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f(x) = cot^-1(\(\frac{x^2}{x^2+1}\))

\(\because\) x² + 1 > x²

⇒ \(\frac{x^2}{x^2+1}<1\)

Also \(\frac{x^2}{x^2+1}\geq0\) (\(\because\) x² \(\geq\) 0 & x² + 1 > 0)

⇒ 0 \(\leq\) \(\frac{x^2}{x^2+1}<1\)

\(\because\) cot^-1x  is a decreasing function.

\(\therefore\) cot^-1 1 < cot^-1(\(\frac{x^2}{x^2+1}\)) \(\leq\) cot^-10

⇒ \(\frac{\pi}4\) < f(x) \(\leq\) \(\frac{\pi}2\) 

\(\therefore\) Range of f(x) is ( \(\frac{\pi}4\), \(\frac{\pi}2\)].

 \(\therefore\) a = π/4 & b = π/2

\(\therefore\) b/a + 2 = \(\frac{\pi/2}{\pi/4}\) + 2 = 4/2 + 2 = 4

f(x) = \( \begin{cases}\frac x2 & {,x = 2n; n\in N }\\0 & {x\, is \,odd}\end{cases}\)

= \( \begin{cases}n, & {n\in N, x \, is \, even \, s.t. \, x = 2n }\\0, & {x\, is \,odd}\end{cases}\)

\(\therefore\) f (x) is a whole number

2sin² \(\cfrac{\pi}6\) + cosec² \(\cfrac{7\pi}6\) cos² \(\cfrac{\pi}3\)

= 2(sin \(\cfrac{\pi}6\))² + (cosec (π + \(\cfrac{\pi}6\)))² (cos\(\cfrac{\pi}3\))²

= 2 (\(\cfrac{1}2\))² + (-cosec\(\cfrac{\pi}6\))² (\(\cfrac{1}2\))² (\(\because\) sin \(\cfrac{\pi}6\) = cos \(\cfrac{\pi}3\) = \(\cfrac{1}2\))

= \(\cfrac{2}4\) + (-2)² x \(\cfrac{1}4\) (\(\because\) cosec \(\cfrac{\pi}6\) = \(\cfrac1{sin{\cfrac{\pi}6}}\) = \(\cfrac1{{\cfrac{1}2}}\) = 2)

= \(\cfrac{1}2\) + 4 x \(\cfrac{1}4\) = \(\cfrac{1}2\) + 1

= \(\cfrac{3}2\) = \(\cfrac{3}{3-1}\)

∴ m = 3 (by comparing)

f(x) = \(\begin{cases}-55x&,x<-5\\ 2x^3-3x^2-120x&,-5\leq x\leq4\\2x^3&,x>4\end{cases}\)

f'(x) = \(\begin{cases}-55&x<-5\\6x^2-6x-120&,-5<4<4\\6x^2-6x-36&,x>4\end{cases}\)

For f'(x) \(\geq\) 0

Case I :- -5 < x < 4

Then f'(x) \(\geq0\) implies

6x² - 6x - 120 \(\geq\) 0

⇒ x² - x - 20 \(\geq\) 0

⇒ (x - 5) (x + 4) \(\geq\) 0

⇒ x \(\leq\) - 4 or x \(\geq\) 5

But -5 <x < 4

\(\therefore\) -5 < x < 4

Hence, function f(x) increases in domain x \(\in\)(-5, -4)

Case II: x > 4

Then f'(x) \(\geq\) 0 implies

6x² - 6x - 36 \(\geq\) 0

⇒ x² - x - 6 \(\geq\) 0

⇒ (x - 3) (x + 2) \(\geq\) 0

⇒ x \(\leq\) - 2 or x \(\geq\) 3

But x > 4

Hence, f(x) increases in (4, \(\infty\))

From both cases, we can say that f(x) is increases in (-5, -4) U (4, \(\infty\)).

A = {x|x \(\in\) (-5, -4) U (4, \(\infty\))}.

\(\bar x = \frac{\sum x_i}{5}=\frac{65}5=13\)

\(\sum x_i^2\) = 9 + 49 + 144 + a² + (43 - a)²

 = 2a² - 86a + 2051

\(\therefore\) variance = \(\frac1n\sum x_i^2-(\bar x)^2\)

 = \(\frac15(2a^2-86a+2051)-169\) 

 = \(\frac15(2a^2-86a+2051-845)\) 

 = \(\frac15(2a^2-86a+1206)\) 

 = \(\frac25(a^2-43a+603)\)

Variance is natural number if a² - 43a + 603 is a multiple of 5.

Correct option  (4)  (8, 6)

Explanation:

(y – 0)² = 4.2(x – 2)

y² = 8(x – 2)

Now check all options (8, 6) does not satisfy

Correct option (1)  (3/4,1/2)

Explanation:

3p + 2q + 4r = 0

px + qy + r = 0  .....(i)

3/4p +q/2 + r = 0  .....(ii)

comparing (i) and (ii) x = 3/4, y = 1/2

Given statement is (P implies q ) and r

i.e., (p ⇒ q) ∧ r

or (~ p ∨ q) ∧ r. (\(\because\) p ⇒ q ≡ ~ p ∧ q)

Now, the negation of (p ⇒ q) ∧ r is

~((~p ∨ q) ∧ r)

or ~ (~p ∨ q) ∨~ r (\(\because\) ~ (p ∧ q) ≡ ~ p ∨ ~ q)

or (~ (~ p) ∧ ~ q) ∨ (~ r) (\(\because\) ~(p ∨ q) ≡ ~p ∧ ~ q)

or (p ∧ ~p) ∨ ~r (\(\because\) ~ (~p) ≡ p)

or p ∧ ~q ∨ ~r

or p ∧ ~(q ∧ r)

Given:

The given Expression is (5¹² + 5¹³ + 5¹⁴)

Concept Used:

To check the divisibility of number the number should be multiple of given number

Calculation:

The given expression is (5¹² + 5¹³ + 5¹⁴) from this expression we take 5¹² common

∴ 5¹² × (5 + 5¹ + 5²) = 5¹² × 31

So, the given expression is multiple of 31

∴ (5¹² + 5¹³ + 5¹⁴) is divisible by 31

Hence, option (1) is correct

Given:

The number is 513 + 512 + 511 + 5¹⁰ 

Concept used:

If the number can be expressed as pa.qb.r^c.s^d, where p, q, r, and s are prime numbers and a, b, c and d are whole numbers,

Then total number of factors of the number = (a + 1) × (b + 1) × (c + 1) × (d + 1)

Calculation:

513 + 512 + 511 + 510 

⇒ 5¹⁰ × (5³ + 5² + 5 + 1)

⇒ 510 × (125 + 25 + 5 + 1)

⇒ 510 × 156

⇒ 510 × 2 × 2 × 3 × 13

⇒ 510 × 2² × 3¹ × 13¹

Total number of factors = (10 + 1) × (2 + 1) × (1 + 1) × (1 + 1)

⇒ Total number of factors = 11 × 3 × 2 × 2

∴ Total number of factors are 132

CONCEPT : 

• The n^th term an of the AP with first term a and common difference d is given by a_n = a + (n – 1) d.
• a_n is also called the general term of the AP. If there are m terms in the AP, then a_m represents the last term which is sometimes also denoted by l.

CALCULATION:

The number of students in the 1st, 2nd, 3rd, . . ., rows are : 23, 21, 19, . . ., 5.It forms an AP as common difference is same i.e. -2.

Let the number of rows in the assembly be n.

Here a = 23, d = – 2, a_n = 5

As, a_n = a + (n – 1) d

⇒ 5 = 23 + (n – 1) × (– 2)

⇒ – 18 = (n – 1)(– 2)

On solving we get n = 10 So, there were 10 rows in the school assembly.

Hence, option (4) is the correct answer.

CONCEPT:

The nth term an of the AP with first term a and common difference d is given by an = a + (n – 1) d.

CALCULATION :

Given A.P = 4, 9, 14, . . .

Here, first term a = 4 and common difference d = 5.

Let consider 289 be the n^th term of the given A.P. Then we have,

⇒ a_n = 289

⇒  a  + (n-1) d = 289

⇒  4 + (n-1) x 5 = 289

⇒  (n-1) x 5 = 289 – 4

On, Solving we get,

⇒  n - 1 = 57

⇒ n = 57 + 1

⇒  n = 58

Hence, option (1) is the correct answer.

CONCEPT : 

Three numbers x, y, and z are in arithmetic progression if and only if : 2y = x + z.

CALCULATION:

Given: Three numbers 3, q and 5 are in arithmetic progression

As we know that three numbers x, y, and z are in arithmetic progression if and only if : 2y = x + z.

⇒ 2q = 3 + 5

⇒ 2q = 8

⇒ q = 4

Hence, option (1) is the correct answer.

A number of steps are taken by the government for increasing the efficiency of the services and to provide service to people in a time bound manner. They are known as administrative reforms. The intention is to make administration people friendly and efficient.

E-governance: 

E-governance is the use of electronic technology in administration. This helped to obtain government services easily in a speedy manner. Its benefits are:

• Need not wait in government offices for services. 
• Can receive service with the help of information technology. 
• Government services offered speedily and with less expense. 
• Efficiency of the offices and quality of the service get enhanced. 
• People can receive the services of Egovemance at home with the spread of E- literacy.

Right to Information Act: 

In India, the Right to Information Act was passed in 2005. This ensures the right of all , citizens of India to receive information. The main 

objectives of this Act are to prevent corruption, create responsibility among employees and make the functioning of the government transparent.

Information Commission: 

To perform the functions under the Right to Information Act, Information Commissions are constituted at national and state levels. There will be a Chief Information Commissioner and not more than ten members in the Information Commission.

Right to Service: 

Right to Service Act is a law which ensures services to the people. This law determines the time limit for every service given by a government office. As per this Act, an officer is appointed in every government office to give guidance and proper help to the applicants.

Lokpaland Lokayukta : 

These are institutions constituted to prevent corruption at administrative, bureaucratic and political levels. The institution constituted at the national level to prevent corruption is Lokpal. Lokayukta is the institution constituted at the state level to hear the corruption cases.

Central Vigilance Commission : 

The Central Vigilance Commission is the institution constituted at the national level to prevent corruption. It is formed to prevent corruption in the central government offices. Based on this model, State Vigilance Commissions have been constituted at state level.

Ombudsman : 

Ombudsman is constituted to file complaints against corruption, nepotism, financial misappropriation or negligence of duties of elected representatives and bureaucrats.

Public administration is related to governmental administration. It is the effective way of utilizing men and materials for implementing the existing law policies and developmental projects. Primary Health Centers, Police station Krishi Bhavan, Village office, Corporation office,Election Commission, and other government organisations are all the parts of public administration.

The responsibilities of these institutions must be strictly executed for the public welfare.There must be an accurate method of public administration for all the sectors of people in our country. The following is a list of a few public administration.

Centers and their duties are given : 

1. Vanitha Commission : 

Ensures the rights and welfare of women.

2. Election Commission : 

Relates with the election procedures. 

3. Human Rights Commission : 

Protects the rights of men. 

4. Village office : 

Ensures the primary needs of people.

Objectives of Right to Information Act, 2005

• To prevent corruption 
• Create responsibility 
• Make the functioning of the government transparent. 
• Information Commissions are constituted at the National and State levels to perform the functions under the Right to Information Act. There will be a Chief Information Commissioner and more than ten members in the Information Commission.

If the authorities do not give, or refused to give the information or gave wrong and unsatisfactory replies, one can approach the Information Commission and can file an appeal. If the Commission is convinced, a fine of ₹ 250 per day can be imposed on the employee concerned till the information is given.

CONCEPT:

The nth term an of the AP with first term a and common difference d is given by an = a + (n – 1) d.

CALCULATION :

Let’s ‘a’ be the first term and ‘d’ be the common difference.

Given that 7^th term of an Arithmetic Progression is 19, i.e., 7^th term = 19

⇒ a + (7 - 1) d = 19

⇒ a + 6d = 19 . . . . . . . . . . . . . (i)

and 13th term of an Arithmetic Progression is 28

⇒ a + (13 - 1)d = 28

⇒ a + 12d = 28 . . . . . . . . . . . .  (ii)

Now, subtract the equation (i) from (ii) we get,

6d = 9

\(\rm\Rightarrow d=\rm\frac{3}{2} \)

Hence, option (3) is the correct answer.

The employees who work under public administrative system and administer the country are together known as bureaucracy. All the services of the government reach people through them. Bureaucracy prepares plan for the maximum and scientific utilisation of human and material resources. The following are the features of bureaucracy.

• Hierarchical organisation: 

Bureaucracy is organised in such a way that there is one employee at the top and the number increases when it reaches the lower levels. This is known as hierarchical organisation. 

• Permanence: 

Persons appointed will continue in service till the age of retirement. 

• Appointment on the basis of qualification: 

Employees are recruited and appointed on the basis of educational qualification. 

• Political neutrality: 

Bureaucrats are liable to implement the policies of whichever party comes to power. Party interests should not reflect in their work. They should act neutrally.

• Professionalism: 

Every government employee must be skilled in their work.

Certain methods are adopted in the selection of employees to the public administrative system in India. The first step is the notification through Public Service Commission of India.Through this, the vacancies in various sections of governments are made known to the public. Later, through the exam conducted on the date notified or on the basis of interviews, candidates are selected and appointed.

Civil Service began during the British rule. It came to be the Civil Service of India after Independence. All the employees who work under the central and state governments and the employees under public sector undertakings are a part of India’s Civil Service. It is divided into All India Service, Central Service and State Service. The selection procedures of employees to these are given below.

All India Service :

• Recruits at national level.
• Appoints in Central Service or State Service. 

eg : IAS, IPS.

Central Service :

• Recruits at national level.
• Appoints in central government department only.

eg : Indian Foreign Service, Indian Railway Service.

State Service :

• Recruits at State level 
• Appoints in state government department

eg : Sales Tax officer.

Candidates to All India Services and Central Services are recruited by the Union Public Service Commission (UPSC). The chairman and members of this commission are selected by the President of India. The UPSC has elaborate mechanisms for the selection based on qualifications.

In the state level, candidates are recruited by the Public Service Commission (PSC) of the state. The governor appoints the Chairman and the member of the State Public Service Commission. UPSC and state PSCs are on the basis of constitutional provisions and can be called constitutional institutions.

Hierarchical organisation : 

Bureaucracy is organised in such a way that there is one employee at the top and the number increases, when it reaches the lower levels. This is known as hierarchical organisation.

Permanence : 

Persons appointed will continue in service till the age of retirement. Appointment based on qualification: Employees are recruited and appointed on the basis of educational qualification.

Political Neutrality :

Bureaucrats are liable to implement the policies of which ever party comes to power. Party interests should not reflect in their work.

Professionalism :

Every government employee must be skilled in their work.

Information Commission : 

To perform the functions under the right to information act, In formation Commissions are constituted at the national and state levels. There will be a chief Information commission and not more than ten members in the Information Com mission.

Central Vigilance Commission : 

The Central Vigilance Commission is the institution constituted at the national level to prevent corruption. It came into effect in 1964. It is formed to prevent corruption in the central government offices.The Central Vigilance Commissioner is the head of the Central Vigilance Commission.

Lokpal and Lokayuktha : 

Lokpal and lokayuktha are institutions constituted to prevent corruption at administrative, bureaucratic and political levels, the institution constituted at the national level to prevent corruption is lokpal.

Lokpal has the power to register cases on issues of corruption against employees and public workers and can suggest necessary actions. Lokayukta is the institution constituted at the state level to hear the corruption cases.

Ombudsman : 

Elected representatives and bureaucrats are part of public administration.

Complaints can be filed against their corruption,nepotism or financial misappropriation or negligence of duties. Ombudsman is constituted for this purpose.

Kerala Public Service Commission notifies for the recruitment of employees to the government service. Then candidates are selected on the basis of competitive examinations and interviews and are appointed in different government sectors. All those appointed in this way become a part of the civil service of India.

The aim of it is to bring welfare programmers speedily to all. Now there are specific services at the central and state levels. All the employees who work under public sector undertakings are the part of India’s civil service. There are All India services, central services and state services.

A number of steps are taken by government for increasing the efficiency of the services and to provide service to people in a time bound manner. They are known as administrative reforms. The intention is to make administration, people friendly and efficient.

Government has taken a number of steps to increase the efficiency of services and to provide service to people within a time limit. This is called as administrative reforms. It makes the administration friendly and effective. Administrative reform commissions are made at national and state levels. Here are some steps taken for the reformation in our country.

E-Governance :

The single window system for Higher Secondary education, online applications for several scholarships etc are example of E-Governance. The use of electronic technology has helped to obtain the services of government effectively in a faster way.

Right to Information :

Every citizen has the right to collect the information from any government office about its working. This is under the Right to Information Act in 2005. The efforts of Mazdoor Kiran Shakthi Samghathan of Rajasthan led to this Act. The interventions of several organisations and social activities helped in passing this Act in 2005. The Right to Information ensures the right of all citizens of India to receive information.

Information Commission :

Files documents, circulars, memos, advice or orders, agreements statistics, reports, log books, press notes, samples, models, information in the form of electronic data,e-mail etc. related to government offices belong to public departments. Information commission helps to know everything.To perform the functions under the Right to information Act, Information Commissions are constituted at National and state levels.

A chief Information Commissioner and members not more than ten are in the Information Commission. If the information given is wrong and unsatisfactory we have the right to approach the Information Commission. If the commission is convinced a fine of Rs. 250 can be imposed on the employee.

Right to Service :

This ensures service to the people. This law determines the time limit for every service given by a government office. If the deserved service is not given within the time limit the responsible employee should pay fine. As per the Right to Service, an officer is appointed to give proper guidance and help to the applicants.

Lokpal and Lokayuktha : 

These are the institutions to prevent corruption at administrative, bureaucratic and political levels. Lokpal has the power to register cases on issues of corruption against public workers and then suggest necessary actions. Lokayuktha hears the corruption cases at the state level. Both follow judicial procedures.

Central Vigilance Commission :

This is constituted at the national level to prevent corruption. It came in to force in 1964. It was formed to prevent corruption at central government offices.

Ombudsman :

Complaints of corruption against the elected representatives and bureaucrats who are a part of public administration are filed to ombudsman. A retired Judge of the High Court is appointed as Ombudsman. People can directly approach the Ombudsman with complaints.

He has the power to summon anyone on receiving the complaints, can order inquiry and recommend actions. Ombudsman began in the banking sector to hear the complaints of clients and rectify them.These actions recommended for the welfare of people as a part of administration maintains the public administration more trans parent.

1. Lokpal and Lokayuktha 

2. Central vigilance commission 

3. Ombudsman.

All the employees who work under the central and the state governments and the employees under the public sector under takings are part of India’s civil service. India’s civil service classification as All India Service, Central Service ans State Service.

All India Service. 

Recruits at National level. 

Appoints in the central or state service. 

The UPSC has elaborate mechanisms for the recruitment of candidates based on qualification. e.g., Indian Administrative Service. 

Indian Foreign Service. 

Indian Police Service.

In 2005, RTI Act was passed by the parliament. To get information is the fundamental right of all citizens. The main objectives of this Act are to prevent corruption, create responsibility and make the functioning of the government transparent. The citizens will get copies of public documents if they apply for them.

• Ensure justice and equal rights to all sections of the society.
• Create corruption free administration and responsibility among the workers. 
• Constitution must be to save the victims and punish culprits.
• Government service must be reached at fixed time.
• Protect the rights of women and ensure their safety.
• Ensure the welfare of all. 

Based on the differences in the form of government, we can find differences in public administration also. During monarchy, the interests of the monarch were the basis of public administration. In democratic system, importance is given to the interests of the people. Democratic administration becomes more effective and efficient through public administration.

Interview with kozhikode District Collector. 

Sir, could you please share with me the inspiration behind your selection this position? 

What were the changes you have thought to bring in Kozhikode before becoming the District Collector? 

Sir, even in the presence of Lokpal, Lokayuktha, Central Vigilance Commission, Ombudsman being launched against corruption still corruption exists in Kerala. Don’t you long for a change ? How do you react as a district collector? 

Sir, hope you have noted that vegetables from Tamil Nadu were sold less here in the Onam season. Do you expect the same for Vishu? As a District Collector, what all steps would you take to improve the agriculture sector of Kerala?

Though Kerala has developed in the educational field, the number of suicidal attempts have also increased. A change in our educational system is unavoidable. Sir, what all changes could be made possible?

Sir, M. C. Noufal was arrested here in Kozhikode for raping a woman from Bangladesh. Though many were arrested they were refused saved. Let me ask you sir with due respect, don’t you long for a change in the system of punishment here?

• Controlled corruption,
• Increased the responsibility of bureaucrats,
• Functioning of government became transparent. 

From,

Madhavan 

Mahal Nivas 

Ollavanna

To,

Secretary 

Grama Panchayath 

Ollavanna

Based on RTI Act of 2005

Sir, 

Sub : Amount spent for the electrification of houses for schedule tribes in the year 2015- 16. 

Could you kindly give details about how many applicants there were and how many money was spent in previous years.

Expecting the reply within 15 days. 

Ollavanna — Madhava

25/08/19  —  sd/-

In 2005, RTI Act was passed by the parliament. To get information is the fundamental right of all citizens. RTI Act is helpful socially, for all the subjects, quickly in our society.

Example: 

Waiting for the results after University Exam if delayed, RTI commission could be approached :

• To get information about certain legal affairs. 
• Any problems concerned with government.

• Ensures welfare of people 
• Formulates government policies 
• Provides goods and services 
• Finds out solution for public grievances 

Public administration is the effective use of men and materials for the implementation of existing laws, governmental policies, programmes and developmental projects.

• Formulate government policies. 
• Ensure welfare of the people. 
• Find out solutions to public issues. 
• Provide goods and services.

The Right Information Act was passed in 2005.

Mazdoor Kisan Shakti Sangathan

• Formulate government policies.
• Ensure welfare of the people.
• Find out solutions to public issues. 
• Provides goods and services. 

UPSC and PSC are constituted on the basis of constitutional provisions.

1. State Public Service Commission – At the state level, candidates are recruited by the public service commission of the state. 

2. Union Public Service Commission Candidates to all India services and central services are recruited by the Indian Public service commission.

• Candidates to central services are recruited by the Union Public Service Commission and to state services by Public Service Commission. 
• UPSC and state PSCs are constituted on the basis of constitutional provisions. So they are called constitutional institutions