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Higher National Certificate/Diploma in Computing
Assessment Brief
Student Name/ID Number Unit Number and Title 11: Maths for Computing Academic Year 2019/ Unit Tutor Darshana Hettiarachchi Assignment Title Application of number theory , probability ,Vector geometry & calculus in computing Issue Date 27 th^ September 2020 Submission Date IV Name & Date Gajhanan V. 26/6/ This assignment should be submitted at the end of your lesson, on the week stated at the front of this brief. The assignment can either be word-processed or completed in legible handwriting. If the tasks are completed over multiple pages, ensure that your name and student number are present on each sheet of paper. Each of the section below would help you get guided in achieving the assessment criteria directly and indirectly.
Unit Learning Outcomes LO1 Use applied number theory in practical computing scenarios LO2 Analyse events using probability theory and probability distributions LO3 Determine solutions of graphical examples using geometry and vector methods LO4 Evaluate problems concerning differential and integral calculus.
Section 1
Consider yourself as the programmer, who is assigned to verify algorithms to solve the below given problem. You’re applying your mathematical knowledge to verify and suggest any remarks accordingly. In addition to the direct answers you may suggest how your derivations can be replicated in coding within what you know. Try to emphasize on importance of prime numbers in computing in your explanations. Part 1
- Mr. John has 120 pastel sticks and 30 pieces of paper to give to his students. a) Find the largest number of students he can have in his class so that each student gets equal number of pastel sticks and equal number of paper. b) Briefly explain the technique you used to solve that question (a).
- Rashmi is making a game board that is 16 inches by 24 inches. She wants to use square tiles. What is the largest tile she can use? Brief your answer accordant to theories. Part 2
- The Class room has 40 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. What is formula can you solve this problem? Using relevant theories with description, find how many seats are there in all 40 rows?
- Suppose you are training to run an 8 km race. You plan to start your training by running 2km a week, and then you plan to add a ½km more every week. At what week will you be running 8km? Brief your answer accordant to theories.
- Uvindu wanted purchase a new laptop worth 98,500. He borrows 100,000 rupees from a bank that charges 12% interest. Using relevant theories, determine how much he will owe the bank over a period of 5 years. Part 3
- Find the multiplicative inverse of 8 mod 11 while explaining the algorithm you learnt in Math for computing? Part 4
- Produce a detailed written explanation of the importance of prime numbers within the field of computing.
- A discrete random variable X has the following probability distribution: x 1 2 3 4 P(X=x) 1/3 1/3 k 1/ Where k is a constant. (a) Find the value of k. (b) Find P(X ≤3). Part 3
- Portable Router manufacturing company quality control analysis given bellow, the random variable X represents the number of defective products per each batch of 100 products produced. Defects (x) (^0 1 2 3 4 ) Batches 95 113 87 64 13 8 (a) Use the frequency distribution above to construct a probability distribution for X. (b) Find the mean of this probability distribution. (c) Find the variance and standard deviation of this probability distribution.
- A surgery has a success rate of 85%. Suppose that the surgery is performed on three Patients. (a) What is the probability that the surgery is successful on exactly 2 patients? (b) Let X be the number of successes. What are the possible values of X? (c) Create a probability distribution for X. (d) Graph the probability distribution for X using a histogram. (e) Find the mean of X. (f) Find the variance and standard deviation of X.
- Computing Research team doing survey of raining in western province. City of Gampaha typically has rain on about 16% of days in April. (a) What is the probability that it will rain on exactly 5 days in April? 15 days? (b) What is the mean number of days with rain in April? (c) What is the variance and standard deviation of the number of days with rain in April? (d) How probability theories relate to Artificial intelligent programming?
- Supermarket CRM system has past records; a supermarket finds that 26% of people who enter the supermarket will make a purchase. 18 people enter the supermarket during a one-hour period. (a) What is the probability that exactly 10 customers, 18 customers and 3 customers make a purchase? (b) Find the expected number of customers who make a purchase. (c) Find the variance and standard deviation of the number of customers who make a purchase. 14.On a recent mathematics test, the mean score was 75 and the standard deviation was 5. Nimal got 93. Would his mark be considered an outlier if the marks were normally distributed? Explain your answer ?.
- For each question, construct a normal distribution curve and label the horizontal axis and answer each question. The shelf life of a dairy product is normally distributed with a mean of 12 days and a standard deviation of 3 days. (a) About what percent of the products last between 9 and 15 days? (b) About what percent of the products last between 12 and 15 days? (c) About what percent of the products last 6 days or less? (d) About what percent of the products last 15 or more days? (e) Brief your answers accordant to theories.
- Statistics held by the Road Safety & computer forensic Division of the Police shows that 78% of drivers being tested for their license pass at the first attempt. If a group of 120 drivers are tested in one center in a year,
find the probability That more than 99 pass at the first attempt, justifying the most appropriate distribution to be used for this scenario. Part 4
- What is hashing and load balancing? Evaluate probability theory to an example involving hashing and load balancing.
Section 4
Part 1
- Find the function whose tangent has slope 4x + 1 for each value of x and whose graph passes through the point (1, 2).
- Find the function whose tangent has slope 3x2 + 6x − 2 for each value of x and whose graph passes through the point (0, 6). Part 2
- It is estimated that r years from now the population of a certain lakeside community will be changing at the rate of 0.6r 2 + 0.2r + 0.5 thousand people per year. Environmentalists have found that the level of pollution in the lake increases at the rate of approximately 5 units per 1000 people. By how much will the pollution in the lake increase during the next 2 years? Brief your answer.
- Large Laser screen printing head is moving so that its speed after t minutes is v(t) = 1+4t+3t 2 meters per minute. How far does the Head move during 3rd minute? Part 3
- Sketch the graph of f(x) = x − 3x 2/3 , indicating where the graph is increasing/decreasing, concave up/down, and any asymptotic behavior.
- Draw the graph of f(x)= 3x4-6X3+3x2 by using the extreme points from differentiation. Part 4
- For the function f(x) = Cos 2x, 0.1 ≤ x ≤ 6, find the positions of any local minima or maxima and distinguish between them. 8.Determine the local maxima and/or minima of the function y = x4 −1/3x3 ,:Brief your answer?
Table of Contents
- Section Contents
- Part
- Part
- Part
- Part
- Importance of prime numbers within the field of computing.
- Section
- Part
- Part
- Part
- Part
- What is Load Balancing?
- What is Hashing?
- Section
- Section
Section 1
Part 1
a) Number of pastel sticks john has = 120 Number of pieces of papers = 30
- Need to find 120 and 30 Greatest Common Factor (GCM) to find the largest number of students receiving the same number of pastel sticks and number of pieces of papers.
- GCF of 120 and 30 Prime factor of 120 and 30 120 = 2 x 2 x 2 x 3 x 5 = 120 30 = 2 x 3 x 5 = 30 Then 2, 3 and 5 is common. GCF of 120 and 30 = 2 x 3 x 5 = 30
- Then 30 students have in the class so that each student can get equal number of pastel sticks and piece of paper. Num of pastel for 1 student = 120/30 = 4 Num of piece of paper for 1 student = 30/30 = 1 b) When Need to equally distribute two or more set of items into the largest group we have to find the GCF. There are few steps we follow 1. Find the key factor in the numbers 2. Multiply the factors that make all the numbers common.
- The task is to divide the game board into cells of equal size at maximum size. It is clear that the side of such a cell should be 16 to 24 inches apart. A suitable number is the largest integer divisor of these numbers. The playing board has only 6 square cells of 8x8 inches. 2 rows on 16 - inch side and 3 rows on 24-inch side. So, Rashmi can use the largest dimension of 8x8 inch.
Part 4
07) Importance of prime numbers within the field of computing.
Definition of prime number A prime number is a natural integer, greater than 1, which can be divided only by 1 or by itself (without generating remainder). Definition of composited number Conversely, a composited number is a natural integer, greater than 1, which can be divided by two or more prime numbers (without generating remainder). Primes: the building blocks of mathematics by the two definitions, we can understand how prime numbers can be considered as the building blocks of mathematics, because with them (along with the number 1, which is a special number) are formed all the natural integers. • 1 (special number = 1 x 1 x 1 x ……) • 2 (prime number) • 3 (prime number) • 4 (composited number) = 2 x 2 • 5 (prime number) • 6 (composited number) = 2 x 3 • 7 (prime number) • 8 (composited number) = 2 x 2 x 2 • 9 (composited number) = 3 x 3 • 10 (composited number) = 2 x 5 • … etcetera … The number 1 is special! Since thousands of years mathematicians have discussed if the number 1 is a prime number or not (as in the modern trend of thought). The number 1 is actually a special number (as the number 0 [zero] is special for other reasons); in fact, it can be at the same time a prime number and a composite number. Here's why: • the number 1 falls in the specificity of prime numbers as it can only be divided by 1 or by itself without generating remainder • the number 1 can, at the same time, be the result of 1x1x..x1 (staying the same) and then fall in the specificity of dialed numbers
Section 2
Part 1
Conditional probability is defined as the likelihood of an event or outcome occurring, based on the occurrence of a previous event or outcome. Conditional probability is calculated by multiplying the probability of the preceding event by the updated probability of the succeeding, or conditional, event. Conditional Probability Formula P(B|A) = P(A and B) / P(A) which you can also rewrite as: P(B|A) = P(A∩B) / P(A)
- A and B
