Higher Nationals
Internal verification of assessment decisions – BTEC (RQF)
INTERNAL VERIFICATION – ASSESSMENT DECISIONS
Programme title
BTEC Higher National Diploma in Computing
Assessor
Unit(s)
Assignment title
Student’s name
Internal Verifier
Unit 11 : Maths for Computing
Importance of Maths in the Field of Computing
Abdul Ahadh
List which assessment
criteria the Assessor has
awarded.
Pass
Merit
Distinction
INTERNAL VERIFIER CHECKLIST
Do the assessment criteria awarded match
those shown in the assignment brief?
Y/N
Is the Pass/Merit/Distinction grade awarded
justified by the assessor’s comments on the
student work?
Y/N
Has the work been assessed
accurately?
Y/N
Is the feedback to the student:
Give details:
• Constructive?
• Linked to relevant assessment
criteria?
Y/N
Y/N
• Identifying opportunities for
improved performance?
Y/N
• Agreeing actions?
Y/N
Does the assessment decision need
amending?
Y/N
Assessor signature
Date
Internal Verifier signature
Date
Programme Leader signature(if
required)
Date
Abdul Ahadh
1
Confirm action completed
Remedial action taken
Give details:
Assessor signature
Date
Internal Verifier
signature
Date
Programme Leader
signature (if required)
Date
Higher Nationals - Summative Assignment Feedback Form
ABDUL AHADH
Student Name/ID
Unit Title
Unit 11 : Maths for Computing
Assignment Number
1
Assessor
Submission Date
Date Received 1st
submission
Date Received 2nd
submission
Re-submission Date
Assessor Feedback:
LO1 Use applied numbe r theory in practical computing scenarios.
Pass, Merit &
Distinction Descripts
LO2 Analyse events
usin g
Pass, Merit &
Distinction Descripts
LO3 Determine
solution s
Pass, Merit &
Distinction Descripts
P1
probabi
P2 theory
lity
robability di
butions
stri
M2
P3
of
graph
geometr
y
examples us P6
ical
ing
and
M3
fferential and
P5
o
D2
P4
LO4 Evaluate problems ncernin
g
co
di
Pass, Merit &
P7
Distinction Descripts
Grade:
and p
D1
M1
vector
m
D3
ds
eth
tegral cal
P8
in
s
M4
culu
D4
Assessor Signature:
Date:
Assessor Signature:
Internal Verifier’s Comments:
Date:
Resubmission Feedback:
Grade:
Signature & Date:
ABDUL AHADH
* Please note that grade decisions are provisional. They are only confirmed once internal and external moderation has
taken place and grades decisions have been agreed at the assessment board.
ABDUL AHADH
Pearson
Higher Nationals in
Computing
Unit 11 : Maths for Computing
General Guidelines
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1. A Cover page or title page – You should always attach a title page to your assignment. Use previous page as
your cover sheet and be sure to fill the details correctly.
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I hereby, declare that I know what plagiarism entails, namely to use another’s work and to present it as my own
without attributing the sources in the correct way. I further understand what it means to copy another’s work.
1. I know that plagiarism is a punishable offence because it constitutes theft.
2. I understand the plagiarism and copying policy of the Edexcel UK.
ABDUL AHADH
3. I know what the consequences will be if I plagiaries or copy another’s work in any of the assignments for this
program.
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I have made use of another’s work, I will attribute the source in the correct way.
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myself and Edexcel UK.
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Student’s Signature:
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ABDUL AHADH
Date:
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Assignment Brief
ABDUL AHADH
Student Name /ID Number
Unit Learning Outcomes:
Unit Number and Title
Unit 11 : Maths for Computing
LO1 Use applied number theory in practical computing scenarios
LO2
Analyse events using probability theory and probability distributions
Academic Year
2017/2018
LO3 Determine solutions of graphical examples using geometry and vector Methods LO4
Unit Tutor
Evaluate problems concerning differential and integral calculus.
Assignment Title
Importance of Maths in the Field of Computing
Issue Date
Submission Date
IV Name & Date
Assignment Brief and Guidance:
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.
ABDUL AHADH
Activity 01
Part 1
1. Mr.Steve 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 (a).
2. 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
3. An auditorium has 40 rows of seats. There are 20 seats in the first row, 21 seats in the second
row, and 22 seats in the third row, and so on. Using 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. Using relevant
theories, determine how much you will owe the bank over a period of 5 years.
Part 3
6. Find 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.
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Activity 02
ABDUL AHADH
ABDUL AHADH
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 0.05 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”.
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.
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) they are all blue
(b)two are blue and one is green
(c) there is one of each colour
ABDUL AHADH
ABDUL AHADH
ABDUL AHADH
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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.
12. 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?
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) 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
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17.Evaluate probability theory to an example involving hashing and load balancing.
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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 R3 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 x2 + y2 + z2 + 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.
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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 3x2 + 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)= 3x4-6X3+3x2 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 = x4 −1/3x3
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.
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Grading Rubric
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.
Achieved
Feedback
M2 Calculate probabilities within both binomially distributed and
normally distributed random variables.
D2 Evaluate probability theory to an example involving hashing and
load balancing.
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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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Activity 01 ..................................................................................................................
17
Part 01 ........................................................................................................................................................ 17
Part 02 ........................................................................................................................................................ 18
Part 03 ........................................................................................................................................................ 19
Activity 02 .................................................................................................................
20
Part 01 ........................................................................................................................................................ 20
Part 02 ......................................................................................................................................................... 24
Part 03 ......................................................................................................................................................... 30
Activity 03 .................................................................................................................
38
Part 01 ......................................................................................................................................................... 38
Part 02 ......................................................................................................................................................... 39
Activity 04 .................................................................................................................
40
Part 01 ........................................................................................................................................................ 40
Part 02 ......................................................................................................................................................... 41
Part 03 ........................................................................................................................................................ 42
29
Part 04 ........................................................................................................................................................ 43
30
Activity 01
Part 01
1.
(A)
H.F.C= 2*3*5
= 30
30 Students
31
(B)
32
I have used the Highest Common Factor – HFC method to solve the previous answer. Whenever we solve this kind
33
of questions using Highest Common Factor, we determine the prime factors of the numbers first, and then multiply
each number that has a common factor.
34
2.
35
H.C.F = 2*2*2
=8
Part 02
3.
•
•
•
•
•
The total of all the seats in the row
= 40
Seats that are available in first row
= 20
Seats that are available in second row = 21
Seats that are available in third row = 22
The common difference
=1
20, 21,22 a1 = 20, n = 40, d = 1 an = a1 + (n-1) d an = 20 + 39
an = 59
all 40 rows, Sn=n/2(a1+an)
Sn=40/2 (20+59)
Sn=20+79
Sn=1580
36
4.
37
1st term = 2
The common difference = 0.5
an = 8 km
an = a +(n-1) d
8 =2 + (n-1) 0.5
6.5 = 0.5n
n = 13
38
13th week
5.
39
a1 = 100,000, r = 15% / 1.5, n = 5
100 – 115, r = 115/100=1, 15 an = a1 rn-1 an = 100,000 * 1.55-4 an =
40
100,000 * 1.54 an = 100,000 * 1.74900
an = 174,900.625
Part 03
6.
8 x 0 mod 11 = 0
8 x 1 mod 11 = 8
8 x 2 mod 11 = 5
8 x 3 mod 11 = 2
8 x 4 mod 11 = 10
8 x 5 mod 11 = 7
8 x 6 mod 11 = 4
8 x 7 mod 11 = 1
41
7.
42
What is Prime number?
43
A prime number is a whole number bigger than one with just one and itself as components. A factor is a full number
44
that may be equally split into another. 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29 are the first few prime numbers.
Composite numbers are those that have more than two components. The number one is neither composite nor
prime.
Activity 02
Part 01
1. Conditional Probability
What is Conditional Probability
The possibility of an event or outcome occurring dependent on the occurrence of a preceding event or outcome is
known as conditional probability. The updated probability of the subsequent, or conditional, event is multiplied by
the probability of the previous, or conditional, event to get conditional probability.
EXAMPLE: • The event A is that a student applying to college is admitted. There is an 80% likelihood that this student will
be admitted into college.
• The event B is that the student will be assigned to a dorm. Only 60% of the approved students will be able to
live in dorms.
45
•
P (Dormitory Housing | Accepted) P (Accepted) = (0.60) *(0.80) = 0.48. P (Accepted) = (0.60) *(0.80) =
46
0.48.
The formula of Conditional Probability
P(B|A) = P (A and B) / P(A)
47
2.
48
Mathematics = A
Biology = B
Mathematics and Biology = A∩B
3.
49
50
51
4.Venn Diagram
52
(A)
H
H∩G
G
D
•
•
•
•
•
•
Student with only graphical calculators: 5-1 = 4
Students who have computers at home: 3-1 = 2
Students who have computers and graphical calculators =1
Students who drive to college: 15-(4+2+1) =8
Students who drive to college every day or student who has graphics calculator: 8+3 =11
Students who have computers and drive to school: 8+3 = 11
Probability of the students who drive to school or has graphical calculator: 13/15
(B)
5
1
𝑃(𝐺) =
=
15
3
3
1
𝑃(𝐺) =
15
5
=
1
1
1
𝑃(𝐻) . 𝑃(𝐺) =
×
=
53
3
5
15
54
55
As a result, the pupils had graphical calculators and the computers were self-contained.
56
5. B=6, G=5, R=4
The total of the balls: 6+5+4 = 15
(A)
Probability Three blue balls are chosen without being replaced.
P(B)*P(B)*P(B)
=6/15*5/14*4/13
=120/2730
=4/91
(B)
The probability of picking two blue balls and one green ball is given by
57
\
58
59
(C)
60
Probability that 1 blue, 1 green and 1 red balls are picked is given by
Answer = 24/91
61
Part 02
62
63
6. The different between Discrete and continues
64
Vensim and other system dynamics software can solve systems of lumped ordinary difference or dif ferential
65
equations on a technical level. Vensim is frequently referred to be a tool for continuous sim ulation. This implies
it works best in cases where the majority of the variables change constantly rat her than in increments. This is in
contrast to discrete event simulation, which tracks individual entiti es and adds up the outcomes to report
behavior. While discrete variables are conceivable in Vensim, they are typically only useful if the number of
discrete variables is modest in comparison to the rest of the model.
Discrete
Complete
Non- overlapping
The term "discrete variable" refers to a
variable with a limited number of isolated
values.
It can only accept values that are different
or distinct from one another.
Continuous
Incomplete
Overlapping
Data that is collected in a continuous
series is referred to as continuous data.
It may take any value within a certain
range.
7.
a) Finding distribution of M.
1
2
3
4
5
6
1
0
-1
-2
-3
-4
-5
2
1
0
-1
-2
-3
-4
3
2
1
0
-1
-2
-3
4
3
2
1
0
-1
-2
5
4
3
2
1
0
-1
6
5
4
3
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
b) Identifying Expected value of M
66
E (M) -5/36 -8/36 -9/36 -8/36 -5/36 + 0 + 5/36 -8/36 +9/36 +8/36 +5/36
67
E (M) = 0
68
c) Finding Var (M).
69
1/36(-5/36)2 = 1/36 * 25/1296 * 2
1/36(-5/36)2 = 2/36 * 64/1296 * 2
1/36(-5/36)2 = 3/36 * 81/1296 * 2
1/36(-5/36)2 = 4/36 * 64/1296 * 2
1/36(-5/36)2 = 5/36 * 25/1296 * 2
70
71
72
(C)
73
74
75
76
8.
(A), (B), (C)
77
78
79
9.
80
(A), (B)
81
82
Part 03
83
10.
84
(A), (B), (C)
85
86
87
11.
a.
X = 2 n = 3 p = ¼ q =1/4
P (x=2) = nc2 p x Qn-x
= 3 c2 (3/4)2 (1/4)3-2
= (3 * 2 / 2 * 1) * (9/16) * ¼
= 27/64
= 0.421
b.
x=0 x=1 x=2 x=3
c.
P (x – 0) = nCx px Qn-x
= 3C0 * (3/4)0 * (1/4)3-0
= 1/64
= 0.015
P (x = 1) = nCx px Qn-x
= 3C1 * (3/4)1 * (1/4)3-1
88
= 3 * ¾ * 1/16
= 9/64
= 0.1406.
89
P (x = 2) = nCx px Qn-x
90
= 3C2 (3/4)2 (1/4)3-2
91
= (3 * 2/2 * 1) * (9/16) * ¼
= 27/64
= 0.421.
P (x = 3) = nCx px Qn-x
= 3C3 (3/4)3 (1/4)3-3
= 27/64
= 0.421.
92
And
12
(A), (B), (C)
93
x
e.
P(x)
mean = n * p = 3 * ¼
0
0.015
1
0.1406
2
0.421
3
0.421
= 0.75
94
f.
95
Variance = n p (1-p)
= (3*3/4)*(1- 3/4)
=9/16
=0.562
e)
Mean =np
Mean = 3*0.75
Mean = 2.25
96
97
98
13.
99
(A), (B), (C)
100
101
102
15.
103
(A), (B), (C), (D)
104
105
106
107
108
Activity 03
Part 01
1. 2. 3. 4.
109
110
111
Part 02 5. 6.
112
113
Activity 04
Part 01
114
1. 2.
115
116
117
Part 02 3. 4.
118
119
120
Part 03
5.
121
122
6
123
Part 04
124
125
•
•
where at; x = 0 (maximum) f / (x) = - 4
x =π/2 (minimum) f / (x) = 4
126
•
x = π (maximum) f / (x) = - 4
127
•
x= 3π/2 (minimum) f / (x) = 4
128
129
130
Conclusion
131
132
Math is essential for our education in order to develop our knowledge and abilities in computers. Math is a
133
great tool for describing and solving difficulties. Calculators and computers are invaluable tools for doing
arithmetic operations. The equipment we use for math processes are significantly more competent, quicker,
and accurate than those we used previously.
134