Place your assignment solutions in the appropriate pigeonhole in the boxes on the third level of the Mathematics Building (PS2, Bundoora campus) before 12.00 noon on Monday 11th May OR hand to Simon Smith (Bendigo) at or before the Monday lecture. Each page of your solutions must carry your name, your demonstrator's name, and the day and time of your second Practice Class of the week.
In submitting your work, you are consenting that it may be copied and transmitted by the University for the detection of plagiarism. You must start your assignment solutions with the following Statement of Originality, signed and dated by you:"
The assignments are designed to help you master the concepts in this subject and also for you to develop your mathematical communication skills. Please note that often it is not the nal answer that is important but your mastery of the required techniques and the way you communicate your ideas and your approach to the problems. Note that your use of language and mathematical symbols
are worth marks in the assignment.
1. Question 1 is a basic survival-skills question.
(a) Simplify the following expression. Cancel all common factors.
16
(k 2)(k + 2)
+
k + 6
k + 2
(b) Show that the following identity holds. [Hint. You should work from the LHS to the RHS. The rst thing to do is to take out any common factors and the fraction: see the calculations in Questions 1 and 3 on NumSys Practice Class 8.]
1
3
k(k + 1)(k + 5) + (k + 1)(k + 4) =
1
3
(k + 1)(k + 2)(k + 6):
2. [See Chap. 6 of Notes on Number Systems, and NumSys Practice Class 7]
(a) Evaluate the following summation:
Σ4
i=1
(1)i i2.
(b) Write the following in summation notation:
2
3
+
4
5
+
6
7
+
8
9
+ +
100
101
.
(c) Use the geometric series to nd a closed form expression for:
(i) 1 +
1
5
+
1
52 +
1
53 + : : : +
1
5n ,
(ii) 1 1
5
+
1
52
1
53 + : : : + (1)n 1
5n ,
(iii)
1
52 +
1
53 +
1
54 + : : : +
1
5n .
3. [See Chap. 1 of Notes on Logic and Proofs, and NumSys Practice Class 8]
Write a very careful proof by mathematical induction that, for all n 2 N, we have
1 21 + 2 22 + 3 23 + + n 2n = 2 + (n 1)2n+1.
A
For Questions 4 and 5 refer to Practice Classes 7 and 8 of Linear Algebra, Lectures 7 and 8, and Chapter 4 of `Notes on Linear Algebra'.
4. (a) (i) Write down a normal vector for the plane S1 = f (x; y; z) 2 R3 j 2x4y +3z = 2 g.
(ii) Determine whether or not the points P1 = (1; 3; 1) and P2 = (1;2;4) belong to S1.
(iii) Give the parametric description of the plane S1 using y and z as parameters. Your answer must be in set notation.
(iv) Give an equational description of the plane S2 that is normal to the vector 2i
~
3j
~
+k and contains the point (1; 0; 1). ~
(b) (i) Give the equational and parametric descriptions of the line L1 through (1;3; 0) and in the direction of the vector i
~
3j
~
+ 2k
~
. Both answers must be in set notation.
(ii) Give the equational and parametric descriptions of the line L2 through (1;3; 0) and in the direction of the vector i
~
+ 2k
~
. Both answers must be in set notation. [Don't divide by 0.]
(c) (i) Find the point (a; b; c) where the line L2 passes through the plane S1. (Refer to Section 4.3 of the printed notes.)
(ii) Find the distance from P1 to S2. (Refer to Section 4.4 of the printed notes.)
5. Recall the labelling of features of a triangle as described on Assignment One. The vertices are A, B and C, where
!
AB = m
~
,
!
BC = n
~
. We de ned a, b and c to be the midpoints of the edges opposite the vertices A, B and C respectively: We found expressions for
!
AC and the median
!
Aa in terms of m
~
and n
~
. You will need to refer to these expressions.
(a) The triangle lies in a plane. What is the normal vector to this plane? (Call it N
~
.)
(Hint: We know two vectors in the plane.)
(b) Write down two set descriptions for the plane S in which the triangle lies: in a form with a single scalar equation, and in a vector form (with two parameters). Both forms will be in terms of the vectors
!
OA, m
~
and n
~
.
(c) (i) The perpendicular bisector of !
BC is the line L lying in the plane of the triangle that passes through a and is perpendicular to
!
BC. Draw a diagram showing this line.
(ii) The line L is perpendicular both to
!
BC and to the vector N
~
. What is its direction?
(iii) Write down a parametric set description of L. In your answer, use the vectors m
~
, n
~ and
!
OA.