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Programme title BTEC Higher National Diploma in Computing Unit(s) Unit 11 : Maths for Computing Assignment title : Importance of Maths in the Field of Computing Pearson Higher Nationals in Computing
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.
Activity 01
Part 1
- Steve has 120 pastel sticks and 30 pieces of paper to give to his students.
- 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.
- Briefly explain the technique you used to solve (a).
- Maya 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?
Part 2
- 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. Using relevant theories, find how many seats are there in all 40 rows?
- 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?
- Suppose you borrow 100,000 rupees from a bank that charges 15% interest. Using relevant theories, determine how much you will owe the bank over a period of 5 years.
Part 3
- Find the multiplicative inverse of 8 mod 11 while explaining the algorithm used.
Part 4
- Produce a detailed written explanation of the importance of prime numbers within the field of computing.
Activity 02
Part 1
- Define ‘conditional probability’ with suitable examples.
- 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.
- 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. Find the 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”.
- 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.
- 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) they are all blue
(b)two are blue and one is green
(c) there is one of each colour
Activity 02
Part 1
- Define ‘conditional probability’ with suitable examples.
- 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.
- 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. Find the 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”.
- 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.
- 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) they are all blue
(b)two are blue and one is green
(c) there is one of each colour
Part 2
- Differentiate between ‘Discrete’ and ‘Continuous’ random variables.
- 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) Write down E(M).
(c) Find Var(M).
- Two 10p coins are tossed. The random variable X represents the total value of each coin lands heads up.
(a)Find E(X) and Var(X).
The random variables S and T are defined as follows:
S = X-10 and T = (1/2)X-5
(b)Show that E(S) = E(T).
(c)Find Var(S) and Var (T).
(d)
Susan and Thomas play a game using two 10p coins. The coins are tossed and Susan records her score using the random variable S and Thomas uses the random variable T. After a large number of tosses they compare their scores.
Comment on any likely differences or similarities.
- A discrete random variable X has the following probability distribution:
x |
1 |
2 |
3 |
4 |
P(X=x) |
1/3 |
1/3 |
k |
1/4 |
where k is a constant.
(a) Find the value of k.
(b) Find P(X ≤3).
Part 3
- 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 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 75%. 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.
- Colombo City typically has rain on about 16% of days in November.
(a) What is the probability that it will rain on exactly 5 days in November? 15 days?
(b) What is 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?
- 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) 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 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 licence 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. What surfaces in R^{3} are represented by the following equations?
z = 3
y = 5
3. Find an 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.
6.
a=(2i+3j) , b=(4i-2j) and c=(1i+4j) evaluate the volume of the shape.
Activity 04
Part 1
1. Find the function whose tangent has slope 4x + 1 for each value of x and whose graph passes through the point (1, 2).
2. Find the function whose tangent has slope 3x^{2} + 6x − 2 for each value of x and whose graph passes through the point (0, 6).
Part 2
3. It is estimated that t years from now the population of a certain lakeside community will be changing at the rate of 0.6t ^{2} + 0.2t + 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?
4. An object is moving so that its speed after t minutes is v(t) = 1+4t+3t ^{2} meters per minute. How far does the object travel during 3rd minute?
Part 3
5. Sketch the graph of f(x) = x − 3x ^{2/3} , indicating where the graph is increasing/decreasing, concave up/down, and any asymptotic behavior.
6. Draw the graph of f(x)= 3x^{4}-6X^{3}+3x^{2} by using the extreme points from differentiation.
Part 4
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. By further differentiation, identify lines with minimum y = 12 x^{ 2} − 2x, y = x ^{2 }+ 4x + 1, y = 12x − 2x ^{2} , y = −3x ^{2} + 3x + 1.
Grading Criteria |
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. |
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