Unit 11 Maths for Computing - BTEC Higher National Diploma (HND) in Computing
You are strongly advised to read "Preparation guidelines of the Coursework Document" before answering your assignment.
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
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 APs 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 5th term of an AP is 17/6 and the 9th term is 25/6. What is the 12th term?
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. We have three numbers in an arithmetic progression, and another three numbers in a geometric progression. Adding the corresponding terms of the two series, we get 120,116,130. If the sum of all the terms in the geometric progression is 342, what is the largest term in the geometric progression?
4. There is a set of four numbers p, q, r and s respectively in such a manner that first three are in G.P. and the last three are in A.P. with a difference of 6. If the first and the fourth numbers are the same, find the value of p.
5. The sum of three numbers in a GP is 26 and their product is 216. Find the numbers.
Part 2
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.
a. What is the probability of getting a sum of 7 when two dice are thrown?
b. A coin is thrown 3 times. what is the probability that at least two head is obtained?
c. From a pack of cards, two cards are drawn at random. Find the probability that each card is numbered and from same suite.
d. A bag contains 10 white, 5 red and 9 blue balls. Four balls are drawn at random from the bag. What is the probability that?
i) all of them are red.
ii) two is white and the rest two is blue.
e. Find the probability distribution of boys and girls in families with 7 children, assuming equal probabilities for boys and girls. Draw the graph of your probability distribution.
f. 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?
a) less than 19 hours?
b) between 19 and 21 hours?
g. The annual salaries of employees in a large company are approximately normally distributed with a mean of £45,000 and a standard deviation of £25,000.
a) What percent of people earn less than £35,000?
b) What percent of people earn between £36,000 and £38,000?
c) What percent of people earn more than £40,000?
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
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
Find the equation of the line passing through (4, 5) and parallel to the line 3x - 2y = 4.
A triangle's vertices have coordinates (0, 0), (0, 7), and (-9, -8) What is its area?
Discover the estimation of m for which the lines 5x+3y +2=0 and 3x-my+6= 0 are
(i) perpendicular to each other.
(ii) parallel to each other
Below is the figure representing different sides with vector x and z. The mid-point of the line AB is D and BE is parallel of AC and half of AC.
(a) Find the vector CD in term of P and Q.
(b) Find the vector DE in term of P and Q.
Two forces F1 and F2 with magnitudes 40N and 70N, respectively, act on an object at a point P as shown. Find the resultant forces acting at P.
To present your 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.
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
Let the profit, P, (in thousands of pounds) earned from producing x items be found by P(x)=2x2-6x+25. Find the average rate of change in profit when production increases from 4 items to 5 items.
An airplane is flying in a straight direction and at a constant height of 5000 meters (see figure below). The angle of elevation of the airplane from a fixed point of observation is a. The speed of the airplane is 500 km / hr. What is the rate of change of angle a when it is 25 degrees?
Find the maximum and minimum value of the function
x3 - 3x2 - 9x + 12
A square sheet of cardboard with each side a centimetre is to be used to make an open-top box by cutting a small square of cardboard from each of the corners and bending up the sides. What is the side length of the small squares if the box is to have as large a volume as possible?
Find the area under the curve using the application of integrals, for the region enclosed by the ellipse x2/36 + y2/25 = 1.
Find the area contained between the two curves x=y2-y-6 and x=2y+4
Attachment:- Maths for Computing.rar