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Question 4461Question →

A maritime shipping contract was enacted on Wednesday, October 14, 1795. The agreement stipulated a mandatory review and renegotiation exactly 14 years later, on October 14, 1809. On what day of the week did the renegotiation fall?

Friday

Sunday

Saturday

Wednesday

Show answer & explanation

Answer: Saturday

Answer

Saturday

The interval is exactly 14 years. Within this period, there are 3 leap years: 1796, 1804, and 1808. The year 1800 is a century year not divisible by 400, so it is not a leap year. This creates a total of 14 + 3 = 17 odd days. Dividing 17 by 7 gives a remainder of 3. Advancing 3 days from Wednesday results in Saturday.

Step-by-Step Solution

Calculate the total number of years elapsed between the two dates.

1809 - 1795 = 14 years.

Each normal year contributes 1 odd day, so determining the year gap gives the base number of odd days.

Identify the number of leap years occurring strictly within this 14-year period.

The leap years are 1796, 1804, and 1808. The year 1800 is not a leap year. Total = 3 leap years.

Century years (ending in 00) must be divisible by 400 to be leap years. Each valid leap year adds 1 additional odd day.

Calculate the total number of odd days.

14 (base odd days) + 3 (additional leap year days) = 17 total odd days.

Combining base year odd days with extra leap year odd days yields the complete shift in the weekly cycle.

Find the remainder when the total odd days are divided by 7, and add it to the starting day.

17 divided by 7 leaves a remainder of 3. Wednesday + 3 days = Saturday.

The days of the week repeat on a 7-day cycle, so only the remainder affects the final day.

Key Concept

Calculating odd days across century boundaries and identifying century leap years.

Question 4462Question →

In a certain coding language, words are translated by shifting each letter forward in the English alphabet based on its positional index within the word. The first letter is shifted by one position, the second letter by two positions, the third letter by three positions, and so on. Based on this translation rule, match the given words on the left with their correctly coded equivalents on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

STAR matches with TVDV, WIND matches with XKQH, LOCK matches with MQFO, and PUMP matches with QWPT.

By successfully applying a variable forward shift (+1 for the first letter, +2 for the second, +3 for the third, and +4 for the fourth) to each original word, we generate their exact coded counterparts.

Step-by-Step Solution

Identify the core logic for the encryption method.

Each letter is shifted forward by its index position in the word (+1, +2, +3, +4).

This establishes the rule needed to decode or encode all the given words.

Apply the progressive shift to the word 'STAR'.

S(+1) = T, T(+2) = V, A(+3) = D, R(+4) = V. The encoded word is TVDV.

To accurately pair the first item.

Apply the progressive shift to the word 'WIND'.

W(+1) = X, I(+2) = K, N(+3) = Q, D(+4) = H. The encoded word is XKQH.

To accurately pair the second item.

Apply the progressive shift to the word 'LOCK'.

L(+1) = M, O(+2) = Q, C(+3) = F, K(+4) = O. The encoded word is MQFO.

To accurately pair the third item.

Apply the progressive shift to the word 'PUMP'.

P(+1) = Q, U(+2) = W, M(+3) = P, P(+4) = T. The encoded word is QWPT.

To accurately pair the final item.

Key Concept

Positional Letter Shifting

Question 4463Question →

Read the given statements and determine which of the conclusions can be logically drawn from them.

Statement I: According to the new administrative guidelines, any public grievance petition that remains pending for more than 45 days is automatically escalated to the State Vigilance Commission.
Statement II: A significant number of petitions regarding rural water supply have remained unresolved for the past two months.

Conclusion I: Several petitions related to rural water supply will be escalated to the State Vigilance Commission.
Conclusion II: The department handling rural water supply is the most inefficient department in the state administration.

Which of the following options 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 evaluates both conclusions. Conclusion I is a direct logical deduction because a period of two months exceeds the 45-day threshold, triggering the automatic escalation detailed in the first statement. Conclusion II is an extreme, unwarranted generalization because the text offers no comparative data about other departments.

Step-by-Step Solution

Analyze Statement I

Establishes a strict rule: if a petition is pending > 45 days, it gets escalated to the State Vigilance Commission.

To identify the necessary condition for an action to occur.

Analyze Statement II

Identifies that rural water supply petitions have been pending for two months.

To compare the provided case against the established rule.

Evaluate Conclusion I

Conclusion I is logically valid.

Since 'two months' (approx. 60 days) is strictly greater than the 45-day threshold, the petitions satisfy the condition in Statement I and must be escalated.

Evaluate Conclusion II

Conclusion II is logically invalid.

The statements only discuss delays in one department. There is no comparative data to deduce that it is the 'most inefficient' department overall.

Key Concept

Statement and Conclusions or Inferences

Question 4464Question →

You are the Chief Engineer of the State Public Works Department, supervising a major bridge construction project. The project is nearing completion and faces intense political pressure to open before the upcoming state elections. During a routine site visit, a junior safety officer privately informs you of a systemic flaw in the concrete curing process used by the primary contractor. While this flaw will not cause an immediate collapse, it significantly compromises the bridge's long-term structural integrity. Fixing the issue requires halting current operations for two weeks. The contractor, who is politically well-connected, insists that the flaw is minor and pushes to continue working, arguing that a halt will cause unacceptable political and financial delays. Which of the following is the most appropriate immediate course of action for you to take?

Temporarily suspend the concrete pouring operations to independently verify the report, and formally mandate the contractor to correct any verified deviations before work resumes.

Immediately terminate the contractor's contract and file criminal charges against their management for endangering public safety, bypassing the verification process to send a strong message.

Permit the construction to proceed on schedule to meet the deadline, but informally advise the contractor to make minor adjustments without documenting the flaw in official records.

Instantly deploy law enforcement to confiscate the contractor's equipment and order the immediate demolition of the affected pillars without waiting for a comprehensive structural assessment.

Show answer & explanation

Answer: Temporarily suspend the concrete pouring operations to independently verify the report, and formally mandate the contractor to correct any verified deviations before work resumes.

Answer

The most appropriate action is to temporarily suspend operations for independent verification and strictly enforce the correction of any verified deviations before continuing.

The correct action demonstrates a balanced, objective, and protocol-driven approach. By temporarily pausing the work, the engineer prevents further potential damage while ensuring the junior officer's report is formally verified. It upholds public safety and administrative transparency without bowing to political pressure or taking premature, disproportionate actions against the contractor.

Step-by-Step Solution

Analyze the core conflict.

The ethical tension is between the long-term structural integrity of a public asset (statutory duty to public safety) and short-term political deadlines (external pressure).

Identifying primary stakeholders and risks is the foundational step in administrative decision-making.

Evaluate procedural requirements.

An unverified claim by a junior officer requires formal validation before taking definitive contractual action, but the risk is serious enough to warrant a temporary pause in operations.

Administrative actions must be based on verified facts, proper documentation, and due process.

Determine the proportional response.

Suspending work to verify the claim balances safety with fairness, ensuring public interest is protected without resorting to premature punitive measures or unethical cover-ups.

Proportionality and transparency are key tenets of ethical public administration.

Key Concept

Administrative propriety, due process, and prioritizing public safety over political expediency

Estimated Time:1m 30s

Question 4465Question →

If the statements 'All expressways are toll roads' and 'Some toll roads are privately maintained' are both true, then it necessarily follows that 'Some expressways are privately maintained'.

True

False

Show answer & explanation

Answer: False

Answer

The statement is False because the logical deduction is invalid.

The proposed logical deduction is invalid because it fails to distribute the middle term in either premise, making the conclusion uncertain.

Step-by-Step Solution

Identify the logical structure and terms of the given premises.

The first premise is 'All E are T' (Universal Affirmative). The second premise is 'Some T are P' (Particular Affirmative). The proposed conclusion is 'Some E are P'.

Standardizing the categorical propositions helps in applying formal syllogistic rules or drawing Venn diagrams.

Analyze the distribution of the middle term ('toll roads' or 'T').

In the universal affirmative statement 'All E are T', the predicate 'T' is undistributed. In the particular affirmative statement 'Some T are P', the subject 'T' is also undistributed.

A valid categorical syllogism requires the middle term to be distributed in at least one of the premises.

Evaluate the validity of the conclusion based on the distribution.

Because the middle term is undistributed in both premises, no definitive relationship can be established between 'E' (expressways) and 'P' (privately maintained).

Without a distributed middle term, the specific toll roads that are expressways might not intersect at all with the specific toll roads that are privately maintained.

Key Concept

Validity of Syllogisms and the Fallacy of the Undistributed Middle

Question 4466Question →

A state-backed wind energy corporation's plan to expand its farm into traditional grazing territories has triggered aggressive blockades by local pastoralists. Concurrently, local wildlife NGOs are threatening litigation over procedural lapses in the project's environmental clearance. As the appointed District Collector handling this volatile situation, arrange the following administrative interventions in the most optimal sequence to de-escalate the standoff and achieve a sustainable negotiated settlement.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The optimal sequence is to first enforce a status quo and conduct fact-finding, followed by separate confidential consultations, then form a joint committee, conduct mediated joint negotiations, and finally ratify a formal MoU.

The correct sequence adheres to the standard public administration protocol for complex stakeholder conflicts: immediate de-escalation (status quo), independent assessment (fact-finding), private trust-building (separate consultations), structured dialogue (joint committee), active mediation (joint negotiation), and final institutionalization (MoU). This progressive approach systematically builds trust and manages volatility before opposing parties face each other.

Step-by-Step Solution

Establish immediate control and gather objective facts.

Temporary status quo enforced and fact-finding team deployed.

Prevents immediate violence and provides an objective administrative baseline for all future discussions.

Understand individual stakeholder positions privately.

Separate, confidential consultations are held.

Allows parties to express genuine concerns and bottom lines without losing face in front of their opponents.

Create a structured, transparent platform for dialogue.

Constitution of a formal multi-stakeholder joint committee.

Brings all parties to a single, officially recognized forum once the necessary groundwork and trust-building are complete.

Conduct the core conflict resolution process.

Mediated joint negotiation sessions occur.

Facilitates direct problem-solving, expectation management, and trade-offs between the conflicting groups.

Formalize the negotiated agreement.

Drafting and ratification of a legally binding MoU.

Ensures the negotiated settlement is recognized by the state and holds all parties accountable to the agreed terms.

Key Concept

Sequential Administrative Conflict Resolution and Negotiation Strategy

Question 4467Question →

If the 9-digit number $94x257y36$ is exactly divisible by $88$ , what is the maximum possible value of the mathematical expression $(5x - 3y)$ ?

Show answer & explanation

Answer: 27

Answer

By applying the divisibility rules of 8 and 11, we isolate a finite set of possible digit combinations for x and y. Evaluating

(5x - 3y)

across all mathematically valid pairs proves that the pair

x=6, y=1

produces the highest numerical output of 27.

Step-by-Step Solution

Determine the co-prime factors of the divisor.

The number must be divisible by 8 and 11, since 88 = 8 * 11 and GCD(8, 11) = 1.

Applying individual divisibility rules for co-prime factors is the standard method for composite divisors.

Apply the divisibility rule for 8 to the last three digits (

y36

The possible values for

y

are 1, 3, 5, 7, and 9.

A number is divisible by 8 if its last three digits form a multiple of 8. 136, 336, 536, 736, and 936 all yield a remainder of 0 when divided by 8.

Apply the divisibility rule for 11 to the entire number.

The expression

(x + y + 4)

must be a multiple of 11.

The rule for 11 requires the difference between the alternating sum of digits to be divisible by 11. Odd positions sum to

(20 + x + y)

and even positions sum to

16

. Difference =

(x + y + 4)

Find all valid

(x, y)

coordinate pairs.

The valid digit pairs are (6, 1), (4, 3), (2, 5), (0, 7), and (9, 9).

Since

x

and

y

are single digits between 0 and 9,

(x + y + 4)

can only equal 11 or 22. This implies

x + y = 7

x + y = 18

Evaluate the target expression

(5x - 3y)

for all valid pairs.

The values obtained are 27, 11, -5, -21, and 18 respectively. The absolute maximum is 27.

Testing all valid edge cases guarantees that the maximum possible value is correctly identified without assumptions.

Key Concept

Solving for unknown digits using combined co-prime divisibility rules and optimizing a linear algebraic expression based on finite digit constraints.

Estimated Time:2m 30s

Question 4468Question →

Analyze the statements below regarding financial governance:

Statement 1: All sovereign wealth funds are state-owned investment vehicles.
Statement 2: Some state-owned investment vehicles prioritize domestic infrastructure.
Statement 3: No entity prioritizing domestic infrastructure is exempt from parliamentary oversight.

Based strictly on the information provided, which of the following conclusions can be logically deduced? (Select all that apply)

Select all that apply

Some sovereign wealth funds are not exempt from parliamentary oversight.

Some state-owned investment vehicles are not exempt from parliamentary oversight.

No entity exempt from parliamentary oversight prioritizes domestic infrastructure.

Sovereign wealth funds typically prioritize domestic infrastructure over foreign investments.

Show answer & explanation

Answer: Some state-owned investment vehicles are not exempt from parliamentary oversight.; No entity exempt from parliamentary oversight prioritizes domestic infrastructure.

Answer

The logically valid conclusions are that some state-owned investment vehicles are not exempt from parliamentary oversight, and that no entity exempt from parliamentary oversight prioritizes domestic infrastructure.

By strictly mapping the categories: since some state-owned investment vehicles prioritize domestic infrastructure, and none of those prioritizing domestic infrastructure are exempt from oversight, it logically follows that those specific state-owned vehicles are not exempt. Additionally, the premise 'no entity prioritizing domestic infrastructure is exempt' logically converts directly to 'no exempt entity prioritizes domestic infrastructure'.

Step-by-Step Solution

Define the categorical groups and their relationships based on the three statements.

Let S = Sovereign wealth funds, V = State-owned investment vehicles, D = Entities prioritizing domestic infrastructure, and E = Entities exempt from oversight. We have: (1) All S are V. (2) Some V are D. (3) No D are E.

Mapping the sets clearly is required to evaluate intersections and logical boundaries.

Test the conclusion that some state-owned investment vehicles are not exempt from oversight.

Because some V are D (Statement 2), and absolutely no D are E (Statement 3), the portion of V that is D cannot be E. Thus, some V are definitely not E. This makes the conclusion valid.

This confirms the validity of the first correct inference by strictly combining Statements 2 and 3.

Test the conclusion regarding exempt entities and domestic infrastructure.

Statement 3 explicitly dictates that the set of D and the set of E have zero overlap (No D are E). By logical conversion, this guarantees that no E are D.

This proves the validity of the second correct inference through direct logical conversion.

Test the remaining conclusions for unwarranted assumptions.

We cannot assume that any S are D because the 'some V' that are D might not include any S. Furthermore, we cannot introduce outside knowledge regarding the general investment priorities of sovereign wealth funds.

Conclusions must be drawn solely from the explicit text provided, without conflating possibilities with certainties.

Key Concept

Statement and Conclusions or Inferences

Question 4469Question →

As the District Forest Conservator, you are managing a dispute over a newly sanctioned state eco-tourism project situated in a wildlife buffer zone. Indigenous communities are blocking access roads, arguing that the project violates their legally recognized traditional foraging rights. Meanwhile, a local transport workers' union is staging a counter-protest, demanding immediate resumption of the project for employment generation. With tensions escalating and minor physical altercations occurring, which of the following is the most appropriate initial administrative strategy to resolve the conflict?

Suspend construction activities temporarily to prevent further violence, and convene a structured multi-stakeholder mediation session to review the statutory foraging claims alongside the project's employment benefits.

Deploy the state reserve police immediately to forcibly clear the blockade and detain the protest organizers, ensuring the government-funded project continues without further financial loss.

Issue a public ultimatum declaring that any party failing to disperse within twelve hours will face permanent legal injunctions and heavy financial penalties.

Unilaterally revoke the eco-tourism project's environmental clearance on the spot and yield full territorial control to the indigenous groups to permanently neutralize the immediate threat.

Show answer & explanation

Answer: Suspend construction activities temporarily to prevent further violence, and convene a structured multi-stakeholder mediation session to review the statutory foraging claims alongside the project's employment benefits.

Answer

The correct strategy balances the immediate need for maintaining law and order with a long-term, legally sound resolution mechanism. By temporarily pausing construction, the administrator effectively de-escalates the physical confrontation. Subsequently initiating a multi-stakeholder dialogue ensures that both the statutory rights of the indigenous community and the economic concerns of the transport union are formally addressed within the framework of the law, avoiding authoritarian overreach.

Step-by-Step Solution

Identify the primary stakeholders and their core interests.

Indigenous groups are defending statutory foraging rights, while the transport union seeks economic and employment opportunities.

Understanding the root drivers of the conflict is essential for mediation.

Assess the immediate operational priority on the ground.

The immediate priority is de-escalating physical tensions and preventing further violence between the two opposing groups.

Administrative action must first ensure public safety before addressing policy nuances.

Select a proportional, protocol-driven intervention.

A temporary halt combined with multi-stakeholder mediation is chosen over extreme force or unilateral capitulation.

This strategy adheres to civil service conduct rules, ensures due process, and fosters a sustainable, negotiated consensus.

Key Concept

Stakeholder Conflict Resolution and Negotiation Strategy

Question 4470Question →

Determine the exact decimal value of the Least Common Multiple (LCM) for the two fractions $\frac{3}{5}$ and $\frac{9}{10}$ .

Show answer & explanation

Answer: 1.8

Answer

1.8

To find the LCM of fractions, we use the standard formula: LCM(numerators) / HCF(denominators). The numerators are 3 and 9, and their LCM is 9. The denominators are 5 and 10, and their HCF is 5. Dividing the LCM of the numerators by the HCF of the denominators gives 9/5. Converting this fraction to a decimal yields exactly 1.8.

Step-by-Step Solution

Identify the correct formula for finding the LCM of fractions.

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

This is the fundamental rule for calculating the least common multiple when dealing with rational numbers.

Calculate the Least Common Multiple of the numerators.

The numerators are 3 and 9. Their LCM is 9.

9 is the smallest integer that is perfectly divisible by both 3 and 9.

Calculate the Highest Common Factor of the denominators.

The denominators are 5 and 10. Their HCF is 5.

5 is the largest integer that perfectly divides both 5 and 10 without leaving a remainder.

Apply the formula and convert the resulting fraction to a decimal.

LCM = 9 / 5 = 1.8

The question specifically asks for the exact decimal value, so the fraction 9/5 must be converted to 1.8.

Key Concept

The Least Common Multiple (LCM) of a set of fractions is found by dividing the LCM of their numerators by the HCF of their denominators.

Question 4471Question →

Study the logical relationship between the first pair of letter-clusters. Applying the identical underlying rule, what will replace the question mark to complete the second pair?

$\text{BIRD} : \text{ERUW} :: \text{FISH} : \text{?}$

ILVK

HRUS

IRVS

ULHK

Show answer & explanation

Answer: IRVS

Answer

IRVS

The underlying pattern alternates between a forward alphabetical shift of 3 positions and finding the reverse alphabetical equivalent letter. Applying this strictly to the word 'FISH' yields 'I' (F+3), 'R' (reverse of I), 'V' (S+3), and 'S' (reverse of H), resulting in 'IRVS'.

Step-by-Step Solution

Analyze the transformation from BIRD to ERUW position by position.

1st letter (B to E) is a +3 forward shift. 2nd letter (I to R) is the reverse alphabetical pair. 3rd letter (R to U) is a +3 forward shift. 4th letter (D to W) is the reverse alphabetical pair.

Establishing the exact sequence of logical operations applied to the first pair is necessary to solve the analogy.

Apply the identical transformation rule to the target word FISH.

1st letter (F) + 3 = I. 2nd letter (I) reverse = R. 3rd letter (S) + 3 = V. 4th letter (H) reverse = S.

The analogy requires the second pair to exhibit the exact same multi-step operational pattern as the first pair.

Key Concept

Alphabet/Letter Pattern Analogy using compound positional rules

Question 4472Question →

An autonomous delivery drone navigation system encodes geographic sector names into numerical strings for secure broadcasting. The sector name "DELTA" is transmitted as 46-44-30-14-52, and the sector name "BRAVO" is transmitted as 50-18-52-10-24. Based on this logic, which numerical string represents the sector name "EAGLE"?

44-52-40-30-44

10-2-14-24-10

22-26-20-15-22

42-50-38-28-42

Show answer & explanation

Answer: 44-52-40-30-44

Answer

The string 44-52-40-30-44 correctly represents the sector name EAGLE.

The correct answer accurately applies the coding logic: it determines the reverse alphabetical position of each letter (found by subtracting the forward position from 27) and multiplies that value by 2. For the word EAGLE: E is 5th, so its reverse is 22 (22 × 2 = 44). A is 1st, reverse is 26 (26 × 2 = 52). G is 7th, reverse is 20 (20 × 2 = 40). L is 12th, reverse is 15 (15 × 2 = 30). Combining these gives 44-52-40-30-44.

Step-by-Step Solution

Analyze the numerical values assigned to the letters in 'DELTA' and 'BRAVO' to find the base transformation.

D corresponds to 46, E to 44, L to 30, T to 14, and A to 52.

Identifying the relationship between the first few letters and their numerical outputs reveals the underlying logic.

Determine the mathematical rule applied to the alphabetical positions.

The forward position of D is 4. Its reverse position (27 - 4) is 23. Multiplying 23 by 2 gives 46. This logic holds for all letters.

A consistent pattern must be established that works for every letter in both provided examples.

Apply the established logic to the target word 'EAGLE'.

E (5) -> reverse 22 -> 44. A (1) -> reverse 26 -> 52. G (7) -> reverse 20 -> 40. L (12) -> reverse 15 -> 30. E (5) -> reverse 22 -> 44.

Applying the exact rule ensures the correct encryption of the target string.

Key Concept

Reverse Alphabetical Indexing and Mathematical Transformation

Estimated Time:1m 30s

Question 4473Question →

Match the following administrative scenarios with the core ethical principle or statutory doctrine that must guide the civil servant's resolution.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The vendor penalty aligns with proportionality and empathy. The election interference aligns with political impartiality. The spouse's trust aligns with conflict of interest and recusal. The corporate job offer aligns with probity and preventing quid pro quo.

Each scenario maps to a distinct pillar of administrative ethics. Proportionality ensures punishments fit the offense. Impartiality protects democratic institutions from political interference. Recusal prevents personal biases from tainting procurement. Probity prevents the misuse of regulatory authority for future personal gain.

Step-by-Step Solution

Analyze the street vendor scenario.

Identified a tension between rigid rule enforcement and the severe socioeconomic impact on a vulnerable individual.

To map the situation to the Doctrine of Proportionality and Empathetic Governance.

Analyze the election polling scenario.

Identified external pressure from the political executive attempting to undermine the democratic process.

To map the situation to Political Impartiality and Statutory Independence.

Analyze the infrastructure bidding scenario.

Identified a personal and familial interest overlapping with official public duties.

To map the situation to the Duty of Recusal and Avoidance of Conflict of Interest.

Analyze the mining conglomerate job offer scenario.

Identified a potential deferred bribe disguised as post-retirement employment.

To map the situation to Probity in Public Life and Prevention of Quid Pro Quo.

Key Concept

Core principles of public service ethics applied to practical administrative dilemmas.

Question 4474Question →

A geologist leaves her base camp and drives a rover $30\text{ km}$ straight North. From there, she turns $135^{\circ}$ to her right and travels $10\sqrt{2}\text{ km}$ in a straight line. She then turns $45^{\circ}$ to her left and travels $20\text{ km}$ . Finally, she takes a $90^{\circ}$ turn to her right and drives another $20\text{ km}$ . What is her final position relative to the base camp?

30\text{ km}

West

10\text{ km}

West

30\text{ km}

East

30\text{ km}

South

Show answer & explanation

Answer: $30\text{ km}$ East

Answer

30\text{ km}

East

By tracing the movements on a Cartesian plane where North is the positive y-axis and East is the positive x-axis, the sequence unfolds as follows: the initial position is

(0, 30)

after the first drive. A

135^{\circ}

right turn points the rover South-East, and moving

10\sqrt{2}\text{ km}

adds a displacement of

(10, -10)

, resulting in position

(10, 20)

. A

45^{\circ}

left turn points the rover East, adding

(20, 0)

to reach

(30, 20)

. The final

90^{\circ}

right turn points the rover South, adding

(0, -20)

to land precisely at

(30, 0)

. The point

(30, 0)

is exactly

30\text{ km}

East of the starting base camp at

(0, 0)

Step-by-Step Solution

Map the first movement from the base camp.

Starting at origin

(0, 0)

, moving

30\text{ km}

North reaches coordinate

(0, 30)

while facing North.

Establishing the initial position and orientation on a Cartesian plane (where North is +y and East is +x) is necessary for calculating subsequent turns.

Calculate the second movement after a

135^{\circ}

right turn.

Turning

135^{\circ}

right from North changes the heading to South-East. Traveling

10\sqrt{2}\text{ km}

South-East yields a displacement of

+10

in the x-direction and

-10

in the y-direction. The new position is

(10, 20)

135^{\circ}

clockwise turn from North equates to a South-East direction. Using trigonometry,

10\sqrt{2} \times \cos(45^{\circ}) = 10

and

10\sqrt{2} \times \sin(45^{\circ}) = 10

for the respective axis displacements.

Calculate the third movement after a

45^{\circ}

left turn.

Turning

45^{\circ}

left (counter-clockwise) from South-East aligns the heading to East. Traveling

20\text{ km}

East shifts the position by

+20

on the x-axis, bringing the coordinates to

(30, 20)

Adjusting the South-East heading by

45^{\circ}

back towards the counter-clockwise direction points exactly East.

Calculate the final movement after a

90^{\circ}

right turn.

Turning

90^{\circ}

right (clockwise) from East changes the heading to South. Traveling

20\text{ km}

South shifts the position by

-20

on the y-axis, resulting in a final coordinate of

(30, 0)

90^{\circ}

right turn from East is South. Moving

20\text{ km}

South precisely negates the remaining

+20\text{ km}

vertical displacement.

Determine the final relative position.

The final coordinate is

(30, 0)

while the base camp is at

(0, 0)

. This represents a displacement of

30\text{ km}

directly East.

Comparing the final position coordinates directly to the origin determines the exact distance and direction.

Key Concept

Vector displacement combining angular turns and cardinal directions

Question 4475Question →

Consider the number $P = 4 + 7 \times 38^{99}$ . What is the remainder when $P$ is divided by $13$ ?

Show answer & explanation

Answer: 10

Answer

The correct remainder is found by applying modular arithmetic step-by-step. First,

38

modulo

13

simplifies to

-1

. Because

99

is an odd power,

(-1)^{99}

equals

-1

. Multiplying this by

7

yields

-7

. Adding

4

gives an intermediate remainder of

-3

. To find the true positive remainder, we add the divisor

13

-3

, resulting in

10

Step-by-Step Solution

Simplify the base of the exponentiated term modulo 13.

38 \equiv -1 \pmod{13}

Using a negative remainder that is close to zero makes evaluating large exponential powers mathematically manageable.

Evaluate the exponentiated term modulo 13.

(-1)^{99} = -1

A negative base raised to an odd power retains its negative sign.

Substitute the simplified term back into the original expression and apply the correct order of operations.

4 + 7 \times (-1) = 4 - 7 = -3

Standard order of operations requires multiplication to be performed before addition.

Convert the negative intermediate remainder into a valid positive remainder.

-3 \equiv -3 + 13 = 10 \pmod{13}

Remainders must be positive integers strictly less than the divisor.

Key Concept

Modular Arithmetic with Negative Bases and Order of Operations

Alternative Method

One could alternatively find the positive remainder of 38 modulo 13, which is 12, and then evaluate

4 + 7 \times 12^{99} \pmod{13}

. However, this requires observing the cyclicity pattern of powers of 12 modulo 13, making the negative base approach (-1) far more direct and efficient.

Estimated Time:1m 0s

Question 4476Question →

A metropolitan public library system launched a project to digitize a total of $12,000$ rare manuscripts. The manuscripts were distributed among three main facilities: the Central Library, the North Branch, and the South Branch in the ratio of $5:4:3$ , respectively. After one month of continuous operations, a progress report revealed that the Central Library had successfully digitized $60\%$ of its allocated manuscripts. The North Branch had completed $45\%$ of its assigned quota, while the South Branch managed to digitize $70\%$ of the manuscripts allocated to it.

Based on the information provided in the passage, which of the following statements are correct? (Select all that apply)

Select all that apply

The North Branch currently has a greater number of manuscripts left to digitize compared to the Central Library.

The three facilities combined successfully digitized a total of

6,900

manuscripts during the first month.

The South Branch accounted for exactly

70\%

of the total manuscripts digitized across all three facilities combined.

The Central Library digitized

1,200

fewer manuscripts than the North Branch during the first month.

Show answer & explanation

Answer: The North Branch currently has a greater number of manuscripts left to digitize compared to the Central Library.; The three facilities combined successfully digitized a total of $6,900$ manuscripts during the first month.

Answer

The correct statements are that the North Branch has more manuscripts left to digitize than the Central Library, and that the three facilities combined digitized a total of 6,900 manuscripts.

Based on the data derived from the caselet, the three facilities digitized a total of

3,000 + 1,800 + 2,100 = 6,900

manuscripts. Additionally, the North Branch has

2,200

manuscripts remaining (

4,000 - 1,800

), which is strictly greater than the Central Library's

2,000

remaining manuscripts (

5,000 - 3,000

Step-by-Step Solution

Calculate the initial allocation of manuscripts for each facility based on the

5:4:3

ratio.

Central Library:

5,000

; North Branch:

4,000

; South Branch:

3,000

The total ratio parts are

5 + 4 + 3 = 12

. Each part is worth

12,000 / 12 = 1,000

manuscripts. Multiplying by each facility's ratio gives their total quota.

Calculate the number of manuscripts digitized by each facility in the first month.

Central:

3,000

; North:

1,800

; South:

2,100

Multiply each facility's allocated quota by their respective completion percentage (e.g.,

60\%

5,000 = 3,000

Calculate the number of manuscripts remaining to be digitized for each facility.

Central:

2,000

; North:

2,200

; South:

900

Subtract the number of digitized manuscripts from the initial quota for each facility.

Evaluate the accuracy of the given statements using the extracted data.

The total digitized is

3,000 + 1,800 + 2,100 = 6,900

. North remaining (

2,200

) is greater than Central remaining (

2,000

Comparing the processed numbers against the claims confirms which statements are factually correct based on the passage.

Key Concept

Extracting proportional allocations and applying distinct percentage rates to multi-entity caselet data.

Estimated Time:2m 0s

Question 4477Question →

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

Statement:
"To encourage private car commuters to shift to public transit, the City Transit Bureau is launching a fleet of premium Wi-Fi-enabled express buses on major IT corridor routes, with fares set at triple the standard bus ticket price."

Assumptions:
I. A significant portion of private car commuters value travel amenities and comfort enough to pay a premium over standard public transit fares.
II. The introduction of these premium buses will successfully reduce vehicular emissions and alleviate peak-hour traffic jams on the IT corridors.

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

Only assumption II is implicit

Neither assumption I nor II is implicit

Only assumption I is implicit

Both assumption I and II are implicit

Show answer & explanation

Answer: Only assumption I is implicit

Answer

Only assumption I is implicit

Assumption I is a valid implicit premise. For the Transit Bureau's strategy—charging triple fare for Wi-Fi buses to attract car users—to be logical, they must assume that the target demographic is willing to pay this higher price for added comfort. Assumption II, however, is an expected future outcome or consequence of the policy, not a foundational premise required for the statement to be made. Therefore, only the first assumption is logically implicit.

Step-by-Step Solution

Analyze the core action and goal presented in the statement.

The goal is to shift car commuters to public transit. The action is providing premium Wi-Fi buses at triple the normal fare.

Identifying the author's primary objective and method is required to deduce what they must be assuming for the plan to work.

Evaluate Assumption I against the statement's logic.

Assumption I is implicit. The Bureau is offering a high-cost, high-comfort option specifically to attract car users.

If the Bureau did not assume car users were willing to pay this premium for comfort, launching this specific service to attract them would be completely illogical.

Evaluate Assumption II against the definition of a logical assumption.

Assumption II is not implicit. It describes the expected future result of the policy, not a premise.

An assumption is an unstated premise necessary for the argument to be valid before action is taken, whereas reducing emissions is an inference or consequence that follows the action.

Key Concept

Distinguishing between implicit foundational premises and intended logical consequences or factual outcomes.

Estimated Time:1m 15s

Question 4478Question →

The product of two positive integers is $2160$ and their Highest Common Factor (HCF) is $12$ . What is their Least Common Multiple (LCM)?

Show answer & explanation

Answer: 180

Answer

180

For any two positive integers, the product of their Highest Common Factor (HCF) and Least Common Multiple (LCM) is always equal to the product of the numbers themselves. Given that the product is

2160

and the HCF is

12

, the LCM is calculated by rearranging the formula to

\text{LCM} = \frac{\text{Product}}{\text{HCF}}

. Substituting the values gives

2160 \div 12 = 180

Step-by-Step Solution

Identify the fundamental formula connecting HCF, LCM, and the product of two positive integers.

\text{HCF} \times \text{LCM} = \text{Product}

This relationship allows you to find one missing value when the other two are known.

Substitute the known values from the problem into the equation.

12 \times \text{LCM} = 2160

The problem explicitly states that the product is

2160

and the HCF is

12

Solve for the LCM by dividing both sides of the equation by

12

\text{LCM} = 180

Isolating the LCM variable provides the final requested value.

Key Concept

Relationship between the product of two numbers and their HCF and LCM

Question 4479Question →

At exactly $4:28$ PM, what is the exact acute angle (in degrees) between the hour hand and the minute hand of a standard analog clock?

Show answer & explanation

Answer: 34

Answer

The angle is determined by finding the exact position of both hands relative to the 12 o'clock mark. The minute hand is at 168 degrees (28 minutes multiplied by 6 degrees). The hour hand is at 134 degrees, which includes its base position at 4 o'clock (120 degrees) plus the drift from the 28 minutes that have passed (14 degrees). The absolute difference between these two positions is exactly 34 degrees.

Step-by-Step Solution

Calculate the angle of the minute hand from the 12 o'clock position.

28 \times 6^\circ = 168^\circ

The minute hand moves at a rate of 6 degrees per minute.

Calculate the base angle of the hour hand exactly at 4 o'clock.

4 \times 30^\circ = 120^\circ

Each hour marking on the clock represents 30 degrees from the top.

Calculate the drift of the hour hand caused by the 28 elapsed minutes.

28 \times 0.5^\circ = 14^\circ

The hour hand continuously moves at a rate of 0.5 degrees per minute.

Determine the true position of the hour hand.

120^\circ + 14^\circ = 134^\circ

Adding the drift to the base position gives the exact current location of the hour hand.

Calculate the acute angle between the two hands.

|168^\circ - 134^\circ| = 34^\circ

The angle between the hands is the absolute difference between their positions from the 12 o'clock mark.

Key Concept

Calculating the precise angle between clock hands by accounting for hour-hand drift

Question 4480Question →

Consider the following statements and the subsequent conclusions.

Statement 1: To address severe groundwater depletion, the state administration has reduced electricity subsidies by 40% for farmers cultivating water-intensive crops.
Statement 2: Concurrently, a direct cash incentive program has been launched specifically for farmers who transition to growing drought-resistant crops like millets.

Conclusion I: The reduction in electricity subsidies will completely eliminate the cultivation of water-intensive crops in the region.
Conclusion II: The state administration's policy aims to promote a shift in agricultural patterns toward less water-dependent farming.

Which of the given conclusions logically follows from the statements?

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 II follows

Answer

Only Conclusion II follows

The correct answer is that only Conclusion II follows. Conclusion II directly synthesizes the two statements: applying a penalty to water-heavy crops and a reward for drought-resistant crops logically infers an administrative goal to shift agricultural patterns. Conclusion I is invalid because it jumps to an extreme absolute ('completely eliminate') that cannot be proven by a mere 40% subsidy reduction.

Step-by-Step Solution

Analyze the core actions described in the statements.

The government is using a 'stick' (reducing subsidies by 40% for high-water crops) and a 'carrot' (cash incentives for drought-resistant crops).

Understanding the factual premises is required before evaluating any conclusions.

Evaluate the logical validity of Conclusion I against the premises.

Conclusion I claims water-intensive crops will be 'completely eliminated'. This is invalid.

A 40% reduction in a subsidy makes cultivation more expensive, but it does not logically dictate that every single farmer will stop growing those crops. Absolute statements require absolute evidence.

Evaluate the logical validity of Conclusion II against the premises.

Conclusion II claims the policy aims to promote a shift toward less water-dependent farming. This is valid.

The combination of discouraging water-intensive crops and explicitly rewarding drought-resistant crops (like millets) is the exact definition of promoting an agricultural shift.

Key Concept

Statement and Conclusions or Inferences

Estimated Time:1m 15s

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