Mathematical Foundations for Computing: Semester 1

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MATHS FOR COMPUTING

Contents
Introduction...........................................................................................................................................3
LO1 Use applied number theory in practical computing scenarios.......................................................4
Part 1: Calculate the greatest common divisor and least common multiple of a given pair of
numbers............................................................................................................................................4
Part 2.................................................................................................................................................4
2.1 The first and tenth and last terms of an arithmetic progression are 9, 40.5, and 425.5
respectively. Find (a) the number of terms, (b) the sum of all the terms and (c) the 70th term.......4
2.2 On commencing employment a man is paid a salary of £7200 per annum and receives annual
increments of £350. Determine his salary in the 9th year and calculate the total he will have received
in the first 12 years................................................................................................................................7
2.3 A drilling machine is to have 6 speeds ranging from 50rev/min to 750rev/min. If the speeds form
a geometric progression determine their values, each correct to the nearest whole number.............9
LO2 Analyse events using probability theory and probability distributions.........................................11
Part 1...............................................................................................................................................11
1.1 The mean height of 500 people is 170 cm and the standard deviation is 9 cm. Assuming the
heights are normally distributed, deduce the number of people likely to have heights between 150
cm and 195 cm................................................................................................................................11
1.2 Determine the probabilities of selecting at random (a) a man and (b) a woman from a crowd
containing 20 man and 33 women..................................................................................................12
Part 2...............................................................................................................................................13
2.1 Identify the expectation of obtaining a 4 upwards with 3 throws of a fair dice........................13
2.2 Calculate the probabilities of having (a) at least 1 girl (b) at least 1 girl and boy in a family of 4
children, assuming equal probability of male and female birth. In addition, you are required to
produce a brief evaluation report on probability theory by giving an example involving hashing
and load balancing...........................................................................................................................13
LO3 Determine solutions of graphical examples using geometry and vector methods.......................15
3.1 The coordinates of a point P on a straight line L1 is given by P (1,3) in the x-y plane. L1 is
perpendicular to a straight line L2:𝑥 − 2𝑦 + 2 = 0. Identify the following: (a) the equation of L1 (b)
Construct the scaling of simple shapes that are described by vector coordinates..........................15
3.2 Find the unit vector perpendicular to the plane 4𝑥+2𝑦+4𝑧=−7........................................17
Part 2...............................................................................................................................................18
3.1 In the following diagram, PR= u and PQ = 2v and M is the midpoint of RQ and N is the midpoint
of RM...............................................................................................................................................18
3.5 Two forces F1 and F2 with magnitudes 30 N and 40 N, respectively, act on an object at a point
P as shown in the figure below. Find the resultant forces acting at P.............................................20
LO4 Evaluate problems concerning differential and integral calculus.................................................21

Part 1...............................................................................................................................................21
1.1 The length l metres of a certain metal rod at temperature θ0 C is given by..............................21
l=1+0.00005θ + 0.0000004θ2. Determine the rate of change of length, in mm/0C, when the
temperature is (a) 1000 C and 4000 C.............................................................................................21
1.2 Supplies are dropped from the helicopter and the distance fallen at time t seconds is given by
.........................................................................................................................................................22
Part 2...............................................................................................................................................22
2.1 Evaluate ∫3𝑥√(2𝑥2+1)𝑑𝑥20 taking positive values of square roots only..........................22
2.2 Show that the function 𝑓(𝑥,𝑦)=𝑥3−3𝑥2−4𝑦2+2 has one saddle point and one
maximum point. Determine the maximum value............................................................................23
Conclusion...........................................................................................................................................25
References...........................................................................................................................................26
List of figures
Figure 1: figure1..................................................................................................................................18
Figure 2: figure 2.................................................................................................................................19
Figure 3: figure 3.................................................................................................................................21

Introduction
The assignment is to understand the concept of mathematics by solving different problems. It
contains the theory of numbers in the scenarios of practical computing to gain knowledge as
by solving the question on number theory it includes the LCM & GCD, series & sequences it
includes the AP & GP. The second task contains the questions related to probability &
distribution of probability it includes trails of independents, distribution randomly. The third
task contains the question on geometry by utilizing the vector & graphical methods; it
includes geometry that provides the knowledge of planes & vectors includes finding the
resultant of force. The fourth task contain the question on integral calculus & differential, it
helps to gain the knowledge in differential calculus & integral calculus by finding the
maximum values & acceleration, etc.

LO1 Use applied number theory in practical computing scenarios
Part 1: Calculate the greatest common divisor and least common multiple of a
given pair of numbers.
Given the cluster of numbers are 2, 3, 5, 7 & 11.
For finding the GCD & LCM:
The above number given are all prime, therefor the GCD for the 2, 3, 5, 7 & 11
The final result is,
= 1
LCM = 2 *3*5 *7* 11
By solving, get
= 2310.
The greatest common divisor is equal to 1.
The least common multiple is equal to 2310 (Beachy, 2019).
Part 2
2.1 The first and tenth and last terms of an arithmetic progression are 9, 40.5, and 425.5
respectively. Find (a) the number of terms, (b) the sum of all the terms and (c) the 70th term
The first term of AP is 9…….. (Given)
The tenth term of AP is 40.5…….. (Given)
The last term of AP is 425.5…….. (Given)
To compute, the total terms in number by adding every term & the 70th term value.
The formula of AP:
The numerical value of the nth of the AP is ………………………………… (eq. 1)
A is 9 & the term 10th is 40.5

This state that
Put the value of n in eq.1
40.5 = [a + [10 - 1] * d]
Putting the value of a in eq.1
40.5 = [9 + [10 - 1] * d]
By solving it,
40.5 =9 + 9* d
Equating the equation,
40.5 – 9 = 9 * d
By solving further,
d = 31.5 / 9
The final result,
d = 3.5
The CD (common difference) of the AP series is 3.5
As, last term of series is 425.5
To calculate the value of n,
Last term is = 425.5
Now calculated the value of n, by putting value of d in eq.1
= 425.5 = 9 + (n-1)* 3.5
Take 9 on the other side to make it positive,
= 425.5 -9 = (n-1) * 3.5
Now, divide it with 3.5 get,
= (425.5 – 9)/ 3.5 = n-1

By further solving,
= 119 = n-1
The final value get,
n = 120.
The value of the n is 120.
The formula of the sum of AP is (Badr, E. and Sadek, 2019)
…………………………… (eq. 2)
Now, putting the value of n, d, & a in eq. 2
= [120/ 2 * [2 * 9 + (120-1)* 3.5] = sn
After dividing get,
= [60*[2*9+ (120-1)*3.5] = sn
By multiplying the values,
= [60* [18+ (119)*3.5] = sn
Now open the brackets to multiply,
= [60* [18+ (416.6)] = sn
By solving further,
= [60* 434.5] = sn
By solving this, get the value of sn
26070 is sn
Now, the 70th in the AP is:
Now, putting a & d values in eq.1
n is 70
=9 + (70 -1)*3.5

By subtracting get,
= 9+ 69 * 3.5
Now multiply the terms,
= 241.4 + 9
The final result is,
a70 is 250.5
The value of the 70th term in the AP is 250.5
2.2 On commencing employment a man is paid a salary of £7200 per
annum and receives annual increments of £350. Determine his salary in the
9th year and calculate the total he will have received in the first 12 years.
Man salary for starting of job is £ 7200…………………………… (Given)
The increment annually is £ 350…………………………… (Given)
Calculate the salary in year ninth.
First year salary is £ 7200, the second year salary is £ 7550 & the salary for the third year is
£7900.
According to the increment the salary is in AP form that has d (common difference) with 350
& a is the 7200.
For calculating the term 9th in AP by using AP formula ……….. (eq. 1)
Put the values in eq.1,
=7200+ (9-1) * 350 = a9
By solving further get,
=7200+ 2800 = a9
By calculating this gets a9 value,
=10,000

The 9th year salary is £ 10,000
Now, compute the total that man gained in starting 12 years
Formula used: …………………………………. (eq.2)
Now putting values of n, a & d in eq.2
= [12/2 (2*a+(12-1)*d] = S12
After putting all values in eq.2
= [6 (2*7200+(12-1)*350] = S12
By multiplying this, get
= [6 (14400+(12-1)*350] = S12
Subtracting d by 1,
= [6 (14400+(11)*350] = S12
By multiple get,
= [6 (14400+3850] = S12
By solving this, value gets for S12
= 109500
The total money gained is £ 109500.

2.3 A drilling machine is to have 6 speeds ranging from 50rev/min to
750rev/min. If the speeds form a geometric progression determine their
values, each correct to the nearest whole number.
The drill machine range is 50 to 750 rev/min………………….. (Given)
Speed is 6………………….. (Given)
The range of the speed is in GP
The term in the GP is ………………………………… (eq.1)
The value of a is 50 rev/min
Put the value of n in eq.1,
The value of n is 6 then ar6-1…………………………………… (eq.1)
Ar5 = 750
Put the value of a =50,
r2 = 750/ 50
by further solving get,
r = (15)1/5
By final result get,
= 1.7188
Now the value of r put in eq.1
First, compute the ar is 50 * 1.7188 is equal to 85.94
Then, ar2 is 50 * 1.7188 ^2 is 147.71
Then for ar3
= 50 * 1.7188 ^ 3 is 253.89
Then for ar4

= 50 * 1.7188 ^ 4 is 436.39
Then for ar5
=50 * 1.7188 ^ 5 is 750.06
Drilling machine speed varies 50, 86, 148, 254, 436 & 750 rev per min.

LO2 Analyse events using probability theory and probability distributions
Part 1
1.1 The mean height of 500 people is 170 cm and the standard deviation is 9 cm.
Assuming the heights are normally distributed, deduce the number of people
likely to have heights between 150 cm and 195 cm.
The 500 people mean height is 170 cm…………………………. (Given)
Standard deviation is 9 cm……………………………………………… (Given)
Compute the height of people that having a height range of 150 to 195 cm
Formula is
= (150- 170)/ 9 = z1
By solving this we get,
= -2.22 is z1
It lies on the negative part as it is left of the z=0
Now, z is 195
= (195- 170)/ 9 = z1
By solving this we get,
= 2.78 is z2
It lies on the positive part as it is the right side of the z=0
Calculate the total by
= z1+z2
Put the values of z1 & z2,
=0.4869 + 0.4973
By adding get,