
(a) The numbers \(3 ^ { 2 n }\), where \(n \in \mathbb { Z }\).
(b) The numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\) and \(n \geqslant 0\).
(c) The numbers \(3 ^ { \left( \pm n ^ { 2 } \right) }\), where \(n \in \mathbb { Z }\).
1 (a) A group \(G\) of order 6 has the combination table shown below.
| \(e\) | \(a\) | \(b\) | \(p\) | \(q\) | \(r\) |
| \(e\) | \(e\) | \(a\) | \(b\) | \(p\) | \(q\) | \(r\) |
| \(a\) | \(a\) | \(b\) | \(e\) | \(r\) | \(p\) | \(q\) |
| \(b\) | \(b\) | \(e\) | \(a\) | \(q\) | \(r\) | \(p\) |
| \(p\) | \(p\) | \(q\) | \(r\) | \(e\) | \(a\) | \(b\) |
| \(q\) | \(q\) | \(r\) | \(p\) | \(b\) | \(e\) | \(a\) |
| \(r\) | \(r\) | \(p\) | \(q\) | \(a\) | \(b\) | \(e\) |
- State, with a reason, whether or not \(G\) is commutative.
- State the number of subgroups of \(G\) which are of order 2 .
- List the elements of the subgroup of \(G\) which is of order 3 .
(b) 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 \(c ^ { 5 }\).
2 Find the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 4 x$$
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 \(\mathbf { r }\). - Give a geometrical description of the locus of \(P\) when \(\mathbf { r }\) satisfies the equation \(\mathbf { r } = \lambda \mathbf { a }\), where \(0 \leqslant \lambda \leqslant 1\).
- Given that \(P\) is a point on the line \(A B\), use a property of the vector product to explain why \(( \mathbf { r } - \mathbf { a } ) \times ( \mathbf { r } - \mathbf { b } ) = \mathbf { 0 }\).
- Give a geometrical description of the locus of \(P\) when \(\mathbf { r }\) satisfies the equation \(\mathbf { r } \times ( \mathbf { a } - \mathbf { b } ) = \mathbf { 0 }\).
4 The integrals \(C\) and \(S\) are defined by
$$C = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \cos 3 x \mathrm {~d} x \quad \text { and } \quad S = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \sin 3 x \mathrm {~d} x$$
By considering \(C + \mathrm { i } S\) as a single integral, show that
$$C = - \frac { 1 } { 13 } \left( 2 + 3 \mathrm { e } ^ { \pi } \right)$$
and obtain a similar expression for \(S\).
(You may assume that the standard result for \(\int \mathrm { e } ^ { k x } \mathrm {~d} x\) remains true when \(k\) is a complex constant, so that \(\left. \int \mathrm { e } ^ { ( a + \mathrm { i } b ) x } \mathrm {~d} x = \frac { 1 } { a + \mathrm { i } b } \mathrm { e } ^ { ( a + \mathrm { i } b ) x } .\right)\)
5 - Find the general solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { y } { x } = \sin 2 x$$
expressing \(y\) in terms of \(x\) in your answer.
In a particular case, it is given that \(y = \frac { 2 } { \pi }\) when \(x = \frac { 1 } { 4 } \pi\).
- Find the solution of the differential equation in this case.
- Write down a function to which \(y\) approximates when \(x\) is large and positive.
6 A tetrahedron \(A B C D\) is such that \(A B\) is perpendicular to the base \(B C D\). 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 \(B C D\) is \(x + 2 y - 2 z = - 1\).
- Find, in either order, the coordinates of \(B\) and the length of \(A B\).
- Find the acute angle between the planes \(A C D\) and \(B C D\).
- (a) Verify, without using a calculator, that \(\theta = \frac { 1 } { 8 } \pi\) is a solution of the equation \(\sin 6 \theta = \sin 2 \theta\).
(b) 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 \(0 < \theta < \frac { 1 } { 2 } \pi\). - Use de Moivre's theorem to prove that
$$\sin 6 \theta \equiv \sin 2 \theta \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 3 \right)$$
- 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.
\section*{Jan 2008}
8 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 \(\left\{ 2 ^ { 0 } , 2 ^ { 1 } , 2 ^ { 2 } , 2 ^ { 3 } \right\}\) under multiplication modulo 15,
\(D\) : the set of numbers \(\left\{ \frac { 1 + 2 m } { 1 + 2 n } \right.\), where \(m\) and \(n\) are integers \(\}\) under multiplication. - Write down the identity element for each of groups \(A , B , C\) and \(D\).
- Determine in each case whether the groups
$$\begin{aligned}
& A \text { and } B ,
& B \text { and } C ,
& A \text { and } C
\end{aligned}$$
are isomorphic or non-isomorphic. Give sufficient reasons for your answers. - Prove the closure property for group \(D\).
- Elements of the set \(\left\{ \frac { 1 + 2 m } { 1 + 2 n } \right.\), where \(m\) and \(n\) are integers \(\}\) are combined under addition. State which of the four basic group properties are not satisfied. (Justification is not required.)
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OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge.
}1 (a) A cyclic multiplicative group \(G\) has order 12. The identity element of \(G\) is \(e\) and another element is \(r\), with order 12.
- Write down, in terms of \(e\) and \(r\), the elements of the subgroup of \(G\) which is of order 4.
- Explain briefly why there is no proper subgroup of \(G\) in which two of the elements are \(e\) and \(r\).
(b) A group \(H\) has order \(m n p\), where \(m , n\) and \(p\) are prime. State the possible orders of proper subgroups of \(H\).
2 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 \(\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) + \mu ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\).
3 - Use the substitution \(z = x + y\) to show that the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x + y + 3 } { x + y - 1 }$$
may be written in the form \(\frac { \mathrm { d } z } { \mathrm {~d} x } = \frac { 2 ( z + 1 ) } { z - 1 }\).
- Hence find the general solution of the differential equation (A).
4
- By expressing \(\cos \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), show that
$$\cos ^ { 5 } \theta \equiv \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
- Hence solve the equation \(\cos 5 \theta + 5 \cos 3 \theta + 9 \cos \theta = 0\) for \(0 \leqslant \theta \leqslant \pi\).
5 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.
- Show that, for all values of \(k\), the lines intersect, and find their point of intersection in terms of \(k\).
- For the case \(k = 1\), find the equation of the plane in which the lines lie, giving your answer in the form \(a x + b y + c z = d\).
6 The operation ○ on real numbers is defined by \(a \circ b = a | b |\).
- Show that ∘ is not commutative.
- Prove that ∘ is associative.
- Determine whether the set of real numbers, under the operation ∘, forms a group.
\section*{June 2008}