Mathematics in software Engineering concepts, Assignments of Mathematics for Computing

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Higher Nationals

Internal verification of assessment decisions – BTEC (RQF)

Student Name/ID

UnitTitle Unit 11 : Maths for Computing

Assignment Number^1 Assessor

Submission Date Date Received 1st submission

Re-submission Date Date Received 2nd submission Assessor Feedback:

LO1 Use applied number theory in practical computing scenarios.

Pass, Merit & Distinction P1 P2 M1 (^) D Descripts LO2 Analyse events using probability theory and probability distributions

Pass, Merit & Distinction P3 P4 M2 D Descripts

LO3 Determine solutions of graphical examples using geometry and vector methods Pass,^ Merit^ &^ Distinction^ P5^ P6^ M3^ D Descripts LO4 Evaluate problems concerning differential and integral calculus

Pass, Merit & Distinction P7 P8 M D4 Descripts

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Pearson

Higher Nationals in

Computing

Unit 11 : Maths For Computing

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Assignment Brief

Student Name /ID Number

Unit Number and Title Unit 11 : Maths for Computing

Academic Year 2020/

Unit Tutor

Assignment Title Importance of Maths in the Field of Computing

Issue Date

Submission Date

IV Name & Date

Submission Format:

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.

Assignment Brief and Guidance :

Activity 01

Part 1

1. Mr.Steve has 120 pastel sticks and 30 pieces of paper to give to his students.

a) Calculate 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 (a).

2. Maya is making a game board that is 16 inches by 24 inches. She wants to

use square tiles. Calculate metrics of the largest tile she can use?

Part 2

3. An auditorium 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. Use relevant theories, find

how many seats are there in all 40 rows?

4. Suppose you are training to run an 8km 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?

5. Suppose you borrow 100,000 rupees from a bank that charges 15%

interest. Use relevant theories, determine how much you will owe the bank

over a period of 5 years.

Part 3

6. Identify the multiplicative inverse of 8 mod 11 while explaining the algorithm used.

Part 4

7. Produce a detailed written explanation of the importance of prime numbers

within the field of computing.

Activity 02

Part 1

1. Define ‘conditional probability’ with suitable examples.

2. A school which has 100 students in its sixth form, 50 students study mathematics,

29 study biology and 13 study both subjects. Find the probability of the student

studying mathematics given that the student studies biology.

3. A certain medical disease occurs in 1% of the population. A simple screening

procedure is available and in 8 out of 10 cases where the patient has the disease, it

produces a positive result. If the patient does not have the disease there is still a 0.

chance that the test will give a positive result. Find the probability that a randomly

selected individual:

(a) Does not have the disease but gives a positive result in the screening test

(b) Gives a positive result on the test

(c) Nilu has taken the test and her result is positive. Deduce the conditional

probability that she has the disease.

Let C represent the event “the patient has the disease” and S represent the event

“the screening test gives a positive result”.

4. In a certain group of 15 students, 5 have graphics calculators and 3 have a computer

at home (one student has both). Two of the students drive themselves to college each

day and neither of them has a graphics calculator nor a computer at home. A student

is selected at random from the group.

(a) Find the probability that the student either drives to college or has a graphics

calculator.

(b) Show that the events “the student has a graphics calculator” and “the student has a

computer at home” are independent.

Let G represent the event “the student has a

graphics calculator” H represent the event “the

student has a computer at home”

D represent the event “the student drives to college each day”

Represent the information in this question by a Venn diagram. Use the above Venn diagram to answer the questions.

10. In a quality control analysis, the random variable X represents the

number of defective products per each batch of 100 products produced.

Defects (x) 0 1 2 3 4 5

Batches 95 113 87 64 13 8

(a) Use the frequency distribution above to calculate the probabilities of X.

(b) Calculate the mean of this probability distribution.

(c) Find the variance and standard deviation of this probability distribution.

11. A surgery has a success rate of 75%. Suppose that the surgery is

performed on three patients.

(a) Calculate 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) Calculate the mean of X.

(f) Find the variance and standard deviation of X.

12. Colombo City typically has rain on about 16% of days in November.

(a) Calculate the probability that it will rain on exactly 5 days in November? 15 days?

(b) Calculate the mean number of days with rain in November?

(c) What is the variance and standard deviation of the number of days with rain in

November?

13. From 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) Calculate 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 math test, the mean score was 75 and the standard deviation was

5. Shan got 93. Would his mark be considered an outlier if the marks were

normally distributed? Explain.

15. 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?

16. Statistics held by the Road Safety 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 Centre 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

17.Evaluate probability theory to an example involving hashing and load balancing.

Activity 03

Part 1

1. If the Center of a circle is at (2, -7) and a point on the circle (5,6) find the formula of

the circle.

2. Identify the surfaces in R^3 that are represented by the

following equations? z = 3

y = 5

3. Determine the equation of a sphere with radius r and center C(h, k, l).

4. Show that x^2 + y^2 + z^2 + 4x – 6y + 2z + 6 = 0 is the equation of a sphere. Also,

find its center and radius.

Part 2

5. 3y= 2x-5 , 2y=2x+7 evaluate the x, y values using graphical method.

7. 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 = x^4 −1/3x^3

9. Justify by further differentiation, the minimum lines of y = 12 x 2 − 2x, y = x 2 + 4x + 1,

y = 12x − 2x 2 , y = −3x 2 + 3x + 1.

Grading Rubric

Grading Criteria Achieve

d

Feedback

LO1 : Use applied number theory in practical

computing scenarios

P1 Calculate 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.

D1 Produce a detailed written explanation of the

importance of prime numbers within the field of

computing.

LO2 Analyse events using

probability theory and

probability distributions

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 random

variables.

D2 Evaluate probability theory to an example

involving hashing and load balancing.

LO3 Determine solutions of graphical

examples using geometry and vector

methods

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.

D3 Construct the scaling of simple shapes

that are described by vector coordinates.

LO4 Evaluate problems

concerning differential and

integral calculus

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.

D4 Justify, by further differentiation, that a value is

a minimum.

Alarm System ........................................................................................................................... .................................................................................................................................................

2. A school which has 100 students in its sixth form, 50 students study mathematics, 29 study biology and 13 study both subjects. Find the probability of the student studying mathematics given that the student studies biology. ..............................................................................................................................
3. A certain medical disease occurs in 1% of the population. A simple screening procedure is available and in 8 out of 10 cases where the patient has the disease, it produces a positive result. If the patient does not have the disease there is still a 0.05 chance that the test will give a positive result. Find the probability that a randomly selected individual: .................................................................................................. a) Does not have the disease but gives a positive result in the screening test .................................... b) Gives a positive result on the test ........................................................................................... c) Nilu has taken the test and her result is positive. Deduce the conditional probability that she has the disease. Let C represent the event “the patient has the disease” and S represent the event “the screening test gives a positive result”. ..........................................................................................................
4. In a certain group of 15 students, 5 have graphics calculators and 3 have a computer at home (one student has both). Two of the students drive themselves to college each day and neither of them has a graphics calculator nor a computer at home. A student is selected at random from the group. ................... a) Find the probability that the student either drives to college or has a graphics calculator. .............. b) Showthat the events “the student has a graphics calculator” and “the student has a computer at home” are independent. Let G represent the event “the student has a graphics calculator” H represent the event “the student has a computer at home” D represent the event “the student drives to college each day” Represent the information in this question by a Venn diagram. Use the above Venn diagram to answer the questions. 28
5. A bag contains 6 blue balls, 5 green balls and 4 red balls. Three are selected at random without replacement. Find the probability that ............................................................................................... a) Probability that they are all blue............................................................................................. b) Probability that two are blue and one is green .......................................................................... c) Probability that there is one of each color ................................................................................

Part 2 ...............................................................................................................................................

6. Differentiate between ‘Discrete’ and ‘Continuous’ random variables. .............................................. Comparison Chart ....................................................................................................................
7. Two fair cubical dice are thrown: one is red and one is blue. The random variable M represents the score on the red die minus the score on the blue die. ........................................................................... a) Find the distribution of M. .................................................................................................... b) Identify the Expected value of M ........................................................................................... c) Find Var (M). ......................................................................................................................
8. Two 10p coins are tossed. The random variable X represents the total value of each coin lands heads up. 33 a) Identify the expected value of the random variable X ............................................................... b) Show that E(S) = E(T). .........................................................................................................

d) Susan and Thomas play a game using two 10p coins. The coins are tossed and Susan records her

a) What is the probability that exactly 10 customers, 18 customers and 3 customers make a purchase?

15. For each question, construct a normal distribution curve and label the horizontal axis and answer

Table of Figures

 - c) Find Var(S) and Var (T)........................................................................................................ - they compare their scores. Comment on any likely differences or similarities. .................................... score using the random variable S and Thomas uses the random variable T. After a large number of tosses 

• 9. A discrete random variable X has the following probability distribution: ........................................
  • a) Find the value of k. ..............................................................................................................
  • b) Find P(X ≤3). .......................................................................................................................
• Part 3 ...............................................................................................................................................
  • each batch of 100 products produced................................................................................................. 10. In a quality control analysis, the random variable X represents the number of defective products per
    • a) Use the frequency distribution above to calculate the probabilities of X. .....................................
    • b) Calculate the mean of this probability distribution ....................................................................
    • c) Find the variance and standard deviation of this probability distribution......................................
  • 11. A surgery has a success rate of 75%. Suppose that the surgery is performed on three patients. .......
    • a) Calculate 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) Calculate the mean of X. .......................................................................................................
    • f) Find the variance and standard deviation of X..........................................................................
  • 12. Colombo City typically has rain on about 16% of days in November. .........................................
    • a) Calculate the probability that it will rain on exactly 5 days in November? 15 days? ......................
    • b) Calculate the mean number of days with rain in November? ......................................................
    • c) What is the variance and standard deviation of the number of days with rain in November? ..........
  • purchase. 18 people enter the supermarket during a one-hour period. .................................................... 13. From past records, a supermarket finds that 26% of people who enter the supermarket will make a
    • b) Calculate 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. .........
  • his mark be considered an outlier if the marks were normally distributed? Explain ................................. 14. On a recent math test, the mean score was 75 and the standard deviation was 5. Shan got 93. Would
  • standard deviation of 3 days. ............................................................................................................ each question. The shelf life of a dairy product is normally distributed with a mean of 12 days and a
    • 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? .............................................................
  • 9. Justify by further differentiation, the minimum lines of y = 12 x 2 − 2x, y = x 2 + 4x + 1, ...................
  • y = 12x − 2x 2 , y = −3x 2 + 3x + 1. ...................................................................................................
• Figure 1 Venn diagram showing that 20 out of 40 alarm buyers purchased bucket seats .........................
• Figure 2 Venn diagram ........................................................................................................................
• Figure 3 Histogram ..............................................................................................................................
• Figure 4 Graph - 1 ...............................................................................................................................
• Figure 5 Graph - 2 ...............................................................................................................................
• Figure 6 Graph - 3 ...............................................................................................................................
• Figure 7 Graph - 4 ...............................................................................................................................
• Figure 8 Graph - 5 ...............................................................................................................................
• Figure 9 involving hashing and load balancing ......................................................................................
• Figure 10 involving hashing and load balancing ....................................................................................
• Figure 11 z=3, a plane in R^3 ..................................................................................................................
• Figure 12 y=5, a plane in R^3 ..................................................................................................................
• Figure 13 Graph 1 ................................................................................................................................
• Figure 14 Graph 2 ................................................................................................................................
• Figure 15 Graph ..................................................................................................................................
• Figure 16 Graph ..................................................................................................................................
• Table 1 Comparison Chart ................................................................................................................... List of Tables
• Table 2 Probability outcomes ...............................................................................................................
• Table 3 Probability distribution ............................................................................................................
• Table 4 Variance calculation ................................................................................................................
• Table 5 Probability distribution ............................................................................................................
• Table 6 Probability distribution ............................................................................................................
• Table 7 Derivative test ..........................................................................................................................
• Table 8 1. First derivative test ..............................................................................................................
• Table 9 2.First derivative test ...............................................................................................................
• Table 10 3.First derivative test ..............................................................................................................
• Table 11 4. First derivative test .............................................................................................................

Activity 01

Part 1

1. Mr. Steve has 120 pastel sticks and 30 pieces of paper to give to his students.

a) Calculate 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.

Common factors = 23

G.C.F = 30

b) Briefly explain the technique you used to solve (a).

I choose the method of GCF- Greatest Common Factor. Remember that factors

are the numbers I multiply together to get another number

 List the factors of each number,

 Look for factors that both lists have in common,

 Pick out the largest factor that both lists have in common and call this GCF.

Remember that factors are the numbers we multiply together to get another number

Example:- 2x 3 x 5= 30

2,3and 5 are factors of 30.

2. Maya is making a game board that is 16 inches by 24 inches. She wants to use square tiles. Calculate metrics of the largest tile she can use?

Common factors = 2*

G.C.F = 8