All practice questions

Question 781Question →

The following criteria must be satisfied for the appointment of a Senior Disaster Management Officer in the State Disaster Management Authority (SDMA) as of July 1, 2026:

1. The candidate must hold a Master's degree in Disaster Management, Environmental Science, or Civil Engineering with at least 60% aggregate marks.
2. The candidate must be between 30 and 45 years of age as of July 1, 2026.
3. The candidate must possess at least 5 years of post-qualification field experience in crisis response management.
4. The candidate must have published at least 2 research papers in indexed journals or completed a certified international training module in flood control.

However, the following exception clauses apply:
(i) If a candidate satisfies all criteria except (1) above, but holds a Bachelor's degree in Civil Engineering with at least 70% aggregate marks and has more than 8 years of field experience, the case is to be referred to the Vice-Chairman of SDMA.
(ii) If a candidate satisfies all criteria except (4) above, but has served in the National Disaster Response Force (NDRF) for a minimum of 3 years, the case is to be referred to the Director-General of SDMA.

Candidate Profile:
Deepa Nair was born on October 12, 1992. She completed her Bachelor's degree in Civil Engineering with 74% aggregate marks and has accumulated 9 years of field experience in crisis response management. She has published 3 research papers in indexed journals.

Based on the criteria and information provided above, what administrative decision should be taken regarding Deepa Nair's application?

The case should be referred to the Vice-Chairman of SDMA.

The candidate should be selected directly for the post of Senior Disaster Management Officer.

The case should be referred to the Director-General of SDMA.

The candidate should be rejected because she lacks a Master's degree.

Show answer & explanation

Answer: The case should be referred to the Vice-Chairman of SDMA.

Answer

The case should be referred to the Vice-Chairman of SDMA.

The applicant satisfies criterion 2 (age 33 falls between 30 and 45), criterion 3 (9 years field experience exceeds the 5-year requirement), and criterion 4 (3 research papers exceeds the minimum requirement of 2). She fails primary criterion 1 because she lacks a Master's degree. However, she meets all conditions of exception clause (i): holding a B.Tech in Civil Engineering with 74% marks (which is

\ge 70\%

) and having 9 years of field experience (which is

> 8

years). Therefore, her case must be referred to the Vice-Chairman of SDMA.

Step-by-Step Solution

1

Evaluate primary criteria against candidate attributes

Age as of July 1, 2026 is 33 years (passes criterion 2: 30-45 years). Experience is 9 years (passes criterion 3: min 5 years). Research papers published is 3 (passes criterion 4: min 2 papers). Educational qualification is B.Tech with 74% (fails primary criterion 1 which requires a Master's degree).

Systematic verification identifies which primary conditions are satisfied and which single condition fails.

2

Cross-check applicable exception clauses for the failed criterion

Exception clause (i) addresses failure of criterion (1). It requires a B.Tech in Civil Engineering with

\ge 70\%

(candidate has 74%) and

> 8

years field experience (candidate has 9 years). Both conditions are met.

Candidates missing primary educational qualifications may qualify for administrative referral if specific exception thresholds are satisfied.

3

Determine the mandated administrative action

Under exception clause (i), the candidate's case must be referred to the Vice-Chairman of SDMA.

Following procedural protocol strictly dictates referral to the designated authority.

Key Concept

Criteria Verification with Exception Overrides

Question 782Question →

Determine the value of the missing term that logically continues the given numerical sequence:

6, 13, 28, 59, 122, ?

Show answer & explanation

Answer: 249

Answer

The missing term in the sequence is 249.

The sequence follows the pattern

T_n = T_{n-1} \times 2 + (n-1)

for

n \ge 2

. Thus, the 6th term is

122 \times 2 + 5 = 249

.

Step-by-Step Solution

1

Examine the transition between consecutive numbers in the series

From 6 to 13:

6 \times 2 + 1 = 13

From 13 to 28:

13 \times 2 + 2 = 28

Establishing the basic recursive function pattern

2

Verify the pattern across remaining terms

From 28 to 59:

28 \times 2 + 3 = 59

From 59 to 122:

59 \times 2 + 4 = 122

Confirming that the multiplier is constant (2) while the added constant increments by 1 at each step

3

Apply the rule to find the missing term

122 \times 2 + 5 = 244 + 5 = 249

Extending the pattern by adding 5 after doubling the previous term

Key Concept

Recursive series with constant multiplier and arithmetic additive offset

Estimated Time:1m 0s

Question 783Question →

A Sub-Divisional Magistrate (SDM) faces an escalating conflict in a commercial zone where shopkeepers demand a complete ban on street vendors due to severe foot-traffic congestion, while street vendor associations demand allocation of vending spots under municipal guidelines. Which of the following actions represents the most appropriate initial negotiation strategy for the officer to resolve this conflict?

Convene a joint consultation with representatives of both associations and municipal officials to identify mutually acceptable designated vending hours and nearby alternative hawking zones.

Issue an immediate administrative order invoking police force to clear all street vendors from the area without prior dialogue.

Concede entirely to the shopkeepers' demands by revoking all vendor licenses to avoid friction with local business owners.

Instruct municipal staff to verbally promise vending spots to the street vendors without holding formal talks or informing the shopkeepers.

Show answer & explanation

Answer: Convene a joint consultation with representatives of both associations and municipal officials to identify mutually acceptable designated vending hours and nearby alternative hawking zones.

Answer

Convene a joint consultation with representatives of both associations and municipal officials to identify mutually acceptable designated vending hours and nearby alternative hawking zones.

Initiating a joint consultation involving both vendor and shopkeeper associations alongside municipal authorities provides a balanced, participatory platform. It addresses shopkeepers' concerns about market access while upholding statutory street vending guidelines through practical compromises like designated hours and alternative zones.

Step-by-Step Solution

1

Bring both conflicting parties (shopkeepers and street vendors) alongside relevant municipal officers into a structured dialogue.

Creates a transparent, multi-stakeholder negotiation framework.

Public administration requires impartial mediation rather than unilateral decision-making.

2

Formulate compromise mechanisms such as staggered vending schedules or nearby designated spaces.

Protects the economic livelihoods of vendors while alleviating shopkeeper concerns regarding congestion.

Sustainable conflict resolution seeks win-win administrative solutions aligned with statutory guidelines.

Key Concept

Stakeholder Conflict Resolution and Negotiation Strategy

Estimated Time:1m 0s

Question 784Question →

Match each urban development or infrastructure initiative in List I with its corresponding direct primary cause or logical outcome in List II.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The correct pairings match each action/phenomenon to its direct primary result: Implementation of HOV lanes and congestion pricing matches with marked modal shift toward public transit; Prolonged drainage blockages match with widespread traffic paralyzation and emergency delays; Electrification of transit buses matches with direct reduction in tailpipe emissions; and Rollout of a unified multi-modal ticketing system matches with improved transit interchange efficiency and lower fare burdens.

Each statement in List I serves as a direct cause or initiating policy intervention for the corresponding physical, behavioral, or economic outcome listed in List II.

Step-by-Step Solution

1

Analyze the cause-effect mechanism for high-occupancy vehicle lanes and congestion pricing.

Imposing financial costs on peak private driving directly encourages public transportation adoption.

Economic disincentives for private driving produce a behavioral shift toward public transit.

2

Analyze the physical impact of blocked storm-water drainage during intense rainfall.

Excess surface runoff cannot drain, creating rapid inundation on arterial roadways.

Physical drainage failure directly leads to urban flooding, gridlock, and delayed emergency response.

3

Assess the environmental outcome of transitioning from diesel buses to electric fleet buses.

Internal combustion engines are eliminated from daily municipal routes.

Removing fossil-fuel burning vehicles directly cuts local air pollutants and acoustic noise.

4

Evaluate the effect of a unified multi-modal digital ticketing system.

Commuters use a single pass across trains, metros, and feeder buses.

Eliminating redundant single-operator ticketing minimizes transfer delay and lowers total multi-leg fare costs.

Key Concept

Cause and Effect Reasoning in Urban Infrastructure and Transport Policy

Question 785Question →

An administrative administrator is assigned to resolve a high-stakes standoff involving indigenous forest dwellers, the state forest department, and a private eco-tourism developer regarding land use rights within a newly notified wildlife buffer zone. Arrange the following conflict resolution and negotiation steps in the optimal sequence to achieve a sustainable, legally compliant settlement.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The optimal administrative order begins with enforcing a temporary moratorium to de-escalate conflict, followed by joint neutral fact-finding and interest mapping, facilitating interest-based negotiation workshops, and concluding with institutionalizing a joint governance committee for long-term monitoring.

The correct sequence follows standard administrative negotiation doctrine: (1) Immediate stabilization via a temporary moratorium to prevent unilateral action, (2) Neutral fact-finding and interest mapping to replace rhetoric with verifiable data, (3) Interest-based negotiation to craft a balanced compromise, and (4) Institutionalization of joint monitoring mechanisms to ensure enduring compliance.

Step-by-Step Solution

1

Immediate Conflict De-escalation & Stabilization

Enforced moratorium on contested activities prevents unilateral advantage and establishes operational trust.

Substantive negotiations cannot proceed under active hostility or ongoing disruption.

2

Objective Baseline Assessment & Interest Mapping

Unbiased data on statutory rights, community livelihoods, and environmental constraints is established.

Conflict resolution requires separating verifiable facts and core interests from emotional positioning.

3

Collaborative Interest-Based Negotiation

Co-created integrative framework balancing eco-tourism revenue, forest rights, and conservation policy.

Engaging stakeholders in solution design ensures buy-in and aligns disparate goals with public welfare.

4

Institutionalization & Governance Formalization

Permanent oversight body with institutional dispute-resolution channels.

Long-term policy success requires legal institutionalization and ongoing participatory governance.

Key Concept

Phased Administrative Stakeholder Negotiation & Dispute De-escalation Model

Question 786Question →

A municipal administrator is managing the development of a new public park. Local shopkeepers express concern regarding the temporary reduction of street parking during construction, while nearby residents strongly advocate for maximizing pedestrian space and urban greenery. Which of the following approaches represents the most effective stakeholder conflict resolution strategy for the administrator?

Facilitate a joint consultative meeting with representatives of both shopkeepers and residents to identify shared priorities and negotiate a phased construction schedule with designated temporary parking solutions.

Issue an immediate administrative order imposing heavy fines and legal sanctions on any shopkeeper who attempts to obstruct or protest the park construction.

Publicly confront the shopkeepers in a press conference, aggressively accusing them of putting commercial profit ahead of community well-being.

Bypass formal public consultation and quietly reduce the planned pedestrian space to satisfy the shopkeepers' parking demands without notifying the residents.

Show answer & explanation

Answer: Facilitate a joint consultative meeting with representatives of both shopkeepers and residents to identify shared priorities and negotiate a phased construction schedule with designated temporary parking solutions.

Answer

Facilitate a joint consultative meeting with representatives of both shopkeepers and residents to identify shared priorities and negotiate a phased construction schedule with designated temporary parking solutions.

The correct strategy focuses on collaborative stakeholder engagement, transparent communication, and finding balanced solutions (such as phased timelines and temporary parking) that preserve public interests while accommodating commercial concerns.

Step-by-Step Solution

1

Identify the competing stakeholder interests and core administrative objective.

Recognized that shopkeepers prioritize commercial accessibility while residents prioritize green space, both of which are valid public concerns.

Effective conflict resolution begins with an objective assessment of all legitimate stakeholder perspectives.

2

Evaluate negotiation strategies based on civil service standards of transparency and multi-party engagement.

Determined that collaborative problem-solving through joint dialogue allows for balanced compromises, such as temporary parking provisions and phased timelines.

Integrative negotiation seeks win-win solutions that satisfy core interests without resorting to authoritarian action or procedural bypasses.

Key Concept

Stakeholder Conflict Resolution and Negotiation Strategy

Question 787Question →

Match each number pair analogy in List-I with its corresponding underlying mathematical rule in List-II.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The pair

8 : 448

matches

x : x^3 - x^2

;

7 : 336

matches

x : x^3 - x

;

6 : 222

matches

x : x^3 + x

; and

5 : 150

matches

x : x^3 + x^2

.

Each pair in List-I represents a cubic transformation of the first number. Evaluating

8^3 - 8^2 = 448

,

7^3 - 7 = 336

,

6^3 + 6 = 222

, and

5^3 + 5^2 = 150

confirms the one-to-one mapping with the respective rules in List-II.

Step-by-Step Solution

1

Analyze the relationship for the first pair

8 : 448

Calculate

8^3 = 512

. Subtracting

8^2 = 64

gives

512 - 64 = 448

, establishing the rule

x : x^3 - x^2

.

Identify the operation involving cubes and squares of the base number

8

.

2

Analyze the relationship for the second pair

7 : 336

Calculate

7^3 = 343

. Subtracting

7

gives

343 - 7 = 336

, establishing the rule

x : x^3 - x

.

Identify the operation involving subtracting the base number from its cube.

3

Analyze the relationship for the third pair

6 : 222

Calculate

6^3 = 216

. Adding

6

gives

216 + 6 = 222

, establishing the rule

x : x^3 + x

.

Identify the operation involving adding the base number to its cube.

4

Analyze the relationship for the fourth pair

5 : 150

Calculate

5^3 = 125

. Adding

5^2 = 25

gives

125 + 25 = 150

, establishing the rule

x : x^3 + x^2

.

Identify the operation involving adding the square of the base number to its cube.

Key Concept

Polynomial operations on base numbers in numerical analogies

Estimated Time:2m 0s

Question 788Question →

As a Sub-Divisional Magistrate (SDM) overseeing land acquisition for a critical national highway expansion, you receive a representation from a group of marginal farmers. They present evidence suggesting that their fertile agricultural land was acquired while adjacent, uncultivated land owned by an influential developer was deliberately bypassed during the alignment survey. Halting work completely will cause severe contractual delays and financial penalties, but ignoring the petition risks violating procedural fairness and public trust. Which of the following is the most ethically sound and administratively appropriate course of action?

Issue an immediate district-wide suspension of all highway construction activities and order criminal proceedings against the survey team.

Dismiss the farmers' petition to prevent project delays and exchequer losses, advising them to file a civil lawsuit if aggrieved.

Constitute a time-bound independent technical committee to re-verify the disputed alignment while allowing construction to proceed on undisputed segments.

Informally direct the field engineers to shift the highway boundary onto the developer's land on-site to resolve the issue quickly.

Show answer & explanation

Answer: Constitute a time-bound independent technical committee to re-verify the disputed alignment while allowing construction to proceed on undisputed segments.

Answer

The most ethically sound and administratively appropriate action is to constitute a time-bound independent technical committee to re-verify the disputed alignment while allowing construction to proceed on undisputed segments.

Constituting a time-bound, independent technical committee to investigate the disputed stretch while permitting work on undisputed segments reflects administrative prudence. It respects natural justice by investigating credible allegations of bias, adheres to proportionality by avoiding an unnecessary project-wide shutdown, and upholds public interest.

Step-by-Step Solution

1

Analyze the administrative dilemma

Identified competing values: procedural justice and public trust versus operational efficiency and avoidance of exchequer loss.

Administrative decisions must balance natural justice with public interest and institutional efficiency.

2

Evaluate options against constitutional standards and administrative propriety

Determined that complete shutdown or informal alterations violate administrative balance or statutory protocol, while dismissing grievances violates natural justice.

Extreme measures or rule violations exacerbate administrative failures, whereas complete denial of grievance redressal damages public credibility.

3

Select the balanced course of action

Instituting an independent, time-bound technical review limited to the disputed section maintains due process while keeping the broader public project on schedule.

This minimizes financial loss while rigorously investigating potential corruption or procedural error.

Key Concept

Principle of Proportionality and Natural Justice in Administrative Decisions

Question 789Question →

While managing an emergency relief camp following a localized flash flood, a District Relief Officer observes that several displaced families lack official government identity cards lost during the disaster, making them unable to complete standard biometric authentication for ration distribution. What is the most ethically sound and administratively appropriate course of action for the officer to take?

Issue provisional relief tokens based on local spot-verification and official record-logging while simultaneously initiating expedited registration for replacement identity cards.

Refuse all ration distribution to the unverified individuals until they obtain fresh formal identity documents from the central administrative office.

Distribute relief supplies informally without creating any provisional logs or recording recipient details in order to speed up the process.

Suspend the entire distribution camp temporarily for all beneficiaries until a complete re-audit of all displaced victims is conducted.

Show answer & explanation

Answer: Issue provisional relief tokens based on local spot-verification and official record-logging while simultaneously initiating expedited registration for replacement identity cards.

Answer

Issue provisional relief tokens based on local spot-verification and official record-logging while initiating expedited registration for replacement identity cards.

The correct action combines humanitarian responsiveness with administrative propriety. Utilizing temporary spot-verification ensures that genuine victims receive life-saving rations immediately, while creating provisional logs maintains public transparency and official accountability.

Step-by-Step Solution

1

Identify the core administrative tension between strict procedural verification and immediate emergency humanitarian relief.

Recognized that rigid adherence to standard paperwork causes immediate harm to flood victims, while completely ignoring records causes audit failure.

Administrative decisions in public welfare must prioritize public interest while preserving basic record accountability.

2

Evaluate options for administrative proportionality and statutory flexibility during disaster management.

Provisional entry with spot-verification permits immediate aid without compromising transparency.

Public administration rules allow for emergency flexibility provided proper provisional documentation is maintained.

Key Concept

Situational Ethics and Administrative Dilemmas

Question 790Question →

What is the value of the mathematical expression $24 + 4 \times \left[ 15 - \left\{ 12 \div 4 + (8 - \overline{5 - 2}) \right\} \right]$ ?

52

196

68

44

Show answer & explanation

Answer: 52

Answer

The simplified value of the expression is 52.

Following the VBODMAS hierarchy: first evaluate the bar line

\overline{5-2} = 3

, then the round brackets

(8-3) = 5

, then division inside the curly braces

12 \div 4 = 3

, followed by addition inside curly braces

3+5 = 8

. Next, subtract within square brackets

15-8 = 7

. Finally, multiply before adding:

4 \times 7 = 28

, and

24 + 28 = 52

.

Step-by-Step Solution

1

Evaluate the vinculum bar

\overline{5 - 2}

5 - 2 = 3

According to VBODMAS rules, operations under a vinculum (bar) have the highest priority among bracket operations.

2

Evaluate the expression inside the round brackets

(8 - 3)

8 - 3 = 5

Round brackets (parentheses) are simplified next.

3

Evaluate the expression inside the curly braces

\{12 \div 4 + 5\}

12 \div 4 = 3

, then

3 + 5 = 8

Inside the curly braces, division takes precedence over addition.

4

Evaluate the expression inside the square brackets

[15 - 8]

15 - 8 = 7

Square brackets are solved after round and curly brackets.

5

Perform multiplication before final addition

4 \times 7 = 28

Multiplication takes precedence over addition outside the brackets.

6

Perform the final addition

24 + 28 = 52

Complete the final addition step to obtain the result.

Key Concept

VBODMAS Rule (Vinculum, Brackets, Orders, Division, Multiplication, Addition, Subtraction)

Question 791Question →

What is the simplified value of the following expression?

25\% \text{ of } 160 - \left[ 12 \div \left\{ 3\frac{1}{5} + \left( 1.8 - \overline{0.6 + 0.4} \right) \right\} \right] \times 5

25

185

27.5

39.4

Show answer & explanation

Answer: 25

Answer

25

Evaluating the expression following the standard BODMAS priority (Vinculum

\rightarrow

Brackets

\rightarrow

Orders/Percentage

\rightarrow

Division

\rightarrow

Multiplication

\rightarrow

Subtraction):
1. Vinculum:

\overline{0.6 + 0.4} = 1

2. Round brackets:

1.8 - 1 = 0.8

3. Curly brackets:

3\frac{1}{5} + 0.8 = 3.2 + 0.8 = 4

4. Square brackets:

12 \div 4 = 3

5. Percentage:

25\% \text{ of } 160 = 40

6. Final calculation:

40 - (3 \times 5) = 40 - 15 = 25

.
Hence, the correct value is 25.

Step-by-Step Solution

1

Evaluate the expression under the vinculum bar

\overline{0.6 + 0.4}

0.6 + 0.4 = 1.0

The vinculum (bar bracket) takes the highest priority in the hierarchy of brackets.

2

Evaluate the innermost round brackets

(1.8 - 1.0)

1.8 - 1.0 = 0.8

Round brackets must be resolved next after the vinculum.

3

Convert mixed fraction

3\frac{1}{5}

to decimal and evaluate the curly brackets

\left\{ 3.2 + 0.8 \right\}

3\frac{1}{5} = 3.2

, so

3.2 + 0.8 = 4.0

Curly brackets are evaluated after resolving all inner terms.

4

Evaluate the square brackets

\left[ 12 \div 4.0 \right]

12 \div 4 = 3

Square brackets enclose the remaining multi-bracket expression.

5

Evaluate the 'of' percentage operation

25\% \text{ of } 160

\frac{25}{100} \times 160 = 40

'Of' / percentage operations take precedence over standard multiplication and subtraction.

6

Perform multiplication before subtraction in

40 - 3 \times 5

3 \times 5 = 15

, then

40 - 15 = 25

According to BODMAS, multiplication precedes subtraction.

Key Concept

Hierarchy of Brackets and BODMAS Rule

Estimated Time:2m 0s

Question 792Question →

A survey was conducted among $500$ civil service aspirants in a coaching institute regarding their daily newspaper reading habits among three publications: *The Hindu*, *The Indian Express*, and *Business Standard*.

The survey revealed the following data:
- The total number of aspirants reading *The Hindu*, *The Indian Express*, and *Business Standard* are $240$ , $210$ , and $180$ respectively.
- Exactly $20$ aspirants read all three newspapers.
- The ratio of the number of aspirants who read ONLY *The Hindu* and *The Indian Express* to those who read ONLY *The Indian Express* and *Business Standard* to those who read ONLY *The Hindu* and *Business Standard* is $3 : 2 : 4$ .
- The number of aspirants who read ONLY *Business Standard* is $70$ .
- The number of aspirants who read ONLY *The Hindu* is equal to the number of aspirants who read ONLY *The Indian Express*.

Based on the data provided, how many aspirants in total do NOT read any of the three newspapers?

45

65

35

55

Show answer & explanation

Answer: $45$

Answer

The total number of aspirants who do not read any of the three newspapers is 45.

By resolving the 7 disjoint regions of the Venn diagram using the given ratio

3k : 2k : 4k

, we find

k = 15

from the total for Business Standard (

180 = 70 + 4k + 2k + 20

). This yields exclusive intersection values of

45

,

30

, and

60

. Substituting these into the total for Indian Express (

210

) yields

\text{Only}(E) = 115

, which also equals

\text{Only}(H)

. Summing all seven disjoint regions gives

115 + 115 + 70 + 45 + 30 + 60 + 20 = 455

. Subtracting from the total population of

500

gives

45

aspirants who do not read any of the three newspapers.

Step-by-Step Solution

1

Define disjoint regions of the Venn diagram using given variables.

Let

H

,

E

, and

B

denote the sets of readers. Let

\text{Only}(H \cap E) = 3k

,

\text{Only}(E \cap B) = 2k

, and

\text{Only}(H \cap B) = 4k

. The center intersection

\text{All Three} = 20

.

Setting up explicit variables for mutually exclusive regions allows algebraic representation of total set sizes.

2

Calculate the ratio constant

k

using the total for set

B

(Business Standard).

|B| = \text{Only}(B) + \text{Only}(H \cap B) + \text{Only}(E \cap B) + \text{All Three} \implies 180 = 70 + 4k + 2k + 20 \implies 90 = 6k \implies k = 15

.

All sub-regions comprising set

B

are known except

k

, making it possible to solve for

k

directly.

3

Determine the exact counts of two-set exclusive intersections.

\text{Only}(H \cap E) = 3(15) = 45

,

\text{Only}(E \cap B) = 2(15) = 30

, and

\text{Only}(H \cap B) = 4(15) = 60

.

Multiplying the ratio multipliers by

k = 15

gives exact counts for each overlap.

4

Calculate

\text{Only}(E)

and

\text{Only}(H)

.

|E| = \text{Only}(E) + 45 + 30 + 20 = 210 \implies \text{Only}(E) = 115

. Since

\text{Only}(H) = \text{Only}(E)

,

\text{Only}(H) = 115

.

Subtracting known intersection counts of set

E

from

|E| = 210

gives

\text{Only}(E)

, which equals

\text{Only}(H)

.

5

Calculate the total number of aspirants who read at least one newspaper and find the complement.

|H \cup E \cup B| = 115 + 115 + 70 + 45 + 30 + 60 + 20 = 455

. Aspirants reading none

= 500 - 455 = 45

.

Summing all 7 disjoint regions gives the union. Subtracting the union from the universal set gives the number of aspirants reading no newspaper.

Key Concept

3-Set Principle of Inclusion-Exclusion and Disjoint Region Analysis

Question 793Question →

What is the unit digit of the expression $432^{216} - 517^{142} - 734^{121} \times 283^{105}$ ?

5

0

1

4

Show answer & explanation

Answer: 5

Answer

The unit digit of the given expression is 5.

Evaluating each component:

432^{216}

ends in 6 (since 216 is divisible by 4, giving

2^4 = 16

),

517^{142}

ends in 9 (

142 \pmod 4 = 2

, giving

7^2 = 49

), and

734^{121} \times 283^{105}

ends in

4 \times 3 = 12 \rightarrow 2

. Combining these gives

(6 - 9 - 2) = -5 \equiv 5 \pmod{10}

. Thus, the option with value 5 is correct.

Step-by-Step Solution

1

Determine the unit digit of the first term

432^{216}

Unit digit is 6

The base unit digit is 2, which has a cyclicity of 4 (pattern: 2, 4, 8, 6). Dividing exponent 216 by 4 gives a remainder of 0. When the remainder is 0, we use the 4th power:

2^4 = 16

, so the unit digit is 6.

2

Determine the unit digit of the second term

517^{142}

Unit digit is 9

The base unit digit is 7, which has a cyclicity of 4 (pattern: 7, 9, 3, 1). Dividing exponent 142 by 4 gives a remainder of 2. Taking

7^2 = 49

, the unit digit is 9.

3

Determine the unit digit of the product term

734^{121} \times 283^{105}

Unit digit is 2

For

734^{121}

, base 4 with an odd exponent gives unit digit 4. For

283^{105}

, base 3 with exponent

105 \pmod 4 = 1

gives unit digit 3. Multiplying their unit digits:

4 \times 3 = 12

, so the unit digit is 2.

4

Combine the unit digits following order of operations (BODMAS)

Unit digit is 5

We compute

(6 - 9 - 2) \pmod{10}

. Evaluating left-to-right:

6 - 9 = -3 \equiv 7 \pmod{10}

(by adding 10 to handle borrowing). Then

7 - 2 = 5

.

Key Concept

Unit digit calculation using cyclicity of numbers and modular arithmetic rules under BODMAS

Estimated Time:2m 0s

Question 794Question →

Evaluate the following mathematical expression by applying the standard order of operations (BODMAS):

18 \div \left[ 4 \frac{1}{2} - \left\{ 2 \frac{1}{4} + \frac{3}{4} \times \left( 4.8 - \overline{2.4 + 0.4} \right) \right\} \right] + 5.5

What is the final numerical value of the simplified expression?

Show answer & explanation

Answer: 29.5

Answer

The simplified numerical value of the given expression is 29.5.

Following the BODMAS rule systematically, we first resolve the vinculum to get 2.8, then evaluate the round brackets (4.8 - 2.8 = 2.0). Next, inside the curly brackets, multiplication yields 1.5 and addition yields 3.75. Evaluating the square brackets yields 4.5 - 3.75 = 0.75. Dividing 18 by 0.75 gives 24, and adding 5.5 gives the final correct answer of 29.5.

Step-by-Step Solution

1

Evaluate the expression under the vinculum bar

\overline{2.4 + 0.4} = 2.8

The vinculum (bar) takes highest priority over standard parentheses and operations.

2

Simplify the innermost round brackets

4.8 - 2.8 = 2.0

Perform subtraction inside the round brackets next.

3

Perform multiplication inside the curly brackets

\frac{3}{4} \times 2.0 = 1.5

Multiplication precedes addition within the curly brackets according to BODMAS.

4

Complete evaluation inside the curly brackets

2 \frac{1}{4} + 1.5 = 2.25 + 1.5 = 3.75

Convert the mixed fraction 2 1/4 to decimal 2.25 and add 1.5.

5

Evaluate the terms inside the square brackets

4 \frac{1}{2} - 3.75 = 4.5 - 3.75 = 0.75

Convert mixed fraction 4 1/2 to decimal 4.5 and subtract 3.75.

6

Perform division before final addition

18 \div 0.75 = 24

Division takes precedence over the final addition operation.

7

Add the final constant

24 + 5.5 = 29.5

Complete the evaluation by adding 5.5 to 24.

Key Concept

Hierarchy of operations in BODMAS including vinculum bar, nested brackets, and precedence of operations.

Estimated Time:1m 30s

Question 795Question →

Calculate the numerical value of the following mathematical expression:

75\% \text{ of } 240 - \left[ 42 \div \left\{ 2\frac{1}{4} + \left( 3.5 - \overline{1.8 - 0.8} \right) \times 1.5 \right\} \right]

Show answer & explanation

Answer: 173

Answer

The simplified numerical value of the expression is 173.

Applying the VBODMAS rule sequentially (vinculum, round brackets, multiplication, addition within curly brackets, square bracket division, percentage evaluation, and final subtraction) yields 173.

Step-by-Step Solution

1

Evaluate the vinculum bar expression

\overline{1.8 - 0.8} = 1.0

Under VBODMAS rules, the vinculum (bar) grouping has the highest priority and must be evaluated first.

2

Evaluate the subtraction inside the round brackets

3.5 - 1.0 = 2.5

Evaluate the parentheses next using the result from the vinculum evaluation.

3

Perform multiplication inside the curly brackets

2.5 \times 1.5 = 3.75

Within the curly brackets, multiplication precedes addition according to BODMAS.

4

Perform addition inside the curly brackets

2\frac{1}{4} + 3.75 = 2.25 + 3.75 = 6.0

Convert the mixed fraction

2\frac{1}{4}

to decimal

2.25

and add to

3.75

.

5

Perform division inside the square brackets

42 \div 6.0 = 7

Resolve the square brackets by dividing 42 by the value of the curly bracket expression.

6

Evaluate the percentage 'of' expression

75\% \text{ of } 240 = 0.75 \times 240 = 180

The 'of' operator indicates multiplication applied to percentage.

7

Subtract the square bracket result from the percentage value

180 - 7 = 173

Perform final subtraction to complete the calculation.

Key Concept

Hierarchical priority of operations in VBODMAS (Vinculum -> Brackets -> Of -> Division -> Multiplication -> Addition -> Subtraction)

Question 796Question →

What is the simplified numerical value of the following mathematical expression when evaluated using the standard BODMAS rule?

60\% \text{ of } 350 - \left[ 5 \frac{1}{4} + \left\{ 36 \div \left( 4.5 - \overline{2.1 + 0.9} \right) \right\} \right] \times 4

93

100

169

723

Show answer & explanation

Answer: 93

Answer

The simplified numerical value of the given expression is 93.

Following the BODMAS rule: first evaluate the vinculum $\overline{2.1 + 0.9} = 3$, then the round bracket $4.5 - 3 = 1.5$, then the curly bracket $36 \div 1.5 = 24$, then the square bracket $5.25 + 24 = 29.25$. Next, evaluate the percentage term $60\% \text{ of } 350 = 210$ and the product $29.25 \times 4 = 117$. Finally, subtracting 117 from 210 gives 93.

Step-by-Step Solution

1

Evaluate the expression under the vinculum bar

\overline{2.1 + 0.9} = 3.0

The vinculum (bar) has the highest priority inside nested grouping symbols.

2

Evaluate the terms inside the round brackets

4.5 - 3.0 = 1.5

Subtract the result of the vinculum from 4.5 inside the round brackets.

3

Evaluate the division inside the curly brackets

36 \div 1.5 = 24

Perform division within the curly braces.

4

Evaluate the addition inside the square brackets

5 \frac{1}{4} + 24 = 5.25 + 24 = 29.25

Convert the mixed fraction to decimal (5.25) and add to 24.

5

Calculate the percentage term and multiplication term

60\% \text{ of } 350 = \frac{60}{100} \times 350 = 210 \quad \text{and} \quad 29.25 \times 4 = 117

According to BODMAS, percentage/off and multiplication must be computed before final subtraction.

6

Perform the final subtraction

210 - 117 = 93

Subtract the multiplied bracket result from the percentage value.

Key Concept

Order of Operations (BODMAS/PEMDAS) with Vinculum and Nested Brackets

Question 797Question →

In a regional administrative office of $80$ officers, $45$ officers completed a training course in Data Analytics, $40$ officers completed a course in Public Policy, and $15$ officers completed both training courses. How many officers completed neither of the two training courses?

10

25

15

20

Show answer & explanation

Answer: $10$

Answer

10 officers completed neither of the two training courses.

To find the number of officers who completed neither course, calculate the union of the two sets using inclusion-exclusion:

n(\text{Data Analytics} \cup \text{Public Policy}) = 45 + 40 - 15 = 70

. The number of officers who completed neither course is the complement of this union relative to the total group:

80 - 70 = 10

.

Step-by-Step Solution

1

Identify the given set values and total universe

Total officers

n(U) = 80

, Data Analytics

n(A) = 45

, Public Policy

n(B) = 40

, Both

n(A \cap B) = 15

.

Establish baseline data to apply the set inclusion-exclusion principle.

2

Calculate the number of officers who completed at least one training course using the principle of inclusion-exclusion

n(A \cup B) = n(A) + n(B) - n(A \cap B) = 45 + 40 - 15 = 70

.

Avoid double-counting the officers who completed both courses.

3

Calculate the number of officers who completed neither course

n(\text{Neither}) = n(U) - n(A \cup B) = 80 - 70 = 10

.

Subtract the union of the two sets from the total population to find the complement set.

Key Concept

Two-set Principle of Inclusion-Exclusion and Complementary Sets

Estimated Time:45s

Question 798Question →

What is the unit digit of the expression $E = 138^{124} + 264^{93} - 177^{54}$ ?

1

6

9

3

Show answer & explanation

Answer: 1

Answer

The unit digit of the expression is 1.

The correct answer is derived by determining the cyclicity of each base's unit digit: 8 has cyclicity 4 (with remainder 0 yielding unit digit 6), 4 has cyclicity 2 (odd exponent yielding unit digit 4), and 7 has cyclicity 4 (remainder 2 yielding unit digit 9). Combining these yields (6 + 4) - 9 = 1.

Step-by-Step Solution

1

Find the unit digit of

138^{124}

Unit digit is 6

The unit digit of the base is 8. The cyclicity of 8 is 4 (8, 4, 2, 6). Dividing the exponent 124 by 4 gives a remainder of 0. A remainder of 0 corresponds to the 4th power in the cycle (

8^4 = 4096

), so the unit digit is 6.

2

Find the unit digit of

264^{93}

Unit digit is 4

The unit digit of the base is 4. The cyclicity of 4 is 2 (

4^1 = 4, 4^2 = 6

). Since the exponent 93 is odd, the unit digit is 4.

3

Find the unit digit of

177^{54}

Unit digit is 9

The unit digit of the base is 7. The cyclicity of 7 is 4 (7, 9, 3, 1). Dividing the exponent 54 by 4 gives a remainder of 2 (

54 = 4 \times 13 + 2

). Thus, the unit digit is

7^2 = 49

, which ends in 9.

4

Combine the unit digits according to the expression

Unit digit is 1

Evaluating

(6 + 4) - 9 = 10 - 9 = 1

. The unit digit of the entire expression is 1.

Key Concept

Unit Digit Cyclicity and Modular Exponentiation Rules

Question 799Question →

What is the unit digit of the composite exponential expression $E = 332^{520} \times 553^{213} - 448^{631}$ ?

1

0

6

4

Show answer & explanation

Answer: 6

Answer

The unit digit of the given expression is 6.

Evaluating each component using cyclicity of unit digits:

332^{520}

has base unit digit 2 and exponent divisible by 4 (

520 \bmod 4 = 0

), so its unit digit is

2^4 \implies 6

.

553^{213}

has base unit digit 3 with

213 \bmod 4 = 1

, giving unit digit

3^1 = 3

.

448^{631}

has base unit digit 8 with

631 \bmod 4 = 3

, giving unit digit

8^3 \implies 2

. Combining these gives

(6 \times 3) - 2 = 18 - 2 = 16

, which yields a final unit digit of 6.

Step-by-Step Solution

1

Determine the unit digit of

332^{520}

using cyclicity of 2

Unit digit is 6

The unit digit of base 332 is 2. The cyclicity of 2 is 4 (2, 4, 8, 6). Dividing the exponent 520 by 4 gives a remainder of 0. When remainder is 0, we take the 4th power:

2^4 = 16

, so the unit digit is 6.

2

Determine the unit digit of

553^{213}

using cyclicity of 3

Unit digit is 3

The unit digit of base 553 is 3. The cyclicity of 3 is 4 (3, 9, 7, 1). Dividing the exponent 213 by 4 gives a remainder of 1 (

213 = 4 \times 53 + 1

). Thus, the unit digit is

3^1 = 3

.

3

Determine the unit digit of

448^{631}

using cyclicity of 8

Unit digit is 2

The unit digit of base 448 is 8. The cyclicity of 8 is 4 (8, 4, 2, 6). Dividing the exponent 631 by 4 gives a remainder of 3 (

631 = 4 \times 157 + 3

). Thus, the unit digit is

8^3 = 512

, which ends in 2.

4

Combine the unit digits according to the expression

E = (332^{520} \times 553^{213}) - 448^{631}

Unit digit is 6

First multiply the unit digits of the first two terms:

6 \times 3 = 18 \implies 8

. Then subtract the unit digit of the third term:

8 - 2 = 6

.

Key Concept

Unit Digit Cyclicity Rules

Estimated Time:1m 30s

Question 800Question →

Determine the exact result of the given mathematical expression after applying the standard order of operations (BODMAS):

32.4 - \frac{4}{5} \times \left[ 45 \div \left\{ 2.5 \times 4 - \left( 7.6 - \overline{4.1 - 1.5} \right) \right\} \right]

Show answer & explanation

Answer: 25.2

Answer

The simplified numerical value of the expression is 25.2.

Following the standard BODMAS priority order: evaluate the vinculum (4.1 - 1.5 = 2.6), round brackets (7.6 - 2.6 = 5), curly brackets (2.5 * 4 - 5 = 5), square brackets (45 / 5 = 9), multiplication ((4/5) * 9 = 7.2), and finally subtraction (32.4 - 7.2 = 25.2).

Step-by-Step Solution

1

Evaluate the sub-expression under the vinculum bar

\overline{4.1 - 1.5} = 2.6

The vinculum acts as the innermost grouping symbol and takes highest priority.

2

Simplify the expression inside the round brackets

(7.6 - 2.6) = 5.0

Parentheses (round brackets) are evaluated next.

3

Evaluate operations inside the curly brackets using operator precedence

\{2.5 \times 4 - 5.0\} = \{10 - 5.0\} = 5

Multiplication precedes subtraction inside the curly brackets.

4

Evaluate the division inside the square brackets

[45 \div 5] = 9

Square brackets are solved after resolving inner curly brackets.

5

Perform multiplication outside the bracket

\frac{4}{5} \times 9 = 0.8 \times 9 = 7.2

Multiplication takes precedence over final addition/subtraction.

6

Perform the final subtraction

32.4 - 7.2 = 25.2

Subtraction is the final step in the BODMAS order of operations.

Key Concept

BODMAS Rule with Vinculum and Nested Brackets

Estimated Time:1m 30s

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