Plane containing line and point/vector

10 The points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }\) and \(\overrightarrow { O B } = \mathbf { i } + \mathbf { j } + 5 \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - \mathbf { k } )\).

1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
2. Find the equation of the plane containing the line \(l\) and the point \(A\). Give your answer in the form \(a x + b y + c z = d\).

9 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\), \(\overrightarrow { O B } = 4 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O C } = 2 \mathbf { i } + 5 \mathbf { j } - \mathbf { k }\). A fourth point \(D\) is such that the quadrilateral \(A B C D\) is a parallelogram.

1. Find the position vector of \(D\) and verify that the parallelogram is a rhombus.
2. The plane \(p\) is parallel to \(O A\) and the line \(B C\) lies in \(p\). Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).

10 The points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + \mathbf { j } + \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + m \mathbf { k } + \mu ( \mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k } )\), where \(m\) is a constant.

1. Given that the line \(l\) intersects the line passing through \(A\) and \(B\), find the value of \(m\).
2. Find the equation of the plane which is parallel to \(\mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k }\) and contains the points \(A\) and \(B\). Give your answer in the form \(a x + b y + c z = d\).

10 The equations of two lines \(l\) and \(m\) are \(\mathbf { r } = 3 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + \lambda ( - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\) respectively.

1. Show that the lines do not intersect.
2. Calculate the acute angle between the directions of the lines.
3. Find the equation of the plane which passes through the point \(( 3 , - 2 , - 1 )\) and which is parallel to both \(l\) and \(m\). Give your answer in the form \(a x + b y + c z = d\).

2 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \mathbf { i } + 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } - 4 \mathbf { k } )\).
The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to the vector \(2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k }\).

1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
   \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-05_2723_33_99_22} The line \(l _ { 2 }\) is parallel to the vector \(5 \mathbf { i } - 5 \mathbf { j } - 2 \mathbf { k }\).
2. Find the acute angle between \(l _ { 2 }\) and \(\Pi\).

1. The line \(l _ { 1 }\) has equation

$$\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { k } )$$ and the line \(l _ { 2 }\) has equation $$\mathrm { r } = 2 \mathbf { i } + s \mathbf { j } + \mu ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$ where \(s\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) both lie in a common plane \(\Pi _ { 1 }\)

1. show that an equation for \(\Pi _ { 1 }\) is \(3 x + y - z = 3\)
2. find the value of \(s\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 3\)
3. Find an equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
4. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer in degrees to 3 significant figures.

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.

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 r.n \(= p\).
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 r.n \(= p\).
3. Find the distance between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
4. State the relationship between the answer to part (iii) and the lines \(l _ { 1 }\) and \(l _ { 2 }\).

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 \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).

10 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$l _ { 1 } : \mathbf { r } = 6 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad l _ { 2 } : \mathbf { r } = 6 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } + \mu ( 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } ) .$$ Find a cartesian equation of the plane \(\Pi\) containing \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vector of the foot of the perpendicular from the point with position vector \(\mathbf { i } + 10 \mathbf { j } + 3 \mathbf { k }\) to \(\Pi\). The line \(l _ { 3 }\) has equation \(\mathbf { r } = \mathbf { i } + 10 \mathbf { j } + 3 \mathbf { k } + v ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )\). Find the shortest distance between \(l _ { 1 }\) and \(l _ { 3 }\).

With respect to an origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }\) and the plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = ( 4 + \lambda + 3 \mu ) \mathbf { i } + ( - 2 + 7 \lambda + \mu ) \mathbf { j } + ( 2 + \lambda - \mu ) \mathbf { k } ,$$ where \(\lambda\) and \(\mu\) are real. The point \(L\) is such that \(\overrightarrow { O L } = 3 \overrightarrow { O A }\) and \(\Pi _ { 2 }\) is the plane through \(L\) which is parallel to \(\Pi _ { 1 }\). The point \(M\) is such that \(\overrightarrow { A M } = 3 \overrightarrow { M L }\).

1. Show that \(A\) is in \(\Pi _ { 1 }\).
2. Find a vector perpendicular to \(\Pi _ { 2 }\).
3. Find the position vector of the point \(N\) in \(\Pi _ { 2 }\) such that \(O N\) is perpendicular to \(\Pi _ { 2 }\).
4. Show that the position vector of \(M\) is \(10 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }\) and find the perpendicular distance of \(M\) from the line through \(O\) and \(N\), giving your answer correct to 3 significant figures.

10 The line \(l _ { 1 }\) is parallel to the vector \(a \mathbf { i } - \mathbf { j } + \mathbf { k }\), where \(a\) is a constant, and passes through the point whose position vector is \(9 \mathbf { j } + 2 \mathbf { k }\). The line \(l _ { 2 }\) is parallel to the vector \(- a \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\) and passes through the point whose position vector is \(- 6 \mathbf { i } - 5 \mathbf { j } + 10 \mathbf { k }\).

1. It is given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
   (a) Show that \(a = - \frac { 6 } { 13 }\).
   (b) Find a cartesian equation of the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
2. Given instead that the perpendicular distance between \(l _ { 1 }\) and \(l _ { 2 }\) is \(3 \sqrt { } ( 30 )\), find the value of \(a\).

9 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have vector equations $$\mathbf { r } = \lambda _ { 1 } ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) + \mu _ { 1 } ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = \lambda _ { 2 } ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) + \mu _ { 2 } ( 3 \mathbf { i } + \mathbf { j } - \mathbf { k } )$$ respectively. The line \(l\) passes through the point with position vector \(4 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\) and is parallel to both \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find a vector equation for \(l\). Find also the shortest distance between \(l\) and the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

9 The plane \(\Pi\) has equation $$\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) + \mu ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ The line \(l\), which does not lie in \(\Pi\), has equation $$\mathbf { r } = 3 \mathbf { i } + 6 \mathbf { j } + 12 \mathbf { k } + t ( 8 \mathbf { i } + 5 \mathbf { j } - 8 \mathbf { k } )$$ Show that \(l\) is parallel to \(\Pi\). Find the position vector of the point at which the line with equation \(\mathbf { r } = 5 \mathbf { i } - 4 \mathbf { j } + 7 \mathbf { k } + s ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\) meets \(\Pi\). Find the perpendicular distance from the point with position vector \(9 \mathbf { i } + 11 \mathbf { j } + 2 \mathbf { k }\) to \(l\).

1. The line \(l _ { 1 }\) has equation

$$\frac { x - 1 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z - 4 } { 3 }$$ The line \(l _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 3 \mathbf { k } + t ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )$$ where \(t\) is a scalar parameter.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) lie in the same plane.
2. Write down a vector equation for the plane containing \(l _ { 1 }\) and \(l _ { 2 }\)
3. Find, to the nearest degree, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\)

1. The line \(l _ { 1 }\) has equation

$$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )$$ and the line \(l _ { 2 }\) has equation $$\mathbf { r } = 5 \mathbf { i } + p \mathbf { j } - 7 \mathbf { k } + \mu ( 6 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant.
The plane \(\Pi\) contains \(l _ { 1 }\) and \(l _ { 2 }\)

1. Show that the vector \(3 \mathbf { i } - 10 \mathbf { j } - \mathbf { k }\) is perpendicular to \(\Pi\)
2. Hence determine a Cartesian equation of \(\Pi\)
3. Hence determine the value of \(p\) Given that
   • the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\)
   • the point \(B\) has coordinates \(( 12 , - 11,6 )\)
   • determine, to the nearest degree, the acute angle between \(A B\) and \(\Pi\)

4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\frac { x - 7 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z - 6 } { 3 }\) and \(\frac { x - 2 } { 1 } = \frac { y - 6 } { 2 } = \frac { z + 2 } { 1 }\) respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
2. Find the cartesian equation of the plane that contains \(l _ { 1 }\) and \(l _ { 2 }\).

7. The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } \alpha \\ - 4 \\ 0 \end{array} \right) + \mu \left( \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right) .$$ If the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find

1. the value of \(\alpha\),
2. an equation for the plane containing the lines \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in the form \(a x + b y + c z + d = 0\), where \(a , b , c\) and \(d\) are constants. For other values of \(\alpha\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect and are skew lines.
   Given that \(\alpha = 2\),
3. find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\).

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.

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 r.n \(= p\).
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 r.n \(= p\).
3. Find the distance between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
4. State the relationship between the answer to part (iii) and the lines \(l _ { 1 }\) and \(l _ { 2 }\).
5. Show that \(\left( z - \mathrm { e } ^ { \mathrm { i } \phi } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \phi } \right) \equiv z ^ { 2 } - ( 2 \cos \phi ) z + 1\).
6. 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.
7. Hence express \(z ^ { 7 } - 1\) as the product of one real linear factor and three real quadratic factors. 8
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.
9. 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 \(\{ 0 , \pm 1 , \pm 2 , \ldots \}\).)
10. Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.)
11. 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.

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

12. State, with a reason, whether or not \(G\) is commutative.
13. State the number of subgroups of \(G\) which are of order 2 .
14. 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 \(\mathbf { r }\).
15. Give a geometrical description of the locus of \(P\) when \(\mathbf { r }\) satisfies the equation \(\mathbf { r } = \lambda \mathbf { a }\), where \(0 \leqslant \lambda \leqslant 1\).
16. Given that \(P\) is a point on the line \(A B\), use a property of the vector product to explain why \(( \mathbf { r } - \mathbf { a } ) \times ( \mathbf { r } - \mathbf { b } ) = \mathbf { 0 }\).
17. 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
18. 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 \(x = \frac { 1 } { 4 } \pi\).
19. Find the solution of the differential equation in this case.
20. 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 \(x + 2 y - 2 z = - 1\).
21. Find, in either order, the coordinates of \(B\) and the length of \(A B\).
22. Find the acute angle between the planes \(A C D\) and \(B C D\).
23. (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 \(0 < \theta < \frac { 1 } { 2 } \pi\).
24. Use de Moivre's theorem to prove that $$\sin 6 \theta \equiv \sin 2 \theta \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 3 \right)$$
25. 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.
26. Write down the identity element for each of groups \(A , B , C\) and \(D\).
27. 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.
28. Prove the closure property for group \(D\).
29. 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.) \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. 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.
30. Write down, in terms of \(e\) and \(r\), the elements of the subgroup of \(G\) which is of order 4.
31. 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
32. 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 \(\frac { \mathrm { d } z } { \mathrm {~d} x } = \frac { 2 ( z + 1 ) } { z - 1 }\).
33. Hence find the general solution of the differential equation (A). 4
34. By expressing \(\cos \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), show that $$\cos ^ { 5 } \theta \equiv \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$$
35. 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.
36. Show that, for all values of \(k\), the lines intersect, and find their point of intersection in terms of \(k\).
37. 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 \(a \circ b = a | b |\).
38. Show that ∘ is not commutative.
39. Prove that ∘ is associative.
40. Determine whether the set of real numbers, under the operation ∘, forms a group. \section*{June 2008}

15 The points \(A ( 7,2,8 ) , B ( 7 , - 4,0 )\) and \(C ( 3,3.2,9.6 )\) all lie in the plane \(\Pi\). 15

1. Find a Cartesian equation of the plane \(\Pi\).
   15
2. The line \(L _ { 1 }\) has equation \(\mathbf { r } = \left[ \begin{array} { c } 5 \\ - 0.4 \\ 4.8 \end{array} \right] + \mu \left[ \begin{array} { c } 15 \\ 3 \\ 4 \end{array} \right]\) 15
3. 1. Show that \(L _ { 1 }\) lies in the plane \(\Pi\).
      15
4. (ii) Show that every point on \(L _ { 1 }\) is equidistant from \(B\) and \(C\).
   15
5. The line \(L _ { 2 }\) lies in the plane \(\Pi\), and every point on \(L _ { 2 }\) is equidistant from \(A\) and \(B\).
   15
6. The points \(A , B\) and \(C\) all lie on a circle \(G\). The point \(D\) is the centre of circle \(G\). Find the coordinates of \(D\).
   \includegraphics[max width=\textwidth, alt={}, center]{b4ba8a08-333d-4efc-a0ed-14fef2d99410-26_2488_1719_219_150}