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MATH105501

Numbers and Vectors


Academic integrity:

● This is an Open Book assessment. You may refer to lecture notes, books, and websites to help you complete the work (as long as you do not disobey the instructions below).

● The work that you submit must be your own work and must represent your own under-standing.

● You must not ask others to help you to complete your assessments.

● You must not discuss this assessment with other students.

● You must attach a completed Academic Integrity form at the end of your solutions before you submit them.


Instructions:

● There are 4 pages to this assessment.

● There will be 48 hours to complete this assessment, and an additional 2 hours to upload your solutions. Late submissions will not be accepted.

● If you believe the assessment contains a mistake or is not clearly written, send an email to the module leader. The module leader will normally respond within four hours of receiving a query, but will not respond outside of normal working hours (09.00-17.00 UK time).

● If you have trouble submitting your work, write to [email protected].

● If a module leader issues a correction this will be posted on Minerva and you will also be notified by email.

● There are two sections to this exam: Section A and Section B.

   Each question in Section A is worth 5 marks.

   Each question in Section B is worth 20 marks.

● Answer all questions.

● You must show all your calculations.

● There is no page limit, but we expect most students’ handwritten solutions to be between 4 and 12 pages long.

● Your student ID is of the form 201ABCDEF with the digits A,B,C,D,E,F∈ {0, 1, 2, . . . , 9}. Please write these digits down and use them as and when instructed.


SECTION A

Your student ID is of the form 201ABCDEF with the digits A,B,C,D,E,F∈ {0, 1, 2, . . . , 9}. Please write these digits down and use them as and when instructed.

A1. Use mathematical induction to prove the following inequality

for all positive integers N. Make sure you replace F with the last digit of your student ID first.

A2. Find the real numbers x and y satisfying the following equation

Make sure you replace A with the 4th digit of your student ID first.

A3. Sketch and describe (in your own words) the following set M in the complex plane

where B and C are the 5th and 6th digits of your student ID, respectively.

A4. Analyse the boundedness, monotonicity and convergence properties of the following sequence

Make sure you replace A with the 4th digit of your student ID first.

A5. Compute the limit lim  for the following sequence

Make sure you replace A and D with the 4th and 7th digits of your student ID first. Explain each step in your calculation.

A6. Describe how the vanishing test is applied covering all possible outcomes of the test. Give an example of an infinite series to demonstrate the fact that “failing the vanishing test implies divergence”.

A7. Find the intersection of the sphere of radius 10 centred at the origin and the line given by the equation (x, y, z) = (At, Bt, Ct + 1), t ∈ R. Make sure you replace A, B, C with the corresponding digits of your student ID first.

A8. Compute the vector product of = (A, B, C) and = (D, E, F + 1) and find a unit vector that is perpendicular to and . Make sure you replace A, B, C, D, E, F with the corresponding digits of your student ID first. Justify your answer by checking the perpendicularity relations.


SECTION B

Your student ID is of the form 201ABCDEF with the digits A,B,C,D,E,F∈ {0, 1, 2, . . . , 9}. Please write these digits down and use them as and when instructed.

B1. (a) Describe (in your own words) the shape obtained by connecting the fifth roots  of a non-zero complex number z in the complex plane.

(b) Pick a complex number which has non-zero imaginary part and calculate its fifth roots to demonstrate what you have written in part (a). Remember that the argument of complex numbers is in the range (−π, π].

(c) Consider the point P corresponding to 2 + 3i in the complex plane. Draw a closed half-disk M with P on its perimeter. Specify M by writing down what relations the numbers z ∈ M (points belonging to M) have to satisfy.

(d) Find all complex solutions of the equation .

(e) State Euler’s formula and use it to derive the trigonometric triple angle formulae for the sine and cosine functions


B2. (a) Draw a circle of unit radius, inscribe a regular octagon, and connect its vertices to the centre. Pick a vertex P1 and drop a perpendicular from P1 onto the next radial segment as you go clockwise around the circle. Call the base point P2. Drop a perpendicular from P2 to the next radial segment as you continue going clockwise. This yield the point P3. Keep repeating this process of dropping perpendiculars to get the points P4, P5, P6, . . . The line segments PnPn+1 (n ∈ N) form a spiral as shown in the figure below.

● Compute the length dn of each line segment PnPn+1 (n ∈ N).

● Calculate the N-th partial sum sN = d1 + d2 + · · · + dN (N ∈ N).

● Find the length of the spiral by computing the limit lim 

(b) Geometric series can be used to obtain expressions for rational numbers, given in decimal form, as the ratio of two integers.

For we can also write

Use this to express r as the ratio of two integers. Make sure you replace A, B, C, D, E, F with the corresponding digits of your student ID first.

(c) Consider the power series

Using the ratio test, determine the values of x for which the series converges. What can you say about the convergence when |x| is equal to the radius of convergence?


B3. A drone is located at position (1, 1.5, 3) (in kilometres) at noon and travelling in a straight line with velocity (in kilometres per hour). There is a small airport at position (3, 4, 0) (in kilometres).

(a) At what time will the drone pass directly over the airport? (Assume that the drone is flying over flat ground and that the vector points straight up.)

(b) How high above the airport will the drone be when it passes?

(c) The airport’s radar can only detect this drone when it’s closer than 5 (kilometres). At what time will the drone appear on the airport’s radar screen? Where is the drone at the moment it disappears from the radar screen?

(d) A car is travelling on the ground directly below the drone at all times. Find the location where they meet and the angle between their paths.

(e) Obtain the equation of the plane containing the paths of both vehicles.