# Unit 11: Maths for Computing - The cost of painting walls of a room is £100 for first m2 and rises by £50 for each m2. Find the charge to paint all the walls of a room if the total area is 150 m2.

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# HND Assignment Brief

Session: September 2021

 Programme title BTEC Higher National Diploma (HND) in Computing Unit number and title 11 Maths for Computing (L4) Assignment number & title 1 of 1 Maths for Computing (L4) Unit Leader - Assessor (s) Issue Date 6/10/2021 Final assignment submission deadline 10-15 January 2022 Late submission deadline 17-22 January 2022 The learners are required to follow the strict deadline set by the College for submissions of assignments in accordance with the BTEC level 4 – 7 submission guidelines and College policy on submissions. Resubmission deadline TBA Feedback Formative feedback will be available in class during the semester. Final feedback will be available within 2 – 3 weeks of the assignment submission date.
 General Guidelines The work you submit must be in your own words. If you use a quote or an illustration from somewhere you must give the source. Include a list of references at the end of your document. You must give all your sources of information. Make sure your work is clearly presented and that you use readily understandable English. Wherever possible use a word processor and its “spell-checker”.

## Unit 11: Maths for Computing (L4) Session: September 2021 Coursework

ASSIGNMENT

Aim & Objective

This assignment is designed so that it enables the student to demonstrate their understanding of the mathematical concepts covered in the module through answering various practical problems; divided into four parts. The coursework should be submitted as one document in a report format in final submission.

### Part 1 Unit 11: Maths for Computing

Number theory

The GCD (greatest common divisor), LCM (lowest common multiple) and prime numbers is used for a variety of applications in number theory, particularly in modular arithmetic and thus encryption algorithms such as RSA. It is also used for simpler applications, such as simplifying fractions. This makes the GCD, LCM and prime numbers a rather fundamental concept to number theory, and as such several algorithms have been discovered to efficiently compute it. Primes are the set of all numbers that can only be equally divided by 1 and themselves, with no other even division possible. Numbers like 2, 3, 5, 7, and 11 are all prime numbers.

Demonstrate the concepts of greatest common divisor and least common multiple of a given pair of numbers with an example. To support the evidence of your understanding on LCM and GCD, you should present with pseudocode and a computer program in python to compute LCM and GCD based on user’s input. It is desirable, to support your findings by identifying multiplicative inverses in modular arithmetic with an example. Produce a detailed written explanation of the importance and application of prime numbers in RSA encryption (Rivest–Shamir–Adleman). To support the evidence of your understanding on the use of prime numbers, you are required to develop a computer program in C/ C++ or python to demonstrate the asymmetric cryptography algorithm.

### Sequences and Series

Arithmetic progressions are used in simulation engineering and in the reproductive cycle of bacteria. Some uses of AP’s in daily life include uniform increase in the speed at regular intervals, completing patterns of objects, calculating simple interest, speed of an aircraft, increase or decrease in the costs of goods, sales and production and so on. Geometric progressions (GP’s) are used in compound interest and the range of speeds on a drilling machine. In fact, GPs are used throughout mathematics, and they have many important applications in physics, engineering, biology, economics, computer science, queuing theory and finance.

To support the evidence of your understanding on AP and GP, solve the following 4 problems.

1. The cost of painting walls of a room is £100 for first m2 and rises by £50 for each m2. Find the charge to paint all the walls of a room if the total area is 150 m2.
2. An Arithmetic Progression has 23 terms, the sum of the middle three terms of this arithmetic progression is 720, and the sum of the last three terms of this Arithmetic Progression is 1320. What is the 18th term of this Arithmetic Progression?
3. Find the sum of the first 20 terms of the GP with first term 3 and common ratio 1.5.
4. The sum of the first 3 terms of a geometric series is 37/8. The sum of the first six terms is 3367/512. Find the first term and common ratio.
5. How many terms in the GP 4, 3.6, 3.24, . . . are needed so that the sum exceeds 35?

### Part 2 Unit 11: Maths for Computing

Probability theory and probability distributions

The probability of something to happen is the likelihood or a chance of it happening. Values of probability lie between 0 and 1, where 0 represents an absolute impossibility and 1 represent an absolute certainty. The probability of an event happening usually lies somewhere between these two extreme values and it is expressed either as a proper a decimal fraction.

2.1 To support the evidence of your understanding on probability theory and probability distributions, solve the following problems.

1. What is the probability of getting a sum of 9 when two dice are thrown?
2. A coin is thrown 3 times. what is the probability that at least one head is obtained?
3. From a pack of cards, two cards are drawn at random. Find the probability that each card is numbered and from different suite.
4. A bag contains 8 white, 4 red and 7 blue balls. Three balls are drawn at random from the bag. What is the probability that

i) all of them are blue.

ii) two is white and the third is red.

1. Find the probability distribution of boys and girls in families with 5 children, assuming equal probabilities for boys and girls. Draw the graph of your probability distribution.
2. The time taken to assemble a car in a certain plant is a random variable having a normal distribution of 20 hours and a standard deviation of 2 hours. What is the probability that a car can be assembled at this plant in a period of time?

a)  less than 19.5 hours?

b)  between 20 and 22 hours?

1. The annual salaries of employees in a large company are approximately normally distributed with a mean of £40,000 and a standard deviation of £20,000.

a) What percent of people earn less than £30,000?

b) What percent of people earn between £35,000 and £45,000?

c) What percent of people earn more than £50,000?

It is also desirable to support your findings by producing a brief evaluation report on the use of probability theory in hashing and load balancing.

### Part 3 Unit 11: Maths for Computing

Forces, velocities, and various other quantities are vectors functions find their applications in engineering, physics, fluid flow, electrostatics, computational maths and so on. The engineer must understand these fields as the basis of the design and construction of systems, such as airplanes, laser generators, thermodynamically system, or robots. In three dimensions, geometrical ideas become influential, enriching the theory. Thus, many geometrical quantities can be given by vectors.

To support the evidence of your understanding on geometry and vectors, solve the following problems

3.1 Find the equation of the line passing through (5, 7) and parallel to the line 5x – 7y = 4.

3.2 What is an equation of the line  that  goes  through  (−1,  −3)  and  is  perpendicular  to  the line 2x+7y+5=0?

3.3 Discover the estimation of h for which the lines 6x+5y+8=0 and 5x-hy+8= 0 are

(i) perpendicular to each other

(ii) parallel to each other

3.4 Explain the concepts of vector graphics in a computer system.

To present your further understanding on coordinate systems, you are required to evaluate the coordinate system used in programming a simple output device (Use a computer screen as an example of the output device in your evaluation). You should also present your findings on how to construct the scaling of simple shapes; like Triangle, circle, or a straight line in vector coordinates and visually by programming and simulating in Python or any other programming language. Unit 11: Maths for Computing

### Part 4:

Calculus (Differential and integral calculus) is deeply integrated in every branch of the physical sciences, such as physics and biology. It is found in computer science, statistics, and engineering, in economics, business, and medicine. Among the many applications in computer science, numerical calculations, systems modelling and problems involving the performance of algorithm can be cited as examples. In this task you will demonstrate your understanding of differential and integral calculus.

To support the evidence of your understanding on Calculus, solve the following problems

4.1  Suppose that x = f(t) = ¼*t2 - t + 2 denotes the position of a bus at time t.

(a)  Find the velocity as a function of time; plot its graph.

(b)  Find and plot the speed as a function of time.

(c) Find the acceleration.

4.2  Sand is pouring out of a tube at 5 cubic cm per second. It forms a pile which has the shape of a cone. The height of the cone is equal to the radius of the circle at its base. How fast is the sandpile rising when it is 10 cm high?

4.3  Find the maximum and minimum value of the function

x3 - 3x2 - 9x + 12

4.4  Find the dimensions of a rectangle with perimeter 1000 metres so that the area of the rectangle is a maximum.

4.5  If the region bounded above by the graph of the function f(x)= x2 - 6x + 9 and below by the graph of the function g(x)=x+3 over the interval [x, y] is represented as R, find the area of Region R.

4.6  Find the area between the two curves y = x2 and y = 2x – x2

### Relevant Information

To gain a Pass in a BTEC HND Unit, you must meet ALL the Pass criteria; to gain a Merit, you must meet ALL the Merit and Pass criteria; and to gain a Distinction, you must meet ALL the Distinction, Merit and Pass criteria. Unit 11: Maths for Computing

1. Learning Outcomes and Assessment Criteria

 Learning Outcomes and Assessment Criteria - Unit 11: Maths for Computing Pass Merit Distinction LO1 Use applied number theory in practical computing scenarios D1 Produce a detailed written explanation of the importance of prime numbers within the field of computing. P1Calculate the greatest common divisor and least common multiple of a given pair of numbers. P2 Use relevant theory to sum arithmetic and geometric progressions. M1 Identify multiplicative inverses in modular arithmetic. LO2 Analyse events using probability theory and probability distributions D2 Evaluate probability theory to an example involving hashing and load balancing. P3 Deduce the conditional probability of different events occurring within independent trials. P4 Identify the expectation of an event occurring from a discrete, random variable. M2 Calculate probabilities within both binomially distributed and normally distributed LO3 Determine solutions of graphical examples using geometry and vector methods D3 Construct the scaling of simple shapes that are described by vector coordinates. P5 Identify simple shapes using co- ordinate geometry. P6 Determine shape parameters using appropriate vector methods. M3 Evaluate the coordinate system used in programming a simple output device. LO4 Reflect on the application of research methodologies and concepts D4 Justify, by further differentiation, that a value is a minimum. P7 Determine the rate of change within an algebraic function. P8 Use integral calculus to solve practical problems involving area. M4 Analyse maxima and minima of increasing and decreasing functions using higher order derivatives.

2.   Preparation guidelines of the Coursework Document

1. All coursework must be word processed.
2. Avoid using “Text box” in writing your assignment.
3. Document margins must not be more than 2.54 cm (1 inch) or less than 1.9cm (3/4 inch).
4. Font size must be within the range of 10 point to 14 points including the headings and body text (preferred font size is 11) in Arial.
5. Standard and commonly used type face, such as Arial and Times New Roman, should be used.
6. All figures, graphs and tables must be numbered.
7. g.     Material taken from external sources must be properly referred and cited within the text using Harvard system
8. Do not use Wikipedia as a reference.

3.  Plagiarism and Collusion

Any act of plagiarism or collusion will be seriously dealt with according to the College regulations. In this context the definitions and scope of plagiarism and collusion are presented below

Plagiarism is presenting somebody else’s work as your own. It includes copying information directly from the Web or books without referencing the material, submitting joint coursework as an individual effort.

Collusion is copying another student’s coursework, stealing coursework from another student and submitting it as your own work.

Suspected plagiarism or collusion will be investigated and if found to have occurred will be dealt with according to the College procedure (For details on Plagiarism & Collusion please see the Student Handbook).

4.  Submission

1. Initial submission of coursework to the tutors is compulsory in each unit of the course.
2. The student must check their assignments on ICON VLE with plagiarism software Turnitin to make sure the similarity index for their assignment stays within the College approved level. A student can check the similarity index of their assignment up to five times in the Draft Assignment submission point located in the home page of the ICON VLE.
3. All Final coursework must be submitted to the Final submission point into the Unit (not to the Tutor). The student would be allowed to submit only once and that is the final submission.
4. Any computer files generated such as program code (software), graphic files that form part of the coursework must be submitted as an attachment to the assignment with all documentation.
5. e.Any portfolio for a Unit must be submitted as an attachment in the assignment

5.  Good practice

1. Make backup of your work in different media (hard disk, memory stick, etc.) to avoid distress due to loss or damage of your original copy.

6.  Extension and Late Submission

1. If you need an extension for a valid reason, you must request one using an Exceptional Extenuating Circumstances (EEC) form available from the Examination Office and ICON VLE. Please note that the tutors do not have the authority to extend the coursework deadlines and therefore do not ask them to award a coursework extension. The completed form must be accompanied by evidence such as a medical certificate in the event of you being sick and should be submitted to the Examination Office.
2. Late submission will be accepted and marked according to the College procedure. It should be noted that late submission may not be graded for Merit and Distinction.
3. All late coursework must be submitted to the Late submission point into the unit (not to the Tutor) in the ICON VLE. A student is allowed to submit only once and that is also treated as the final submission.
4. If you fail in the Final or Late submission, you can resubmit in the Resubmission window.

 Formative feedback Week 12 Final Submission 10-15 January 2022 Late submission 17-22 January 2022

Submit to: Online to the ICON VLE only

### Glossary

Analyse: Break an issue or topic into smaller parts by looking in depth at each part. Support each part with arguments and evidence for and against (Pros and cons)

Evaluate: When you evaluate you look at the arguments for and against an issue.

Critically Evaluate/Analyse: When you critically evaluate you look at the arguments for and against an issue. You look at the strengths and weaknesses of the arguments. This could be from an article you read in a journal or from a textbook

Discuss: When you discuss you look at both sides of a discussion. You look at both sides of the arguments. Then you look at the reason why it is important (for) then you look at the reason why it is important (against).

Explain: When you explain you must say why it is important or not important.

Describe: When you give an account or representation of in words. Identify: When you identify you look at the most important points. Define: State or describe the nature, scope or meaning.

Implement: Put into action/use/effect Compare: Identify similarities and differences Explore: To find out about

Recommend: Suggest/put forward as being appropriate, with reasons why

The price includes Python software work as well.

Unit 11: Maths for Computing

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