P1.1�A1. Solve the following linear and quadratic equations:

(i)�� 2(3 � 5x) = 15 | (ii)����� x2 + x -20 = 0 |

P1.1�A2. Solve the following sets of simultaneous equations by (a) algebraic method (b) graphical method.

(i)�������� y=2x; y=-2x+1 (ii)������� y = 5x + 1; y =?5x + 1 | (iii)������ ?6y = 3x ? 4; 2y = x + 5 |

P1.2�A3. Find the volume of the following shapes to three significant figures by showing your work step by step:

(i) a cube with a length of one side 27 metres��������� (ii) a sphere with radius 20 inches

P1.2�A4.Using Pythagoras� theorem, proof that triangle ?ABC (9:12:15) is a right-angled triangle

(i) Calculate sine, cosine and tangent for each angles of ?ABC. (ii) Using an appropriate Excel function, demonstrate on a spreadsheet that ?ABC is a right-angled triangle.

P1.3�A5. Two robots, Alice and Bob are pulling a box as shown on the figure

i. Calculate vector c = a+b.

ii. Calculate magnitude of vector c.

iii. Write a Pseudocode for calculating magnitude of vector c.

P2.1�B1. A certain British company has three departments. Following sets are showing departments, surnames and annual salaries of employees of this company:

A={ Martin, Marriott, Boast, Preston, Kans}

B= {24k, 25k, 26k, 27k, 30k}

C= {Production, Sales, Finance}

Mr Martin and Mrs Marriott are working at production department, Mrs Boast and Mrs Preston working at sales department and Mr Kans works at Finance department.

a. Find the Cartesian product of set A and set B. (R=A�B)

b. Find the Natural join of R and C. ( RC)

c. Fill in the below table by using provided information:

(Note: explain your work step by step)

Employee name | Salary | Department |

� | � | � |

� | � | � |

P2.1�B2. A small ICT firm, has three branches in

1. Redbridge,

2. Enfield and

3. Barnet.

Five technicians with following details are working at this company;

Ali (Location: Barnet, age: 25, salary: �21,000), Steve (Location: Redbridge, age: 45, salary: 23,000), Mike (Location: Enfield, age: 50, salary: 19,000), Linda (Location: Barnet, age: 55 , salary: 24,000 ), Carol (Location:� Redbridge, age: 43, salary: 27,000),

- Draw required number of tables and fit in the above information there.
- List individuals satisfying the conditions below:

1. (Age<46) AND (Salary> � 23,000)

2. (Age> 26 ) OR (Salary < �24,000)

3. (Age< 53) AND (Salary>29) OR (Location=1)

4. (Age> 25) XOR (Salary>30) OR (Location=2) Explain how you took the above steps.

P2.2�B3. Create a magic square by identifying values of p, q, r, s, t, u, x, y, z in matrix A.

A =�

�

[Show your work step by step]

B4. Show that if

P = 1��� 2

� � � 3��� 4

Q = ?2 �1

� � � 1.5��?0.5

Then P is the inverse of Q.

P3.1�C1. Suppose that two sets are A and B, defined by

A���� =� { g, e, r, m, a, n, i }

B���� =� { p, o,� l, a,� n, d }

Identify the following statements as true or false:

(i) | a ? A, |

(ii) | b ? B, |

(iii) | d ? B, |

(iv) | u ? A, |

(v) | a ? A?B, |

(vi) | |A| = |B|, |

(vii) | { i, r, a, n} ? A, |

(viii) | |A?B| = 8, |

P3.1�C2. Suppose we have a universal set

{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27} and consider two sets P and O defined as follows:

P���� =� �all multiples of 3�

O��� =� �the first ten even numbers�

Represent all of the elements in a Venn diagram and identify the elements in P?O, P?O and P?O.

P3.1�C3. For all of the following sets defined in set?theoretic notation, list out all of the elements:

S1� =� {*x*:�*x*= 2*n*, where 1 ?��*n*?� 6}

S2 =� {*x*:�*x*= 3*n**2*, where 1 ?��*n*?� 5}

S3 =� {*y*:�*y*= 5*n**3*, where 1 ?��*n*?� 4}

S4�� =� {*x*:�*x*= ?*n*, where 3 <��*n�*< 5}

P3.2�C4. For the circuit shown below, construct a truth table for each intermediate function. Hence, find the output function X.

P3.2�C5. Suppose that a salesman has 4 differently-located customers.

- Find the number of different ways that the salesman can leave home, visit two different customers and then return home.
- Write a pseudocode for calculating the answer for the previous section.

P4.1�D1. A research in 157 households found that the number of children per household is

- Calculate the Mean of frequency distribution for the above case.
- What is the Mode value of number of children�s per household?

P4.1�D2. A company has ten sales territories with approximately the same number of sales people working in each territory.� Last month the sales orders achieved were as follows:

Area | A | B | C | D | E | F | G | H | I | J |

Sales | 150 | 130 | 140 | 150 | 140 | 300 | 110 | 120 | 140 | 120 |

For these sales calculate the following:

- Arithmetic mean
- Mode
- Median
- Lower quartile
- Upper quartile
- Quartile deviation
- Standard deviation
- Mean deviation

Show all the steps you took to complete your answer.

P4.1�D3. Identify a topic in one of the following areas and conduct a research on its application in software development.

- Boolean algebra
- Propositional logic
- Relations and functions
- Probability, sets, reliability

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