Assignment # 01 MTH403 (Fall 2022)

 

 

 

Maximum Marks: 10

Due Date: November 25, 2022

 

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Question # 1:                                                                                                     

Find real 𝑥 and 𝑦 if (𝑥 – 𝑖𝑦) (3 + 5𝑖) is the conjugate of – 6 – 24𝑖.

SOLUTION:                                                                                                     

According to condition

(𝑥 − 𝑖𝑦)(3 + 5𝑖) = ̅−̅̅6̅̅−̅̅̅2̅̅4̅̅𝑖

 

3𝑥 + 5𝑥𝑖 − 3𝑦𝑖 − 5𝑦𝑖² = −6 + 24𝑖

 

3𝑥 + 5𝑥𝑖 − 3𝑦𝑖 − 5𝑦(−1) = −6 + 24𝑖                                     ∴ 𝑖² = −1

 

3𝑥 + 5𝑥𝑖 − 3𝑦𝑖 + 5𝑦 = −6 + 24𝑖 (3𝑥 + 5𝑦) + 𝑖(5𝑥 − 3𝑦) = −6 + 24𝑖 By comparing

3𝑥 + 5𝑦 = −6 … … … … (1)

 

5𝑥 − 3𝑦 = 24 … … … . . (2)

 

Multiply equation (1) by 3 and equation (2) by 5 then add

 

(9𝑥 + 15𝑦) + (25𝑥 − 15𝑦) = −18 + 110

 

9𝑥 + 15𝑦 + 25𝑥 − 15𝑦 = 92

 

34𝑥 = 92

 

𝑥 = 92 = 3

34

 

𝑥 = 3

 

Putting x = 3 in equation (1)

 

3(3) + 5𝑦 = −6

 

9 + 5𝑦 = −6

 

5𝑦 = −6 − 9

 

5𝑦 = −15

 

𝑦 = −15 = −3

5

 

𝑦 = −3

 

 Question # 2:                                                                                                    

Simplify (√3 + 3𝑖)³¹ by using De-Moiver’s theorem.

SOLUTION:                                                                                                     

Here 𝑧 = (√3 + 3)     𝑎𝑛𝑑 𝑛 = 31

We know that

𝑧^𝑛 = 𝑟^𝑛(𝑐𝑜𝑠𝑛𝜃 + 𝑖𝑠𝑖𝑛𝑛𝜃)

So, first we find r and 𝜃

 

𝑟 = √(√3)² + (3)² = √12 = 2√3

 

𝛼 = 𝑡𝑎𝑛⁻¹ ( 3 ) = 𝑡𝑎𝑛⁻¹(√3) = 𝜋

 

𝜃 = 𝛼 = 𝜋

3

√3                                                3

 

(𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑧 𝑙𝑖𝑒 𝑖𝑛 1𝑠𝑡 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡)

So, the above equation become

(√3 + 3)³¹ = (2√3 )31(cos (31) 𝜋 + 𝑖𝑠𝑖𝑛(31) 𝜋)

3                                3

(√3 + 3)³¹ = (2√3 )31(cos 31𝜋 + 𝑖𝑠𝑖𝑛 31𝜋)

3                         3

 

(√3 + 3)³¹ = (2√3 )31 (cos(10π + 𝜋) + 𝑖𝑠𝑖𝑛(10𝜋 + 𝜋))

3                                          3

(√3 + 3)³¹ = (2√3 )31 (𝑐𝑜𝑠 𝜋 + 𝑖𝑠𝑖𝑛 𝜋)

3                    3

 

(√3 + 3)³¹ = (2√3 )31 (1 + 𝑖 √3)

2            2

 

 

 

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Introduction to Sociology (SOC101) Assignment No. 01 Solution

 

                SEMESTER FALL 2022

Introduction to Sociology (SOC101)

Assignment No. 01 Solution

Due Date: 30-11-2022                                                             Assignment marks: 20 “Origin of Sociology”

 

Assignment Objective

Upon completion of this assignment, the students will be able to understand the contribution of sociologists in the development of the society.

Background

Sociology is the scientific study of human social life, groups and societies. Sociologists study the different aspects of society and the role of human beings in society. The scope of Sociology begins with the connection between human beings and society and expands further towards the social institutions that impact our lives and the factors that dominate the proper functioning of these institutions. It sets out to explain how human beings have constructed these institutions, how they perceive them and the role they play in our lives. Thus, Sociologists contribute and work as professionals in different departments and segments of the society.

Question:

 

Following is given a case study explaining the prospective role of sociologists, read it carefully and identify any TEN CAREERS for sociology students in future in the given Format.

 

Solution:

                  

1.  Social Mobilizer                               

 

2.  Rural Development Officer             

 

3.  Population Planning Officer            

 

4.  Rural Health Planner                        

 

5.  Environmental Analyst                    

 

6.  Community development officer     

 

7.  Parole officer                                   

 

8.  Rehabilitation counselor                  

 

9.  Probation officer                              

 

10.  Social Welfare officers                    

 

Marking scheme:

2 marks for Identification of Career: 1*10 (20)

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