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Question 381Question →

When the positive integer $N$ is divided by $15$ , $20$ , and $24$ , it leaves a remainder of $8$ in each case. If $N$ is the smallest such number that is also exactly divisible by $13$ , what is the value of $N$ ?

832

128

728

1664

Show answer & explanation

Answer: 728

Answer

728

The Lowest Common Multiple (LCM) of 15, 20, and 24 is 120. Any number leaving a remainder of 8 when divided by these numbers can be expressed in the form 120k + 8. By testing integer values for k, we find that when k = 6, the number is 120(6) + 8 = 728. Since 728 is perfectly divisible by 13 (728 / 13 = 56), it satisfies all conditions and is the correct answer.

Step-by-Step Solution

Find the Lowest Common Multiple (LCM) of the given divisors: 15, 20, and 24.

LCM(15, 20, 24) = 120.

The base number without the remainder must be exactly divisible by all three numbers.

Set up a general algebraic expression for the number

N

N = 120k + 8, where k is a positive integer.

Because the number leaves a remainder of 8 in each case, it must be 8 more than a multiple of their LCM.

Apply the final divisibility condition to find the smallest valid

k

Test values of k in the expression (120k + 8) to see which result is divisible by 13.

The problem specifies that the final number

N

must be exactly divisible by 13.

Evaluate

120k + 8

for

k = 1, 2, 3...

until a multiple of 13 is reached.

For k = 6, N = 120(6) + 8 = 728. Checking divisibility: 728 / 13 = 56 (an exact integer).

Sequential testing guarantees we find the smallest possible value for

N

Key Concept

Applications of LCM and Modular Arithmetic

Question 382Question →

Compute the exact Highest Common Factor (HCF) for the fractions $\frac{6}{25}$ and $\frac{9}{10}$ . Express your final result as a decimal.

Show answer & explanation

Answer: 0.06

Answer

0.06

The Highest Common Factor (HCF) of a set of fractions is found by dividing the HCF of their numerators by the LCM of their denominators. In this case, HCF(6, 9) is 3, and LCM(25, 10) is 50. This gives the fraction 3/50, which perfectly evaluates to the decimal 0.06.

Step-by-Step Solution

Identify the standard rule for calculating the HCF of fractional numbers.

HCF = (HCF of numerators) / (LCM of denominators)

This is the mathematical formula required to find the greatest common divisor of multiple fractions.

Find the Highest Common Factor (HCF) of the two numerators, 6 and 9.

HCF(6, 9) = 3

3 is the largest integer that divides both 6 and 9 without leaving a remainder.

Find the Least Common Multiple (LCM) of the two denominators, 25 and 10.

LCM(25, 10) = 50

50 is the smallest positive integer that is a multiple of both 25 and 10.

Substitute the results into the formula and convert the fraction to a decimal.

3 / 50 = 0.06

Dividing 3 by 50 yields the terminating decimal 0.06, which is the final required format.

Key Concept

HCF and LCM of fractions

Question 383Question →

As the District Collector of a region severely affected by groundwater arsenic contamination, you are finalizing a critical government tender for a large-scale water purification facility. During the final review, you discover that the lowest bidder, whose technical proposal is genuinely superior and highly needed, has submitted falsified past-experience certificates to qualify for the bid. The local Member of Legislative Assembly (MLA) privately advises you to overlook this discrepancy because canceling the tender will delay the facility by at least a year, severely threatening public health. Which of the following is the most administratively and ethically appropriate course of action?

Award the contract to the lowest bidder given the severe public health emergency, but issue a stern internal warning to the company regarding their documentation practices.

Immediately suspend the entire water purification project, blacklist all companies that participated in the tender, and launch a sweeping criminal investigation into the district procurement office.

Disqualify the fraudulent bidder, initiate legal proceedings for forgery, and immediately invoke fast-track or emergency re-tendering rules to ensure the project proceeds with minimal delay.

Approve the bid to avoid a one-year delay, but mandate that the company provide an additional year of free maintenance as an unofficial penalty for the document falsification.

Show answer & explanation

Answer: Disqualify the fraudulent bidder, initiate legal proceedings for forgery, and immediately invoke fast-track or emergency re-tendering rules to ensure the project proceeds with minimal delay.

Answer

The correct course of action is to disqualify the fraudulent bidder, initiate appropriate legal action for the forgery, and use emergency procurement protocols to restart the project quickly.

This action perfectly balances administrative propriety with public duty. By disqualifying the bidder and recommending legal action, the administrator upholds the rule of law and demonstrates zero tolerance for fraud. Concurrently, by invoking emergency re-tendering rules, the administrator fulfills the duty of care towards the public health crisis without resorting to illegal compromises.

Step-by-Step Solution

Identify the core ethical tension in the scenario.

The conflict is between procedural integrity (adhering to due process and the rule of law) and immediate public welfare (addressing a severe health crisis).

Understanding the competing values is the first step in resolving situational dilemmas without compromising fundamental duties.

Evaluate the consequences of ignoring the forgery.

Approving the contract based on fraudulent documents compromises the integrity of the administration, encourages future corruption, and violates statutory procurement laws, even if done for a 'good' cause.

Public welfare initiatives cannot be sustainably built on a foundation of fraud and illegality.

Determine a balanced administrative response that addresses both concerns.

The administrator must disqualify the bidder to uphold the law, but simultaneously invoke emergency protocols or fast-track processes to minimize the delay in providing safe water.

An effective civil servant must uphold statutory laws while actively seeking legal provisions within the system to address pressing humanitarian concerns.

Key Concept

Balancing Procedural Integrity with Public Welfare in Crisis Management

Question 384Question →

A global climate summit was inaugurated on Tuesday, March 15, 2016. Based on the standard Gregorian calendar, on what day of the week will the same date fall in the year 2027?

Show answer & explanation

Answer: Monday; monday; MONDAY

Answer

Monday

To find the day of the week, we track the odd days. Between 2016 and 2027, there are 11 years, providing a baseline of 11 odd days. The interval strictly crosses two leap days (in 2020 and 2024), adding 2 more odd days. Crucially, the 2016 leap day is not counted because the interval starts in March, after February 29th. The total is

13 \text{ odd days}

, and since

13 \pmod{7} = 6

, shifting 6 days forward from Tuesday gives Monday.

Step-by-Step Solution

Calculate the total number of years between the two dates.

2027 - 2016 = 11 \text{ years}

Each standard year shifts the calendar forward by

1 \text{ odd day}

because

365 \pmod{7} = 1

Identify the leap days (February 29th) that actually fall within this specific date range.

Only

2

leap days are crossed: February 29, 2020, and February 29, 2024.

Although 2016 is a leap year, its February 29th occurred before the start date of March 15, 2016, so it does not add an extra day to this interval.

Calculate the total number of odd days accumulated over the interval.

11 \text{ base days} + 2 \text{ leap days} = 13 \text{ total odd days}

Combining the normal year shifts and the extra days from leap years gives the total day shift.

Determine the final day of the week by taking the total odd days modulo 7.

13 \pmod{7} = 6 \text{ odd days}

. Tuesday

+ 6 \text{ days} = \text{Monday}

The days of the week operate on a 7-day repeating cycle.

Key Concept

The inclusion of a leap year's extra day in a calculation strictly depends on whether February 29th falls within the exact given date interval, not just whether the year itself is a leap year.

Alternative Method

Instead of grouping odd days at the end, you can calculate the day year-by-year: 2016 to 2017 (+1, Wednesday), 2017 to 2018 (+1, Thursday), 2018 to 2019 (+1, Friday), 2019 to 2020 (+2 due to Feb 2020, Sunday), and so on, arriving at Monday for 2027.

Estimated Time:1m 0s

Question 385Question →

Consider the integer $N = 4 - 2 \times 2^{68} + 7 \times 14^{85}$ . If $N$ is divided by $13$ , which of the following represents the correct remainder?

Show answer & explanation

Answer: 6

Answer

The correct remainder is 6.

By applying Fermat's Little Theorem,

2^{68} \equiv 2^8 \equiv 9 \pmod{13}

, and clearly

14^{85} \equiv 1^{85} \equiv 1 \pmod{13}

. Substituting these values gives the expression

4 - 2(9) + 7(1) = 4 - 18 + 7 = -7

. To find the valid positive remainder, we add the divisor 13 to the negative result, yielding

-7 + 13 = 6

Step-by-Step Solution

Simplify the term

14^{85} \pmod{13}

Since

14 \equiv 1 \pmod{13}

, any power of 14 will also be congruent to

1^{85} = 1

Simplify the term

2^{68} \pmod{13}

using Fermat's Little Theorem

Fermat's theorem states

a^{12} \equiv 1 \pmod{13}

. Dividing 68 by 12 gives a quotient of 5 and a remainder of 8. Thus,

2^{68} \equiv 2^8 \pmod{13}

. Since

2^8 = 256

, and

256 = 13 \times 19 + 9

, the value is 9.

Substitute the simplified modulo values back into the original expression

4 - 2(9) + 7(1) \pmod{13}

Replacing the large exponent terms with their modulo 13 equivalents allows for simple arithmetic evaluation.

Evaluate the simplified arithmetic expression from left to right

-7

Following the correct order of operations:

4 - 18 + 7 = -14 + 7 = -7

Convert the negative result into a valid positive remainder

Remainders must be non-negative. Add the divisor (13) to the negative result:

-7 + 13 = 6

Key Concept

Modular Arithmetic, Fermat's Little Theorem, and Negative Remainders

Question 386Question →

Consider a sequence of integers generated by the mathematical formula $T_n = 17^n - 40$ . What is the positive remainder obtained when the $123^{\text{rd}}$ term of this sequence is divided by $9$ ?

Show answer & explanation

Answer: 4

Answer

Applying the rules of modular arithmetic, we first reduce the base

17

modulo

9

, giving

-1

. Raising

-1

to the odd power of

123

yields

-1

. The expression then simplifies to

-1 - 40 = -41

. Because remainders must be non-negative, we add the next largest multiple of

9

(which is

45

) to

-41

, resulting in a final positive remainder of

4

Step-by-Step Solution

Set up the expression for the

123^{\text{rd}}

term modulo

9

T_{123} \equiv (17^{123} - 40) \pmod 9

We need to find the remainder of the specific sequence term when divided by

9

Reduce the base of the exponent modulo

9

17 \equiv -1 \pmod 9

Using a base of

-1

simplifies the calculation of large powers.

Evaluate the exponent part using the reduced base.

(-1)^{123} = -1

Raising a negative number to an odd power retains the negative sign.

Substitute back into the expression and compute the intermediate remainder.

-1 - 40 = -41

Combining the evaluated power with the rest of the sequence formula.

Convert the negative value into a valid positive remainder modulo

9

-41 = 9 \times (-5) + 4 \implies \text{Remainder is } 4

A true remainder must be a positive integer strictly less than the divisor. We achieve this by adding multiples of

9

until the value is positive.

Key Concept

Modular arithmetic reduction, exponent parity rules, and conversion of negative remainders.

Question 387Question →

As a District Magistrate, you are tasked with resolving a tense standoff between a state highway construction company and local farmers. The farmers have blockaded the machinery because construction debris accidentally blocked their primary irrigation canal during the critical sowing season. The company is citing heavy financial losses for project delays, while the farmers are demanding immediate restoration of water flow. Arrange the following administrative steps in the most appropriate and effective sequence for conflict de-escalation and long-term resolution.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence begins with immediate de-escalation and temporary relief, followed by joint fact-finding, then mediation of a settlement, and concludes with the formalization and monitoring of the agreement.

In public administration, effective stakeholder conflict resolution follows a logical sequence: (1) Immediate crisis containment and maintenance of law and order, (2) Systematic fact-finding and consultation with all affected parties, (3) Mediating a proportional and mutually acceptable compromise, and (4) Formalizing the agreement with built-in accountability mechanisms. This sequence ensures that emotions are cooled before rational negotiations begin, and that the final agreement is both legally sound and practically enforceable.

Step-by-Step Solution

Identify the immediate crisis management step.

Providing a temporary water bypass and defusing the blockade is selected as the first step.

Administrative conflict resolution dictates that restoring public order and providing immediate necessary relief must precede long-term negotiations.

Determine the first step toward a permanent solution.

Convening a joint fact-finding meeting is selected as the second step.

Before mediating a solution, the administration must officially assess the damage and hear from all stakeholders in a neutral setting.

Identify the core negotiation step.

Mediating a structured agreement for canal reconstruction is selected as the third step.

With facts established, the administrator negotiates a win-win solution where both parties' primary concerns (water access and project resumption) are met.

Determine the final institutionalization step.

Issuing a formal order and establishing a monitoring committee is selected as the final step.

Agreements must be formalized into binding directives with oversight mechanisms to guarantee implementation and prevent the conflict from recurring.

Key Concept

Sequential De-escalation and Multi-Tier Negotiation in Public Administration

Question 388Question →

In a single row of books on a library shelf, an Encyclopedia is $28^{\text{th}}$ from the left end and a Dictionary is $35^{\text{th}}$ from the right end. If there are exactly 12 books placed strictly between them, what is the minimum possible total number of books in this row?

Show answer & explanation

Answer: 49

Answer

The minimum possible total number of books is 49.

The minimum total is found using the overlapping queue formula:

L + R - M - 2

. Substituting the given values yields

28 + 35 - 12 - 2 = 49

Step-by-Step Solution

Identify the queue configuration for the minimum capacity.

The minimum possible number of books occurs in an overlapping scenario.

In an overlapping scenario, the item counted from the right (Dictionary) is placed to the left of the item counted from the left (Encyclopedia), yielding the smallest overall total length.

Apply the minimum capacity formula.

\text{Total} = L + R - M - 2

We add the rank from the left (

L

) and the rank from the right (

R

), then subtract the number of items strictly between them (

M

) and the 2 reference items to correct for double-counting.

Substitute the given values and calculate the final count.

\text{Total} = 28 + 35 - 12 - 2 = 49

This yields the exact minimum count of books on the shelf.

Key Concept

Overlapping Minimum Capacity in Ordering and Ranking

Estimated Time:1m 0s

Question 389Question →

In the following question, three of the four letter-clusters follow a specific logical pattern based on their alphabetical positions and thus form a group. Which of the following letter-clusters does NOT belong to that group?

F J H M

P T R W

K O M R

C G E I

Show answer & explanation

Answer: C G E I

Answer

The correct answer is the letter-cluster C G E I, because its final alphabetical shift deviates from the established pattern of the other clusters.

The letter-cluster C G E I is the odd one out. While the other three clusters follow a strict alphabetical position shift of +4, -2, and +5, this cluster follows a shift of +4, -2, and +4 (since E is the 5th letter and I is the 9th, the final shift is only +4).

Step-by-Step Solution

Analyze the positional relationship between the letters in the first cluster (F J H M).

The alphabetical shifts are: F(6) to J(10) is +4; J(10) to H(8) is -2; H(8) to M(13) is +5.

To identify the underlying logical rule governing the clusters in the group.

Test the remaining clusters against this +4, -2, +5 pattern.

P T R W (16, 20, 18, 23) and K O M R (11, 15, 13, 18) both follow the +4, -2, +5 rule exactly.

To verify that this is indeed the shared pattern forming the group.

Evaluate the final cluster (C G E I).

C(3) to G(7) is +4; G(7) to E(5) is -2; but E(5) to I(9) is +4.

To confirm which specific cluster deviates from the established logical group.

Key Concept

Letter-Cluster Classification and Positional Shift Sequencing

Question 390Question →

Read the following administrative circular carefully:

Statement:
"To curb the escalating instances of financial cyber fraud, the central banking authority has directed all commercial banks to immediately implement a mandatory two-step biometric verification process for all digital transactions exceeding $10,000."

Based on the statement provided, which of the following assumptions are implicit? (Select all that apply)

Select all that apply

Two-step biometric verification is an effective technological measure to mitigate or prevent cyber frauds in high-value transactions.

Digital transactions below the $10,000 threshold are entirely secure and do not face any risk of cyber fraud.

Commercial banks either already possess or can reasonably deploy the necessary technical infrastructure to support biometric verification.

Traditional password-based security systems are the sole cause of all recent financial cyber frauds in the banking sector.

Show answer & explanation

Answer: Two-step biometric verification is an effective technological measure to mitigate or prevent cyber frauds in high-value transactions.; Commercial banks either already possess or can reasonably deploy the necessary technical infrastructure to support biometric verification.

Answer

The implicit assumptions are that biometric verification is an effective measure against cyber fraud and that commercial banks have the capability to implement this infrastructure.

An implicit assumption is an unstated premise that must be true for the author's argument or directive to make logical sense. In this case, for the central bank to mandate biometric verification to stop fraud, it must assume two things: first, that the technology is actually effective at stopping fraud, and second, that it is practically possible for the banks to implement this technology.

Step-by-Step Solution

Analyze the core objective and the proposed action in the statement.

Objective: Curb cyber fraud. Action: Mandate biometric verification for transactions over $10,000.

Understanding the cause-and-effect relationship intended by the author helps identify what must be assumed for the action to succeed.

Evaluate the choices testing the effectiveness and feasibility of the action.

The authority must assume the new system works (effectiveness) and that banks can implement it (feasibility), validating the assumptions regarding system efficacy and infrastructure.

An action is only mandated if the issuer believes it will solve the problem and can be executed.

Filter out extreme statements and unsupported external inferences.

Reject the ideas that transactions under $10,000 are perfectly safe, or that passwords are the only cause of fraud.

Implicit assumptions must be strictly necessary for the statement to hold true, without relying on extreme generalizations or outside knowledge.

Key Concept

Identifying Implicit Premises and Feasibility Assumptions

Question 391Question →

Consider the following statements regarding the classification and properties of numbers:

I. The number $0$ is the smallest positive even integer, and it can be produced by calculating the sum of two distinct irrational numbers.
II. The rational fraction $\frac{22}{7}$ represents the exact value of the mathematical constant $\pi$ on the real number line.
III. Every prime number greater than $2$ is an odd integer, and the square of any odd integer always leaves a remainder of $1$ when divided by $8$ .

Which of the statements given above is/are correct?

Only III

Only I and III

Only II and III

I, II, and III

Show answer & explanation

Answer: Only III

Answer

Only Statement III is correct. Statement I fails because 0 is not positive, and Statement II fails because pi is irrational and not exactly equal to 22/7.

The correct answer accurately isolates Statement III as the only mathematically true proposition. It correctly identifies the parity of prime numbers and the modulo 8 behavior of odd squares, while dismissing the flawed definitions of zero and

\pi

present in the other statements.

Step-by-Step Solution

Evaluate Statement I regarding the properties of the number

0

and the sum of irrationals.

Statement I is false.

While the sum of two distinct irrational numbers (e.g.,

\sqrt{2}

and

-\sqrt{2}

) can indeed be

0

, the number

0

itself is neither positive nor negative. Describing it as the 'smallest positive even integer' is incorrect. The smallest positive even integer is

2

Evaluate Statement II regarding the relationship between

\pi

and

\frac{22}{7}

Statement II is false.

The constant

\pi

is an irrational number (a non-terminating, non-repeating decimal), while

\frac{22}{7}

is a rational number. They are not exactly equal;

\frac{22}{7}

is merely a convenient fractional approximation used in basic calculations.

Evaluate Statement III regarding primes and the square of odd integers.

Statement III is true.

The only even prime is

2

, so all primes greater than

2

are odd. For any odd integer

n = 2k + 1

, its square is

(2k+1)^2 = 4k^2 + 4k + 1 = 4k(k+1) + 1

. Since

k(k+1)

is the product of two consecutive integers, it must be even (let

k(k+1) = 2m

). Thus,

4(2m) + 1 = 8m + 1

, proving it always leaves a remainder of

1

when divided by

8

Determine the correct option based on the evaluations.

The option stating 'Only III' is the correct choice.

Since Statements I and II contain mathematical misconceptions, Statement III is the only accurate assertion.

Key Concept

Classification of numbers including rational versus irrational properties, integer parity, and fundamental prime characteristics.

Question 392Question →

Consider the positive integers $M = 360$ and $N = 504$ . Based on the mathematical properties of factors and multiples, which of the following statements are correct?

Select all that apply

The Least Common Multiple (LCM) of

M

and

N

2520

The Highest Common Factor (HCF) of

M

and

N

72

The integer

M

has exactly

24

distinct positive prime factors.

The Highest Common Factor (HCF) of the fractions

\frac{M}{7}

and

\frac{N}{5}

2520

Show answer & explanation

Answer: The Least Common Multiple (LCM) of $M$ and $N$ is $2520$ .; The Highest Common Factor (HCF) of $M$ and $N$ is $72$ .

Answer

The statements confirming the Least Common Multiple (LCM) is 2520 and the Highest Common Factor (HCF) is 72 are mathematically correct.

Based on the prime factorizations

M = 2^3 \times 3^2 \times 5

and

N = 2^3 \times 3^2 \times 7

, the HCF is derived from the lowest common powers (

2^3 \times 3^2 = 72

), and the LCM is derived from the highest powers of all prime bases (

72 \times 5 \times 7 = 2520

). Thus, the statements asserting these two values are correct.

Step-by-Step Solution

Determine the prime factorizations of M and N.

M = 2^3 \times 3^2 \times 5^1

and

N = 2^3 \times 3^2 \times 7^1

Prime factorization is the foundational step required to find factors, HCF, and LCM accurately.

Calculate the Highest Common Factor (HCF) and Least Common Multiple (LCM) of M and N.

HCF =

2^3 \times 3^2 = 72

. LCM =

2^3 \times 3^2 \times 5 \times 7 = 2520

This confirms the validity of the first two statements.

Evaluate the number of factors for integer M.

Total positive factors =

(3+1)(2+1)(1+1) = 24

. However, distinct prime factors are only 2, 3, and 5 (total of 3).

This shows the statement claiming 24 prime factors is factually incorrect due to a specific modifier (prime).

Calculate the HCF of the given fractions.

HCF =

\frac{\text{HCF}(360, 504)}{\text{LCM}(7, 5)} = \frac{72}{35}

This invalidates the final statement, which improperly applied the LCM fraction formula.

Key Concept

Applying prime factorization to calculate HCF, LCM, total factor counts, and properties of fractions.

Question 393Question →

The state agriculture department has mandated the relocation of the city's largest traditional wholesale produce market to a new, fully automated agri-tech facility on the outskirts. The transition aims to eliminate middlemen and reduce food wastage. However, the local porters' union and traditional vendors have called for an indefinite strike, citing potential mass unemployment and their inability to operate the new digital bidding systems. The situation is severely disrupting the city's fresh food supply chain. As the Special Nodal Officer appointed to resolve this deadlock, which of the following is the most appropriate initial negotiation strategy?

Invoke emergency maintenance laws to declare the strike illegal and utilize law enforcement to forcibly clear the old market premises to rapidly restore the food supply chain.

Establish a joint transition task force with union leaders and vendors to co-create a phased relocation schedule, incorporating digital literacy programs and guaranteeing operational roles for existing porters.

Issue a final public ultimatum to the striking factions, explicitly threatening the permanent cancellation of their trading licenses if they do not commence relocation within 48 hours.

Suspend the state department's relocation mandate indefinitely and allow the traditional market to operate as usual to immediately appease the unions and restore the food supply.

Show answer & explanation

Answer: Establish a joint transition task force with union leaders and vendors to co-create a phased relocation schedule, incorporating digital literacy programs and guaranteeing operational roles for existing porters.

Answer

The correct answer demonstrates a balanced, participatory approach to conflict resolution. By establishing a joint task force, the administrator validates the stakeholders' concerns while maintaining the ultimate goal of the policy. Integrating digital literacy and job guarantees addresses the specific root causes of the strike (fear of unemployment and lack of skills), creating a win-win scenario that facilitates a smooth administrative transition.

Step-by-Step Solution

Identify the core underlying interests of both parties in the conflict.

The state wants modernization and efficiency; the vendors and porters want job security and technological support.

Understanding underlying motivations is the first step in finding a mutually beneficial resolution.

Determine an approach that bridges the gap between state mandates and stakeholder fears.

A phased transition supported by skill development bridges this gap.

Sudden changes trigger resistance, whereas structured support mechanisms build confidence and compliance.

Select the administrative action that facilitates collaborative problem-solving.

Forming a joint transition task force ensures all voices are heard while moving toward the policy goal.

Stakeholder participation transforms adversaries into partners, ensuring the long-term sustainability of the public initiative.

Key Concept

Collaborative Conflict Resolution and Interest-Based Negotiation

Question 394Question →

Five colleagues—K, L, M, N, and O—are seated in a single straight row, with some facing North and others facing South. The following conditions apply to their seating arrangement:

- M sits at the exact center of the row and faces North.
- L sits second to the right of M.
- K sits to the immediate left of L.
- N and O are immediate neighbors, and O does not sit at an extreme end of the row.
- K and N face the same direction, which is opposite to the direction L faces.
- The persons sitting at the extreme ends of the row face opposite directions.
- O faces South.

Based on this arrangement, what is the position of N with respect to K?

Third to the left

Second to the right

Third to the right

Second to the left

Show answer & explanation

Answer: Third to the right

Answer

N sits third to the right of K.

The correct answer accurately places K at position 4 facing South and N at position 1. Because K faces South, the seats to K's right are positions 3, 2, and 1. Being three seats away, N is third to the right of K.

Step-by-Step Solution

Determine the exact positions of M, L, and K.

Assume positions are numbered 1 to 5 from left to right. M is at position 3 facing North. L is at position 5 (second to the right of M). Because K is to the immediate left of L, L must face North (if L faced South, L's left would be outside the row). Thus, K is placed at position 4.

Establishing the anchor points using explicit positioning rules and boundary logic.

Place N and O in the remaining seats.

Positions 1 and 2 are empty. Since N and O are neighbors and O cannot be at the extreme end (position 1), O must be placed at position 2 and N at position 1.

Using negative constraints to lock down the remaining positions.

Determine the facing directions of K, N, and O.

L faces North. K and N face opposite to L, so both face South. O faces South (given). The ends are N (facing South) and L (facing North), which satisfies the rule that extreme ends face opposite directions.

Applying the relative facing direction conditions to complete the setup.

Calculate N's position with respect to K.

K is at position 4 and faces South. N is at position 1. From K's perspective (facing South), positions 3, 2, and 1 are towards the right. Since N is three seats away, N is third to the right.

Resolving the final question by orienting left/right correctly based on K's facing direction.

Key Concept

Linear seating arrangements with mixed facing directions (North/South) requiring spatial orientation.

Question 395Question →

You are the Chief Executive Officer (CEO) of a Zila Parishad overseeing a state-funded initiative to install solar micro-grids in remote public schools. After a transparent online bidding process, the evaluation committee recommends the lowest bidder (L1), who meets all technical and financial criteria perfectly. Just before the final contract is signed, local activists reveal that the L1 company is owned by the close relative of a prominent local politician who strongly lobbied for this project's funding. Although there is no immediate evidence that the bidding portal was compromised or the technical committee was influenced, the activists threaten a district-wide protest against alleged nepotism. Which of the following is the most appropriate and ethically sound administrative course of action?

Temporarily hold the final award and institute an urgent, independent review of the bidding process to verify strict procedural compliance and formally rule out any conflict of interest.

Immediately cancel the entire bidding process, blacklist the L1 company for its political affiliations, and float a fresh tender to maintain the appearance of absolute public transparency.

Bypass formal grievance procedures and privately request the L1 bidder to voluntarily withdraw their application to avoid media backlash and political controversy.

Disqualify the L1 bidder on the grounds of moral hazard and immediately award the contract to the second-lowest bidder (L2) to ensure the school project is not delayed.

Show answer & explanation

Answer: Temporarily hold the final award and institute an urgent, independent review of the bidding process to verify strict procedural compliance and formally rule out any conflict of interest.

Answer

Temporarily hold the final award and institute an urgent, independent review of the bidding process to verify strict procedural compliance and formally rule out any conflict of interest.

The correct answer balances the need to address public concerns regarding nepotism with the necessity of upholding due process. By instituting an independent review rather than acting impulsively on unproven allegations, the administrator ensures transparency, protects the integrity of the procurement process, and avoids arbitrary or legally indefensible decisions.

Step-by-Step Solution

Identify the core ethical tension in the scenario.

The conflict is between honoring the results of a procedurally fair online bidding process and addressing credible public concerns regarding a potential conflict of interest and political nepotism.

Understanding the competing values (efficiency vs. transparency) is essential for formulating a balanced administrative response.

Evaluate the immediate risks of extreme actions.

Awarding the contract immediately risks public trust and potential legal challenges from activists, while immediately canceling the tender or blacklisting the company without proof violates natural justice and procedural fairness for the bidder.

Administrative decisions must be proportionate and grounded in verifiable facts rather than public pressure or assumptions.

Determine the most appropriate procedural course of action.

An independent review must be instituted to objectively verify whether the conflict of interest actually tainted the procurement process.

This approach upholds administrative transparency, respects due process, and relies on an evidence-based mechanism to resolve the dilemma.

Key Concept

Administrative Propriety, Conflict of Interest Management, and Due Process in Public Procurement

Estimated Time:1m 30s

Question 396Question →

Evaluate the validity of the logical deductions drawn from the provided assertions.

Provided Assertions:
• All subsidized housing projects constructed after 2024 must feature integrated solar power systems.
• Some private residential complexes constructed after 2024 feature integrated solar power systems.
• No building featuring an integrated solar power system relies entirely on the standard municipal grid.

Drawn Conclusions:
I. Some private residential complexes constructed after 2024 do not rely entirely on the standard municipal grid.
II. The standard municipal grid lacks the capacity to independently support subsidized housing projects.

Based on the information above, which of the following is correct?

Only conclusion I follows

Only conclusion II follows

Both conclusion I and conclusion II follow

Neither conclusion I nor conclusion II follows

Show answer & explanation

Answer: Only conclusion I follows

Answer

Only conclusion I follows

The correct answer accurately validates the first conclusion while rejecting the second. Conclusion I is a direct syllogistic deduction from the second and third assertions: since some private complexes have solar systems, and no building with solar systems relies entirely on the municipal grid, those specific private complexes do not rely entirely on the grid. Conclusion II is invalid because the text contains no information regarding the actual capacity of the municipal grid; assuming the grid lacks capacity constitutes the introduction of external real-world assumptions, which is not permitted in strict logical deduction.

Step-by-Step Solution

Analyze Conclusion I against the provided assertions to check for direct logical validity.

Assertion two states that 'some private complexes have solar systems', and assertion three states that 'no building with solar systems relies entirely on the grid'.

Establishing the premises for a direct syllogistic deduction.

Combine the premises from step 1 to evaluate Conclusion I.

Since the specific private complexes with solar systems fall under the rule in assertion three, it must be true that those specific complexes do not rely entirely on the grid. Therefore, Conclusion I logically follows.

Applying standard categorical logic (Some A are B; All B are C; therefore, Some A are C).

Analyze Conclusion II against the provided assertions.

The assertions mandate solar power for subsidized housing and describe grid reliance, but they do not state the reason behind the mandate or mention the grid's actual capacity.

Checking if the conclusion can be directly derived without external context.

Determine the validity of Conclusion II.

Conclusion II introduces external reasoning about 'grid capacity'. This is an assumption, not a logical deduction from the provided text. Therefore, Conclusion II does not follow.

Conclusions must be drawn strictly from the provided information, ignoring external knowledge.

Key Concept

Statement and Conclusions or Inferences

Question 397Question →

A secure digital vault requires a 2-digit numerical access code. The code is determined by calculating the value of the expression $(P + Q) - (R \times S)$ , where the variables are defined based on the mathematical classification of numbers:

* $P$ : The sum of all single-digit prime numbers.
* $Q$ : The product of the smallest positive composite number and the smallest odd prime number.
* $R$ : The total count of whole numbers strictly less than $10$ that are classified as neither prime nor composite.
* $S$ : The smallest positive integer $n$ for which the expression $\sqrt{12n}$ results in a rational number.

What is the final numerical access code?

Show answer & explanation

Answer: 23

Answer

The correct calculation evaluates each subset definition perfectly: P = 17 (sum of primes 2, 3, 5, 7), Q = 12 (4 * 3), R = 2 (0 and 1 are neither prime nor composite), and S = 3 (making 12 * 3 = 36 a perfect square). Plugging these into the equation (17 + 12) - (2 * 3) yields exactly 23.

Step-by-Step Solution

Identify single-digit primes and sum them to find P.

P = 17

The single-digit primes are 2, 3, 5, and 7. Summing them yields 2 + 3 + 5 + 7 = 17.

Identify the smallest composite and smallest odd prime to find Q.

Q = 12

The smallest positive composite number is 4, and the smallest odd prime number is 3. Multiplying them yields 4 * 3 = 12.

Count whole numbers less than 10 that are neither prime nor composite to find R.

R = 2

The set of whole numbers strictly less than 10 is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Only 0 and 1 fit the classification of being neither prime nor composite.

Find the smallest positive integer n making \sqrt{12n} rational.

S = 3

For the square root to be rational, 12n must be a perfect square. Since 12 factors to 2^2 * 3, the smallest integer n to pair the remaining 3 is 3.

Evaluate the final expression.

(P + Q) - (R * S) = (17 + 12) - (2 * 3) = 29 - 6 = 23.

Key Concept

Classification properties of primes, composites, whole numbers, and rational numbers.

Question 398Question →

Six friends (P, Q, R, S, T, and U) are seated in a straight line facing North. Read the following information carefully:

1. P sits third to the left of S.
2. Neither P nor S sits at the extreme ends of the line.
3. R sits to the immediate right of S.
4. T is not an immediate neighbor of P.
5. Q sits to the left of U.

Based on the final seating arrangement, match the individuals (List I) with their correct positional descriptions (List II).

Click a left item, then click its matching right item

Show answer & explanation

Answer

The final arrangement from left to right is Q, P, U, T, S, R. Therefore, Q sits at the extreme left, U sits between P and T, T sits immediately left of S, and S sits second to the right of U.

The fully deduced seating arrangement from left to right is Q, P, U, T, S, R. Matching each person to their derived positional fact yields the correct alignment without any contradictions.

Step-by-Step Solution

Determine the exact positions of P and S.

P is at position 2 and S is at position 5.

Since P is third to the left of S, the possible pairs for (P, S) are (1, 4), (2, 5), and (3, 6). However, neither P nor S can sit at the extreme ends (positions 1 and 6). Thus, the only valid placement is P at 2 and S at 5.

Place R in the arrangement.

R is at position 6.

The condition states R is to the immediate right of S. Since S is at position 5, R must take position 6.

Place T in the arrangement.

T is at position 4.

The remaining empty positions are 1, 3, and 4. T cannot be an immediate neighbor of P (who is at 2), meaning T cannot be at 1 or 3. Therefore, T must take position 4.

Place Q and U in the arrangement.

Q is at position 1 and U is at position 3.

The only remaining spots are 1 and 3. Since Q must sit to the left of U, Q takes position 1 and U takes position 3.

Key Concept

Deducing linear seating arrangements using fixed reference points and elimination.

Question 399Question →

Consider the following statement and the two candidate assumptions that follow it.

Statement:
In an effort to promote digital literacy and expand market reach, the state handicraft board has announced a free three-month online marketing training program for rural artisans, which will be exclusively accessible via smartphones.

Assumptions:
I. A sufficient proportion of rural artisans currently have access to smartphones and basic internet connectivity.
II. The completion of this online program will guarantee an immediate and substantial increase in the sales revenue of the participating artisans.

Which of the given assumptions is/are implicit in the statement?

Only assumption I is implicit

Only assumption II is implicit

Neither assumption I nor II is implicit

Both assumptions I and II are implicit

Show answer & explanation

Answer: Only assumption I is implicit

Answer

The statement relies on the assumption that rural artisans have the necessary devices to participate, meaning only the first assumption is implicit.

The correct answer states that only the first assumption is implicit. When an authority launches a program through a specific medium (smartphones), it inherently assumes that the target demographic has the ability to access that medium, making the first assumption logically necessary. The second assumption is rejected because predicting a 'guaranteed immediate increase' in revenue is an extreme logical leap and an inference about a future outcome, rather than an unstated premise required to launch the program.

Step-by-Step Solution

Analyze the core action in the given statement.

The state handicraft board is launching an online marketing training program exclusively via smartphones for a specific group: rural artisans.

Identifying the primary action helps in determining what underlying conditions must exist for the proposed action to be viable.

Evaluate Assumption I against the statement.

Assumption I is implicit. For the board to invest in and launch a program exclusively via smartphones, they must take for granted that the intended users (rural artisans) actually have access to these devices.

An assumption is a hidden, foundational premise. The program cannot function as described without this premise being true.

Evaluate Assumption II against the statement.

Assumption II is not implicit. The phrasing 'guarantee an immediate and substantial increase' makes this an extreme prediction, not a foundational requirement for initiating the training.

An implicit assumption must be something the author relies on to make the statement logical, not a speculative future outcome or ultimate goal.

Key Concept

Identifying Implicit Premises vs. Extreme Inferences

Question 400Question →

A mechanical watch, which gains time at a continuous and uniform rate, is observed to be exactly 5 minutes slow at 8:00 AM on a Sunday. By 8:00 AM on the immediately following Tuesday, the same watch is exactly 7 minutes fast. How many hours after 8:00 AM on that Sunday did the watch display the true, correct time?

Show answer & explanation

Answer: 20

Answer

The correct answer is derived by mapping the total error shift over the total elapsed time. The watch gains a total of 12 minutes over a span of 48 hours. This establishes a constant gain rate of 1 minute every 4 hours. Because the watch started exactly 5 minutes slow, it requires

5 \times 4 = 20

hours of elapsed true time for it to catch up and display the correct, synchronized time.

Step-by-Step Solution

Calculate the total true time elapsed between the two observations.

48 hours

To find the rate at which the watch gains time, we first need to determine the total duration of the period. From Sunday 8:00 AM to Tuesday 8:00 AM is precisely two full days, which equals 48 hours.

Calculate the total amount of time the watch gained over this period.

12 minutes

The watch transitions from being 5 minutes slow (-5) to being 7 minutes fast (+7). The total change in its display relative to true time is

7 - (-5) = 12

minutes.

Determine the uniform rate at which the watch gains time.

1 minute gained every 4 hours

By dividing the total elapsed time by the total minutes gained (

48 \div 12

), we discover that the watch gains exactly 1 minute for every 4 hours of true time that pass.

Calculate the hours needed to make up the initial 5-minute deficit.

20 hours

In order to display the correct time, the watch must gain precisely the 5 minutes it was initially lagging behind. At a steady rate of 1 minute per 4 hours, this process will take

5 \times 4 = 20

hours.

Key Concept

Uniform time gain and synchronization of faulty clocks

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