All practice questions

Question 341Question →

A tourist is exploring a historical monument. He starts from the main gate and walks $30\text{ m}$ towards the West. He then turns $135^{\circ}$ to his left and walks $20\sqrt{2}\text{ m}$ . Next, he turns $45^{\circ}$ to his right and walks $10\text{ m}$ . Finally, he turns $90^{\circ}$ to his left and walks $50\text{ m}$ . What is his shortest distance and direction from the main gate?

50\text{ m}

, South-East

50\text{ m}

, North-East

50\text{ m}

, North-West

70\text{ m}

, South-East

Show answer & explanation

Answer: $50\text{ m}$ , South-East

Answer

The shortest distance is

50\text{ m}

and the direction is South-East.

By tracking the movements sequentially on a 2D coordinate plane from an origin of (0, 0), the tourist's positions are (-30, 0), (-10, -20), (-10, -30), and finally (40, -30). The straight-line distance from the origin to (40, -30) is the hypotenuse of a right triangle with legs of 40 and 30, which equals 50. The coordinate (40, -30) lies in the lower-right quadrant, indicating a South-East direction.

Step-by-Step Solution

1

Map the starting point as the origin (0, 0) and execute the first movement.

Walking

30\text{ m}

West places the tourist at (-30, 0) facing West.

Establishing a Cartesian coordinate system allows for precise tracking of movements.

2

Calculate the second movement: turn

135^{\circ}

left and walk

20\sqrt{2}\text{ m}

.

A

135^{\circ}

left turn from West changes the facing direction to South-East. Moving

20\sqrt{2}\text{ m}

South-East adds

+20

to the X-axis and

-20

to the Y-axis. New position: (-30 + 20, 0 - 20) = (-10, -20).

A left turn is counter-clockwise. West minus

135^{\circ}

equals South-East. Breaking this diagonal vector into components yields equivalent

+20\text{ m}

East and

-20\text{ m}

South.

3

Calculate the third movement: turn

45^{\circ}

right and walk

10\text{ m}

.

A

45^{\circ}

right (clockwise) turn from South-East points South. Walking

10\text{ m}

South subtracts 10 from the Y-axis. New position: (-10, -20 - 10) = (-10, -30).

Right turns add clockwise rotation, shifting the direction from diagonal to a cardinal line.

4

Calculate the final movement: turn

90^{\circ}

left and walk

50\text{ m}

.

A

90^{\circ}

left (counter-clockwise) turn from South points East. Walking

50\text{ m}

East adds 50 to the X-axis. Final position: (-10 + 50, -30) = (40, -30).

Standard orthogonal rotation maps the final displacement vector.

5

Calculate the direct distance and direction from the origin (0,0) to the final point (40, -30).

Distance =

\sqrt{40^2 + (-30)^2} = \sqrt{1600 + 900} = 50\text{ m}

. Since X is positive (East) and Y is negative (South), the direction is South-East.

The Pythagorean theorem determines the shortest straight-line Euclidean distance, and the quadrant dictates the cardinal direction.

Key Concept

Vector displacement combining angular turns, Cartesian mapping, and the Pythagorean theorem.

Question 342Question →

As the Sub-Divisional Magistrate of a remote block, you are facing a severe crisis regarding the newly implemented Biometric Public Distribution System (PDS). Frequent network connectivity issues and fingerprint recognition failures among manual laborers have caused many legitimate beneficiaries to be denied their designated food rations. Out of frustration, the local village council has locked down all PDS outlets, demanding an immediate return to the traditional manual ledger system. Conversely, the State Civil Supplies Directorate has issued a strict circular mandating zero deviations from the biometric process to prevent corruption, warning of immediate disciplinary action for any officer who bypasses the system. Which of the following is the most appropriate administrative strategy to resolve this conflict?

Initiate a dialogue with the village council to peacefully reopen the outlets, while simultaneously submitting a formal proposal to the Directorate for a temporary, authorized authentication bypass during network failures.

Immediately deploy local law enforcement to forcefully break the locks on the PDS outlets, disperse the protesting villagers, and detain the council leaders for obstructing essential public services.

Unilaterally override the state circular by issuing a sub-divisional order that allows all PDS dealers to permanently revert to manual ledgers, prioritizing immediate food distribution over technological mandates.

Issue a harsh public ultimatum to the village council, explicitly stating that any further obstruction of the biometric system will be treated as subversion and result in the permanent cancellation of their community ration cards.

Show answer & explanation

Answer: Initiate a dialogue with the village council to peacefully reopen the outlets, while simultaneously submitting a formal proposal to the Directorate for a temporary, authorized authentication bypass during network failures.

Answer

Initiate a dialogue with the village council to peacefully reopen the outlets, while simultaneously submitting a formal proposal to the Directorate for a temporary, authorized authentication bypass during network failures.

The correct strategy demonstrates emotional intelligence, procedural adherence, and effective negotiation. By engaging in dialogue, the officer addresses the community's immediate anger without resorting to force. By formally petitioning the State Directorate for an authorized bypass mechanism (like a local official's override), the officer solves the technical problem without violating the anti-corruption mandate or insubordinately breaking the chain of command.

Step-by-Step Solution

1

Analyze the competing interests and constraints.

The public has a genuine grievance (denial of food due to tech failure), but the administration is bound by a strict anti-corruption directive from higher authorities.

Understanding the legitimate concerns of all stakeholders is the first step in public conflict resolution.

2

Identify the boundaries of administrative authority.

As a Sub-Divisional Magistrate, you cannot unilaterally cancel state-level mandates, nor should you ignore citizen starvation or employ excessive force against valid grievances.

Civil servants must operate within statutory limits and procedural guidelines.

3

Formulate a proportional, sequential negotiation strategy.

Engage the local council to de-escalate the physical blockade, while acting as an institutional bridge to secure a legal, temporary technical exception from the State Directorate.

This strategy resolves the immediate tension peacefully while creating a sustainable, officially sanctioned compromise.

Key Concept

Stakeholder Conflict Resolution and Negotiation Strategy

Question 343Question →

Read the policy intervention below designed to address urban congestion, and identify the unstated premises required for this policy to function as intended.

Policy Statement:
"To reduce traffic congestion during peak morning hours, the city administration will offer a 50% property tax rebate to corporate offices that shift their daily operational start time to 11:00 AM."

Which of the following assumptions are implicit in the administrative statement? (Select all that apply)

Select all that apply

A significant number of corporate offices currently have operational start times that overlap with peak morning traffic.

The financial incentive of a property tax rebate is substantial enough to motivate corporate offices to change their established schedules.

Shifting corporate office start times to 11:00 AM will successfully eliminate all traffic jams in the city.

The city administration currently collects more property taxes from corporate offices than it requires for road maintenance.

Show answer & explanation

Answer: A significant number of corporate offices currently have operational start times that overlap with peak morning traffic.; The financial incentive of a property tax rebate is substantial enough to motivate corporate offices to change their established schedules.

Answer

The correct assumptions are that corporate offices currently contribute to peak morning traffic, and that the proposed tax rebate is a strong enough incentive to make them change their schedules.

The correct assumptions establish the necessary foundation for the policy. First, corporate offices must currently be contributing to the morning peak traffic; otherwise, changing their schedules would be pointless. Second, the proposed incentive (a 50% property tax rebate) must be perceived by the administration as substantial enough to actually motivate these offices to alter their established routines. Both are unstated premises required for the policy to be logical.

Step-by-Step Solution

1

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

The objective is to reduce morning traffic congestion. The method is incentivizing corporate offices with a tax rebate to start at 11:00 AM.

Identifying the goal and mechanism helps uncover what must be true for the plan to be viable.

2

Evaluate the first candidate assumption regarding current corporate schedules.

The assumption that corporate offices currently operate during peak hours is valid; if they did not, shifting their hours would not impact the congestion.

An assumption is an underlying premise necessary for the argument or policy to make sense.

3

Evaluate the second candidate assumption regarding the financial incentive.

The assumption that a 50% property tax rebate is a sufficient motivator is valid; without this premise, the administration would not have chosen it as the mechanism.

Policies offering incentives inherently assume the incentive is valuable enough to drive the desired behavioral change.

4

Evaluate the remaining candidate assumptions for logical fallacies.

The assumptions about eliminating all traffic or comparing tax collection amounts to road maintenance are rejected.

They represent extreme inferences or external factual claims rather than necessary implicit premises.

Key Concept

Identifying valid implicit assumptions behind administrative policies.

Question 344Question →

Six historic ships��Bounty, Cutty Sark, Discovery, Endeavour, Mayflower, and Victoria—are anchored in a single straight row facing North (towards the harbor). Their left-to-right positions are determined by the following conditions:

1. The Discovery is anchored at the extreme left end of the row.
2. The Victoria is anchored exactly midway between the Mayflower and the Endeavour.
3. The Bounty anchors immediately to the right of the Mayflower.
4. The Cutty Sark anchors immediately to the left of the Endeavour.
5. The Cutty Sark and the Victoria are anchored next to each other.

Arrange the ships in the correct order from the extreme left (1st position) to the extreme right (6th position).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from left to right is: Discovery, Mayflower, Bounty, Victoria, Cutty Sark, Endeavour.

The correct sequence is established by recognizing how the adjacency rules merge into larger blocks. By deducing that the Cutty Sark and Endeavour form a 'C-E' pair, and Victoria must sit next to Cutty Sark to form 'V-C-E', we establish the exact distance between Victoria and Endeavour (2 positions). Because Victoria must be midway between Mayflower and Endeavour, the Mayflower must also be exactly 2 positions away on the opposite side. This forces the 'Mayflower-Bounty' pair to sit immediately left of Victoria, yielding a unified 5-ship block 'M-B-V-C-E' that perfectly slots in next to the Discovery.

Step-by-Step Solution

1

Establish the given anchor.

Position 1 is Discovery (D). The sequence is currently D, _, _, _, _, _.

Condition 1 explicitly states Discovery is at the extreme left end.

2

Combine adjacency conditions to form fixed spatial blocks.

We identify a two-ship block (M - B) and a three-ship block (V - C - E).

Condition 4 requires Cutty Sark (C) to be immediately left of Endeavour (E), forming 'C - E'. Condition 5 requires Victoria (V) to be next to C. Since placing V to the right of C would break the 'C - E' block, V must be immediately to the left of C, forming 'V - C - E'. Condition 3 requires Bounty (B) to be immediately right of Mayflower (M), forming 'M - B'.

3

Apply the midway condition to combine the blocks into a single sequence.

The combined sequence becomes M - B - V - C - E.

Condition 2 states V is exactly midway between M and E. In our 'V - C - E' block, E is exactly two positions to the right of V. Therefore, M must be exactly two positions to the left of V. Placing the 'M - B' block immediately before V perfectly satisfies this midway requirement.

4

Finalize the arrangement by placing the unified block.

The final arrangement is D, M, B, V, C, E.

We have a solid 5-ship sequence (M - B - V - C - E). With Discovery anchored at position 1, the only way to fit the remaining 5 ships is consecutively in positions 2 through 6.

Key Concept

Linear seating arrangement combining relative positional blocks and distance/midpoint constraints.

Estimated Time:2m 30s

Question 345Question →

Arrange the following numerical values in ascending order (from smallest to largest).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct ascending order is

\frac{2}{5}

,

0.6

,

\frac{7}{10}

, and

0.75

.

When converted to decimals, the values are

0.40

,

0.60

,

0.70

, and

0.75

. Arranging these from smallest to largest yields the correct sequence:

\frac{2}{5}

,

0.6

,

\frac{7}{10}

,

0.75

.

Step-by-Step Solution

1

Convert the fractions into decimal format to establish a common baseline for comparison.

\frac{2}{5} = 0.4

and

\frac{7}{10} = 0.7

.

Decimals are often much easier to compare directly digit by digit than mixed formats.

2

Append trailing zeros to the decimal values to ensure they all have the same number of decimal places (two places, matching 0.75).

The values to compare are

0.40

,

0.60

,

0.70

, and

0.75

.

Aligning decimal places prevents magnitude confusion and makes visual comparison straightforward.

3

Order the standardized decimals from smallest to largest.

0.40 < 0.60 < 0.70 < 0.75

.

The question explicitly requires an ascending order.

4

Map the ordered decimals back to their original given forms.

\frac{2}{5} < 0.6 < \frac{7}{10} < 0.75

.

The final answer must be presented using the exact numbers provided in the question.

Key Concept

Converting fractions to decimals to evaluate and compare their magnitudes.

Estimated Time:45s

Question 346Question →

In a cryptographic system used for secure communications, the word 'CROWN' is coded as 'ETNYP'. Based on this same coding logic, match the following words on the left with their correct coded forms on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

PAINT matches with RZHPV, SHARK matches with UJZTM, MOUSE matches with ONTUD, and BRICK matches with DTHEM.

Each word perfectly aligns with its encrypted code when applying the specific rule derived from the stem: shifting consonants forward by 2 letters in the alphabet, and shifting vowels backward by 1 letter.

Step-by-Step Solution

1

Analyze the example 'CROWN' to identify the shift pattern for each letter.

C becomes E (+2), R becomes T (+2), O becomes N (-1), W becomes Y (+2), and N becomes P (+2).

Identifying how the example word is transformed is the first step in decoding the underlying encryption logic.

2

Generalize the coding rule based on the distinct letter types.

Consonants (C, R, W, N) are shifted forward by 2 positions (+2). The vowel (O) is shifted backward by 1 position (-1).

A consistent rule must be established that applies differently to vowels and consonants in order to decode other words.

3

Apply the derived rule to the target words on the left.

Applying the rule yields: PAINT -> RZHPV, SHARK -> UJZTM, MOUSE -> ONTUD, BRICK -> DTHEM.

Encoding each word accurately allows us to find its correct corresponding match on the right side.

Key Concept

Coding based on letter categorization (vowels vs. consonants) with distinct positional shifts.

Question 347Question →

A university establishes a new scholarship fund on September 15, 2195, which happens to be a Friday. The founding document states that a major administrative review of the fund will occur exactly $12$ years later on September 15, 2207. What day of the week will this review take place?

Show answer & explanation

Answer: Friday; friday; FRIDAY

Answer

Friday

The

12

-year period from 2195 to 2207 contains only

2

leap years (2196 and 2204). The year 2200 is a century year not divisible by

400

, meaning it is a common year with

365

days. The total shift in days of the week is equal to the number of years (

12

) plus the number of extra leap days (

2

), resulting in a total shift of

14

days. Since

14

is a perfect multiple of

7

, there are

0

odd days. The day of the week shifts by exactly

0

days, meaning the review will also take place on a Friday.

Step-by-Step Solution

1

Identify the total number of years in the given interval.

From 2195 to 2207 is exactly

12

years.

This sets the baseline shift, as every standard year shifts the calendar by at least

1

day.

2

Determine the number of leap years in this 12-year interval.

The years divisible by

4

in this range are 2196, 2200, and 2204. However, century years are only leap years if divisible by

400

. Since 2200 is not divisible by

400

, it is not a leap year. Thus, there are only

2

leap years: 2196 and 2204.

Each leap year adds an additional

1

day to the shift because it has

366

days instead of

365

.

3

Calculate the total number of odd days (total shift in days of the week).

Each normal year contributes

1

odd day, and each leap year contributes

2

odd days. For

12

years with

2

leap years, the easiest calculation is:

12

(one for each year)

+ 2

(extra day for each leap year)

= 14

days.

This gives the absolute number of days the weekday will advance.

4

Determine the final day of the week.

A shift of

14

days is exactly

2

full weeks (

14

divided by

7

leaves a remainder of

0

). Therefore, the day of the week remains the same. Friday

+ 0

days

=

Friday.

Days of the week repeat every

7

days, so we use modulo

7

arithmetic.

Key Concept

Calculating day of the week shifts across century boundaries, accounting for the rule that non-400 century years are not leap years.

Question 348Question →

If the letter-group SOUL transforms into the code VLFO according to a specific alphanumeric rule, which of the following letter-groups represents the word DIRT under the identical rule?

GRUW

GLUW

WRIG

WLIG

Show answer & explanation

Answer: GRUW

Answer

GRUW

The underlying logical rule distinguishes between consonants and vowels. Consonants are shifted forward by 3 positions in the English alphabet (e.g., S becomes V, L becomes O). Vowels are replaced by their opposite/mirror letter from the reverse alphabetical order (e.g., O becomes L, U becomes F). Applying this rule to DIRT: The consonants D, R, and T shift forward by 3 to become G, U, and W. The vowel I is replaced by its reverse alphabetical counterpart, R. Therefore, the resulting letter-group is GRUW.

Step-by-Step Solution

1

Analyze the transformation from the reference word SOUL to its corresponding code VLFO to deduce the underlying rule.

By examining the alphabetical positions, consonants (S, L) shift forward by 3 positions to become (V, O). Vowels (O, U) are replaced by their opposite/mirror letters from the reverse alphabet (L, F).

Establishing the precise logical coding pattern is necessary to solve the analogy.

2

Categorize the letters in the target word DIRT into consonants and vowels.

Consonants: D, R, T. Vowel: I.

The established rule relies on treating consonants and vowels differently.

3

Apply the respective rules to the categorized letters of DIRT.

D (+3) yields G. I (reverse mirror) yields R. R (+3) yields U. T (+3) yields W. Combining these results in the string GRUW.

Applying the exact established rule provides the final encoded letter-cluster.

Key Concept

Letter Pattern Analogies and Positional Coding

Question 349Question →

Read the following statements carefully:

1. All carbon-neutral manufacturing zones strictly prohibit the use of fossil-fuel generators.
2. Some heavily subsidized industrial parks are designated as carbon-neutral manufacturing zones.
3. No facility that prohibits fossil-fuel generators permits the entry of high-emission transport vehicles.

Which of the following conclusions logically follow from the given statements? (Select all that logically apply)

Select all that apply

Some heavily subsidized industrial parks do not permit the entry of high-emission transport vehicles.

Facilities that permit the entry of high-emission transport vehicles are not carbon-neutral manufacturing zones.

All heavily subsidized industrial parks strictly prohibit the use of fossil-fuel generators.

The restriction on high-emission transport vehicles is the most effective policy for maintaining carbon-neutral status.

Show answer & explanation

Answer: Some heavily subsidized industrial parks do not permit the entry of high-emission transport vehicles.; Facilities that permit the entry of high-emission transport vehicles are not carbon-neutral manufacturing zones.

Answer

The logically valid conclusions are that some heavily subsidized industrial parks do not permit the entry of high-emission transport vehicles, and that facilities permitting high-emission vehicles are not carbon-neutral zones.

The valid inferences rely on transitive logic and the contrapositive. First, combining all three statements proves that since some subsidized industrial parks are carbon-neutral zones, they must follow the chain of prohibiting fossil fuels and thus banning high-emission vehicles. Second, the rule that carbon-neutral zones ban high-emission vehicles inherently means that any facility allowing such vehicles cannot be a carbon-neutral zone.

Step-by-Step Solution

1

Map the logical relationships given in the three statements.

Statement 1: Carbon-neutral zone → Prohibits fossil-fuel generators. Statement 2: Some subsidized parks → Carbon-neutral zones. Statement 3: Prohibits fossil-fuel generators → Does not permit high-emission vehicles.

Establishing clear categorical and conditional relationships is necessary to draw complex inferences.

2

Chain the relationships from Statements 1 and 3 together.

Carbon-neutral zone → Prohibits fossil-fuel generators → Does not permit high-emission vehicles. (Therefore, Carbon-neutral zone → Does not permit high-emission vehicles).

Transitive logic allows us to connect the beginning of a conditional chain directly to the end.

3

Evaluate the conclusions regarding 'some subsidized parks'.

Since 'some subsidized parks' are carbon-neutral zones, and carbon-neutral zones do not permit high-emission vehicles, it logically follows that 'some subsidized parks' do not permit high-emission vehicles.

This confirms the validity of the first correct conclusion.

4

Apply the contrapositive to the chained relationship.

If a Carbon-neutral zone → Does not permit high-emission vehicles, the contrapositive is: If a facility permits high-emission vehicles → It is not a Carbon-neutral zone.

The contrapositive of a true statement is always logically true, confirming the second correct conclusion.

5

Eliminate conclusions that rely on outside knowledge or overgeneralize.

Identify that claiming 'all' parks prohibit generators is an overgeneralization of 'some', and claiming a policy is 'effective' relies on unstated outside knowledge.

Logical conclusions must be strictly bound by the provided text parameters.

Key Concept

Applying syllogistic reasoning and contrapositive logic to evaluate multi-statement chains.

Question 350Question →

If the number $11^{2025}$ is decreased by $16^{2025}$ , and the resulting difference is divided by $9$ , what is the remainder?

7

2

4

0

Show answer & explanation

Answer: 7

Answer

7

To find the remainder of

11^{2025} - 16^{2025}

divided by

9

, we first simplify the bases using modular arithmetic:

11 \equiv 2 \pmod 9

and

16 \equiv -2 \pmod 9

. Next, we evaluate the powers. We know

2^3 = 8 \equiv -1 \pmod 9

. By rewriting the exponent,

2^{2025} = (2^3)^{675} \equiv (-1)^{675} = -1 \pmod 9

. For the second term,

(-2)^{2025} = - (2^{2025}) \equiv -(-1) = 1 \pmod 9

. Subtracting the two terms yields

-1 - 1 = -2

. Because a remainder must be positive, we add the divisor to the negative result:

-2 + 9 = 7

.

Step-by-Step Solution

1

Reduce the base numbers modulo 9.

11 \equiv 2 \pmod 9

and

16 \equiv 7 \equiv -2 \pmod 9

.

Reducing the bases simplifies the expression to

2^{2025} - (-2)^{2025} \pmod 9

.

2

Determine the cyclicity pattern for powers of 2 modulo 9.

2^1=2

,

2^2=4

,

2^3=8 \equiv -1 \pmod 9

.

Finding a power that leaves a remainder of

1

or

-1

allows us to easily compute extremely large exponents.

3

Evaluate the first term

2^{2025} \pmod 9

.

2^{2025} = (2^3)^{675} \equiv (-1)^{675} = -1 \pmod 9

.

Breaking the exponent

2025

into

3 \times 675

utilizes the known value of

2^3 \pmod 9

.

4

Evaluate the second term

(-2)^{2025} \pmod 9

.

(-2)^{2025} = - (2^{2025}) \equiv -(-1) = 1 \pmod 9

.

An odd power preserves the negative sign, allowing substitution of the result from the previous step.

5

Subtract the remainders and map to a positive equivalent.

(-1) - 1 = -2

. To make it positive, add

9

:

-2 + 9 = 7

.

Remainders must be strictly positive and less than the divisor. Adding the divisor to a negative modular result finds its positive equivalent.

Key Concept

Modular Arithmetic Cyclicity and Negative Remainders

Question 351Question →

A decentralized network consists of $13$ primary verification nodes. A smart contract generates a total of $37^{105}$ encrypted tokens that must be distributed equally among these $13$ nodes. The leftover tokens that cannot be distributed equally are sent to a burn address. How many tokens will be sent to the burn address?

Show answer & explanation

Answer: 8

Answer

8

The correct remainder when

37^{105}

is divided by

13

is

8

. This is found by reducing the base

37

to

-2

modulo

13

, applying Fermat's Little Theorem to reduce the exponent

105

to

9

modulo

12

, and computing

(-2)^9 \pmod{13} = -18 \equiv 8 \pmod{13}

.

Step-by-Step Solution

1

Formulate the problem using modular arithmetic.

Evaluate

37^{105} \pmod{13}

.

Since tokens are distributed equally among 13 nodes, the tokens sent to the burn address represent the remainder when the total is divided by 13.

2

Simplify the base of the exponent.

37 = 13 \times 2 + 11

. Using a negative remainder,

11 \equiv -2 \pmod{13}

.

Working with a smaller absolute base value like -2 makes successive exponentiation much easier than working with 11 or 37.

3

Apply Fermat's Little Theorem.

Since 13 is prime,

(-2)^{12} \equiv 1 \pmod{13}

.

This theorem allows us to significantly reduce large exponents by finding their remainder when divided by

p-1

.

4

Reduce the exponent using the theorem.

105 = 12 \times 8 + 9

. Thus,

(-2)^{105} = ((-2)^{12})^8 \times (-2)^9 \equiv 1^8 \times (-2)^9 \equiv (-2)^9 \pmod{13}

.

The multiples of 12 in the exponent evaluate to 1 modulo 13, leaving only the remainder of the exponent.

5

Calculate the final reduced power.

(-2)^4 = 16 \equiv 3 \pmod{13}

. Therefore,

(-2)^8 = 3^2 = 9 \pmod{13}

. Finally,

(-2)^9 = 9 \times (-2) = -18 \pmod{13}

.

Breaking the calculation into smaller powers prevents the need to calculate very large numbers directly.

6

Convert the negative result to a positive remainder.

-18 = 13 \times (-2) + 8

. The positive remainder is

8

.

Physical quantities, such as the number of leftover tokens, must be represented by a positive remainder.

Key Concept

Modular arithmetic, Fermat's Little Theorem, and negative remainders

Question 352Question →

A water storage tank is currently filled to $0.65$ of its total capacity. If the tank currently contains $130$ liters of water, what is the total capacity of the tank in liters?

Show answer & explanation

Answer: 200

Answer

The total capacity of the tank is 200 liters.

The correct capacity is found by dividing the current volume of water (130 liters) by the decimal that represents the filled portion (0.65). This calculation,

130 \div 0.65

, yields 200 liters.

Step-by-Step Solution

1

Identify the relationship between the filled portion and the total capacity.

Let

C

be the total capacity. We establish the equation:

0.65 \times C = 130

.

Translating the word problem into a mathematical equation allows us to solve for the unknown whole amount.

2

Rearrange the equation to solve for the total capacity.

C = \frac{130}{0.65}

Isolating

C

on one side of the equation gives us the expression needed to find the total capacity.

3

Perform the division by clearing the decimal in the denominator.

C = \frac{13000}{65} = 200

Multiplying the numerator and denominator by 100 eliminates the decimal point, making the division straightforward.

Key Concept

Calculating the total amount when a specific decimal fraction of that amount is known.

Question 353Question →

Consider the following pair of character sequences that share a specific structural relationship:

PLANT : MQBOI

Applying the exact same underlying logic, which sequence correctly completes the analogous pair below?

YIELD : ?

BRVOW

AKGNF

DTXQY

ZPTMU

Show answer & explanation

Answer: DTXQY

Answer

The correct sequence is DTXQY.

The established relationship relies on a two-step transformation: first, finding the alphabetical opposite (mirror) of each letter, and second, shifting that new letter forward by 2 positions in the alphabet. Applying this logic to YIELD first gives its alphabetical mirror BRVOW. Shifting each letter in BRVOW forward by 2 positions produces DTXQY.

Step-by-Step Solution

1

Analyze the transformation from the first word (PLANT) to the second word (MQBOI) by examining the primary letter relationship.

The opposite (mirror) letters for P, L, A, N, T are K, O, Z, M, G respectively.

A common structural pattern in alphabet analogies is reverse indexing, where A=Z, B=Y, C=X, etc.

2

Determine the secondary transformation required to get from the intermediate mirror letters to the final code.

Shifting K, O, Z, M, G forward by 2 positions (+2) results in M, Q, B, O, I.

This establishes the complete two-step rule: find the alphabetical opposite, then shift forward by 2.

3

Apply the first step of the established rule to the target word (YIELD).

The opposite letters for Y, I, E, L, D are B, R, V, O, W.

Consistency in applying the exact same logical framework to the analogous pair.

4

Apply the second step of the rule to the intermediate result.

Shifting B, R, V, O, W forward by 2 positions yields D, T, X, Q, Y.

Completing the logic gives the final analogous character group.

Key Concept

Dual-Step Alphabet Analogy (Reverse Indexing and Positional Shifting)

Question 354Question →

Evaluate the following mathematical expression:

1.2 + 2.4 \div \frac{4}{5} \times 0.5

What is the correct resulting value?

7.2

2.7

2.25

9.0

Show answer & explanation

Answer: 2.7

Answer

2.7

The correct sequence of operations is to first standardize the numbers (converting

\frac{4}{5}

to

0.8

), perform the division (

2.4 \div 0.8 = 3

), perform the multiplication (

3 \times 0.5 = 1.5

), and finally add the result to the initial value (

1.2 + 1.5 = 2.7

).

Step-by-Step Solution

1

Convert the fraction to a decimal to make all terms uniform.

\frac{4}{5} = 0.8

. The expression becomes

1.2 + 2.4 \div 0.8 \times 0.5

.

Working entirely in decimals (or entirely in fractions) simplifies the arithmetic process.

2

Apply the order of operations (BODMAS/PEMDAS) by resolving division and multiplication from left to right. First, divide.

2.4 \div 0.8 = 3

. The expression becomes

1.2 + 3 \times 0.5

.

Division and multiplication take precedence over addition, and must be performed sequentially from left to right.

3

Perform the multiplication step.

3 \times 0.5 = 1.5

. The expression becomes

1.2 + 1.5

.

Multiplication is the next operation in the left-to-right sequence.

4

Perform the final addition.

1.2 + 1.5 = 2.7

.

Addition is the last operation remaining in the expression.

Key Concept

Order of Operations (BODMAS) with Mixed Number Formats

Question 355Question →

In a regional sports academy, a group of $70$ trainees were surveyed about their participation in two sports: Cricket and Football. $45$ trainees play Cricket, $30$ trainees play Football, and $10$ trainees play neither sport. How many trainees play both Cricket and Football?

15

5

25

10

Show answer & explanation

Answer: $15$

Answer

15 trainees play both Cricket and Football.

The total number of trainees participating in at least one sport is found by subtracting the

10

non-participants from the total

70

, giving

60

. Summing the individual sport counts gives

45 + 30 = 75

. The overlap represents those counted twice, which is

75 - 60 = 15

.

Step-by-Step Solution

1

Calculate the total number of trainees who play at least one sport.

Trainees playing at least one sport =

70 - 10 = 60

.

Subtract trainees playing neither sport from the total group size.

2

Apply the Principle of Inclusion-Exclusion for two sets.

N(\text{Cricket} \cup \text{Football}) = N(\text{Cricket}) + N(\text{Football}) - N(\text{Cricket} \cap \text{Football})

.

The sum of individual sets double-counts the intersection.

3

Substitute the known values into the equation and solve for the intersection.

60 = 45 + 30 - N(\text{Cricket} \cap \text{Football}) \implies N(\text{Cricket} \cap \text{Football}) = 75 - 60 = 15

.

Algebraic simplification yields the number of trainees playing both sports.

Key Concept

Two-Set Inclusion-Exclusion Principle

Question 356Question →

What is the remainder when $3^{53}$ is divided by $7$ ?

5

2

1

4

Show answer & explanation

Answer: $5$

Answer

The remainder when

3^{53}

is divided by

7

is

5

.

By applying modular arithmetic rules,

3^3 = 27 \equiv -1 \pmod{7}

. Expanding

3^{53}

gives

(3^3)^{17} \times 3^2 \equiv (-1)^{17} \times 2 = -2 \pmod{7}

. Adding

7

to the negative remainder yields the standard positive remainder of

5

.

Step-by-Step Solution

1

Express the base

3

in terms of powers close to a multiple of

7

3^3 = 27 = 7 \times 4 - 1 \equiv -1 \pmod{7}

Finding a power that yields

\pm 1 \pmod{7}

simplifies modular exponentiation.

2

Rewrite

3^{53}

using

3^3

3^{53} = (3^3)^{17} \times 3^2

Break down the exponent

53

as

3 \times 17 + 2

using laws of indices.

3

Evaluate the expression modulo

7

3^{53} \equiv (-1)^{17} \times 9 \equiv -1 \times 2 = -2 \pmod{7}

Since

(-1)

raised to an odd power is

-1

, and

9 \equiv 2 \pmod{7}

.

4

Convert negative remainder to positive equivalent

-2 + 7 = 5

Remainders in standard arithmetic must be non-negative integers in the range

[0, \text{divisor}-1]

.

Key Concept

Remainder Theorem and Modular Arithmetic

Question 357Question →

In a state policy research institute, a survey of $200$ research analysts was conducted regarding their domain specializations in three sectors: Health ( $H$ ), Education ( $E$ ), and Agriculture ( $A$ ). The data revealed that $90$ analysts specialize in Health, $85$ in Education, and $80$ in Agriculture. Furthermore, $35$ analysts specialize in both Health and Education, $30$ in both Education and Agriculture, $25$ in both Health and Agriculture, and $10$ analysts specialize in all three sectors. How many research analysts specialize in corporate policy research analysts who specialize in exactly two of these sectors?

60

70

80

90

Show answer & explanation

Answer: $60$

Answer

The number of research analysts who specialize in exactly two sectors is

60

.

To find the number of analysts specializing in exactly two sectors, we must determine the exclusive intersection of each pair of sectors. The given intersections represent analysts in at least two sectors.
- Analysts specializing in ONLY Health and Education =

35 - 10 = 25

- Analysts specializing in ONLY Education and Agriculture =

30 - 10 = 20

- Analysts specializing in ONLY Health and Agriculture =

25 - 10 = 15

Summing these exclusive regions gives

25 + 20 + 15 = 60

.

Step-by-Step Solution

1

Identify given set values and intersections

n(H \cap E) = 35

,

n(E \cap A) = 30

,

n(H \cap A) = 25

, and

n(H \cap E \cap A) = 10

.

Each pairwise intersection given includes both analysts specializing in exactly two sectors and analysts specializing in all three sectors.

2

Calculate the number of analysts specializing ONLY in each specific pair of two sectors

Only Health & Education =

35 - 10 = 25

; Only Education & Agriculture =

30 - 10 = 20

; Only Health & Agriculture =

25 - 10 = 15

.

Subtracting

n(H \cap E \cap A)

isolates the analysts specializing in exactly two sectors for each pair.

3

Sum the three isolated regions of exactly two specializations

Total =

25 + 20 + 15 = 60

.

Adding these three mutually exclusive regions yields the overall count of analysts specializing in exactly two sectors.

Key Concept

Logical Venn Diagrams - Intersection Exclusion Principle

Alternative Method

Using the general formula:

\text{Exactly two sets} = n(H \cap E) + n(E \cap A) + n(H \cap A) - 3 \times n(H \cap E \cap A) = 35 + 30 + 25 - 3(10) = 90 - 30 = 60

.

Estimated Time:1m 30s

Question 358Question →

As a District Collector, you are facing a volatile situation where a proposed government solar park on communal grazing land has sparked intense protests from local pastoralist communities. Meanwhile, state energy officials are pressing for immediate land clearance. The situation is rapidly escalating with road blockades. Arrange the following administrative actions in the most appropriate logical sequence to de-escalate the conflict and achieve a sustainable negotiated resolution.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The optimal sequence is to first stabilize the crisis through de-escalation and communication, followed by objective joint fact-finding, then structured mediation based on those facts, and concluding with the formalization of a binding agreement.

Effective conflict resolution in public administration follows a progressive hierarchy. The administration must first stabilize the volatile environment (de-escalation), then create a shared baseline of truth to anchor discussions (fact-finding), followed by bargaining based strictly on those established facts (mediation), and finally locking in the consensus through institutional protocols (formalization).

Step-by-Step Solution

1

Analyze the immediate crisis environment and prioritize public order.

Identify that the active road blockades and escalating tension must be addressed first by freezing provocative actions and opening a dialogue channel.

Meaningful negotiations cannot take place under the duress of active protests or unilateral administrative aggression.

2

Determine the required foundation for productive negotiations.

Recognize that a neutral, evidence-based assessment is necessary to understand the true impact on grazing lands.

Without a mutually accepted factual baseline, conflicting parties will entrench themselves in extreme, unsupported positions.

3

Position the mediation and bargaining phase.

Place the structured negotiation phase after the fact-finding process is complete.

Mediation relies on the objective data gathered during fact-finding to propose and debate realistic compromises.

4

Establish the conclusion of the conflict resolution strategy.

Conclude the sequence with signing and implementing a transparent rehabilitation pact.

A negotiated settlement remains fragile unless it is formally codified with clear compliance and grievance redressal mechanisms.

Key Concept

Sequential Stakeholder Conflict Resolution (De-escalation, Fact-Finding, Mediation, Formalization)

Question 359Question →

Consider the following statements and the subsequent conclusions.

Statements:
1. A recent municipal decree mandates that all manufacturing units generating hazardous chemical waste must relocate to the designated special industrial zone outside the city limits by the end of the current year.
2. 'NovaChem' and 'PolyTech' are manufacturing units currently operating well within the city limits.

Conclusions:
I. If NovaChem generates hazardous chemical waste, it is legally required to relocate to the designated special industrial zone by the end of the current year.
II. If PolyTech relocates to the designated special industrial zone by the end of the current year, it implies that PolyTech definitely generates hazardous chemical waste.

Which of the given conclusions logically follows from the statements?

Only conclusion II follows

Neither conclusion I nor conclusion II follows

Only conclusion I follows

Both conclusion I and conclusion II follow

Show answer & explanation

Answer: Only conclusion I follows

Answer

Only conclusion I follows.

Conclusion I is a direct and valid logical deduction (Modus Ponens) from Statement 1. If NovaChem meets the criteria of generating hazardous waste, it is subject to the relocation mandate. Conclusion II, however, does not logically follow. It commits the formal fallacy of affirming the consequent. Statement 1 dictates what hazardous waste generators must do, but it does not state that *only* hazardous waste generators relocate. PolyTech could choose to relocate to the special industrial zone voluntarily for other reasons, such as tax incentives or better infrastructure.

Step-by-Step Solution

1

Analyze Statement 1 to determine the logical condition established.

The decree establishes a one-way conditional rule: IF a unit generates hazardous chemical waste, THEN it must relocate to the special industrial zone.

Understanding the directional nature of conditional statements is required to evaluate logical inferences properly.

2

Evaluate Conclusion I based on the established condition.

Conclusion I applies the premise exactly: If NovaChem meets the 'IF' condition (generates hazardous waste), it faces the 'THEN' outcome (must relocate). This logically follows as a valid deduction.

To test if the conclusion strictly adheres to the stated rule without assumptions.

3

Evaluate Conclusion II based on the established condition.

Conclusion II assumes the reverse: If a unit meets the 'THEN' outcome (relocates), it must satisfy the 'IF' condition (generates hazardous waste). This does not logically follow, as units might relocate for other unstated reasons.

To check for the logical fallacy of affirming the consequent.

Key Concept

Conditional Logic and the Fallacy of Affirming the Consequent

Question 360Question →

Consider the fractions $\frac{12}{25}$ , $\frac{18}{35}$ , and $\frac{24}{55}$ . Which of the following statements are mathematically correct regarding their Highest Common Factor (HCF) and Least Common Multiple (LCM)?

Select all that apply

The HCF of the given fractions is

\frac{6}{1925}

.

The LCM of the given fractions is

\frac{72}{5}

.

The HCF of the given fractions is

\frac{72}{5}

.

The LCM of the given fractions is

\frac{6}{1925}

.

Show answer & explanation

Answer: The HCF of the given fractions is $\frac{6}{1925}$ .; The LCM of the given fractions is $\frac{72}{5}$ .

Answer

The correct statements are that the HCF of the fractions is

\frac{6}{1925}

and the LCM of the fractions is

\frac{72}{5}

.

The correct statements properly evaluate the formulas for fractions: the HCF of a set of fractions is the HCF of their numerators divided by the LCM of their denominators, and the LCM of a set of fractions is the LCM of their numerators divided by the HCF of their denominators.

Step-by-Step Solution

1

Find the HCF and LCM of the numerators (

12, 18, 24

).

HCF(

12, 18, 24

) =

6

, and LCM(

12, 18, 24

) =

72

.

These values are required for the numerator positions in the fraction HCF and LCM formulas.

2

Find the HCF and LCM of the denominators (

25, 35, 55

).

HCF(

25, 35, 55

) =

5

, and LCM(

25, 35, 55

) =

5^2 \times 7 \times 11 = 1925

.

These values are required for the denominator positions in the fraction HCF and LCM formulas.

3

Calculate the HCF of the fractions.

HCF =

\frac{\text{HCF of numerators}}{\text{LCM of denominators}} = \frac{6}{1925}

.

This determines the validity of the HCF statements.

4

Calculate the LCM of the fractions.

LCM =

\frac{\text{LCM of numerators}}{\text{HCF of denominators}} = \frac{72}{5}

.

This determines the validity of the LCM statements.

Key Concept

Highest Common Factor (HCF) and Least Common Multiple (LCM) of Fractions

Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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