All practice questions

4581 questions

All types4,581 Multiple choice2,372 Multiple select712 True / False174 Numeric188 Matching670 Ordering415 Fill in the blank20 Short answer30

Question 4541Question →

During a graduation ceremony, graduates are standing in a single file line facing the stage. Liam is standing $18^{\text{th}}$ from the front, while Sophia is standing $31^{\text{st}}$ from the rear. After exchanging their places, Liam becomes $42^{\text{nd}}$ from the front. What is Sophia's new position from the rear?

54^{\text{th}}

72^{\text{nd}}

55^{\text{th}}

7^{\text{th}}

Show answer & explanation

Answer: $55^{\text{th}}$

Answer

Sophia's new position from the rear is

55^{\text{th}}

When Liam and Sophia exchange places, Liam moves from the

18^{\text{th}}

position to the

42^{\text{nd}}

position from the front, representing a shift of

24

places backward (

42 - 18 = 24

). Since Sophia takes Liam's old spot, she shifts

24

places forward (towards the rear end of the line). Therefore, her new position from the rear is her old position plus the shift:

31 + 24 = 55

. Alternatively, finding the total number of people (

42 + 31 - 1 = 72

) and applying the formula for rank from the opposite end (

72 - 18 + 1 = 55

) yields the exact same correct result.

Step-by-Step Solution

Determine the total number of graduates in the line.

Total =

42

(Liam's new position from front) +

31

(Sophia's initial position from rear) -

1

72

After swapping, Liam occupies Sophia's original position. The sum of positions from both ends minus one gives the total count.

Identify Sophia's new position from the front.

Sophia is now at Liam's original position, which is

18^{\text{th}}

from the front.

Because they exchanged places, Sophia takes over Liam's initial spot.

Calculate Sophia's new position from the rear.

New position =

72

(Total) -

18

(Position from front) +

1

55^{\text{th}}

The position from the rear is calculated by taking the total number of people, subtracting the position from the front, and adding one.

Key Concept

Position Interchange and Total Queue Calculation

Question 4542Question →

When a positive integer $x$ is divided by $17$ , the remainder is $12$ . When another positive integer $y$ is divided by $17$ , the remainder is $15$ . What is the positive remainder when the expression $4x - 5y$ is divided by $17$ ?

Show answer & explanation

Answer: 7

Answer

Using the properties of modular arithmetic, we substitute the remainders into the expression:

4(12) - 5(15) = 48 - 75 = -27

. To find the equivalent positive remainder modulo

17

, we add the nearest larger multiple of

17

, which is

34

. Thus,

-27 + 34 = 7

Step-by-Step Solution

Express the given properties using modular arithmetic.

x \equiv 12 \pmod{17}

and

y \equiv 15 \pmod{17}

This allows us to substitute the remainders directly into the expression.

Substitute the remainders into the expression

4x - 5y

4(12) - 5(15)

By the properties of modular arithmetic, the remainder of a sum or difference is the sum or difference of the individual remainders.

Evaluate the arithmetic expression.

48 - 75 = -27

Simplifying the multiplication and subtraction gives the intermediate negative remainder.

Convert the negative remainder into a valid positive remainder.

-27 \equiv -27 + 2(17) \equiv -27 + 34 \equiv 7 \pmod{17}

A valid remainder must be a positive integer between

0

and the divisor minus

1

Key Concept

Modular Arithmetic Operations and Negative Remainders

Question 4543Question →

Six architects—Kael, Lin, Mateo, Nadia, Omar, and Priya—are sitting around a circular table at equal distances. Kael, Mateo, and Omar are facing the center, while Lin, Nadia, and Priya are facing away from the center.

Read the following information carefully:
1. Mateo sits second to the left of Priya.
2. Lin sits to the immediate right of Mateo.
3. Kael sits second to the left of Lin.
4. Nadia is not an immediate neighbor of Priya.

Based on the seating arrangement, which of the following statements are correct?

Select all that apply

Priya sits to the immediate left of Omar.

Nadia sits second to the right of Mateo.

Kael sits second to the right of Lin.

Omar sits second to the left of Kael.

Show answer & explanation

Answer: Priya sits to the immediate left of Omar.; Nadia sits second to the right of Mateo.

Answer

The statements confirming that Priya sits to the immediate left of Omar, and that Nadia sits second to the right of Mateo, are both correct.

Based on the completed arrangement (Clockwise: Priya, Kael, Nadia, Lin, Mateo, Omar), Omar is at position 6 facing the center, making his left clockwise (position 1), where Priya sits. Mateo is at position 5 facing the center, making his right anti-clockwise (position 3), where Nadia sits. Therefore, both of these positional statements perfectly match the deduced layout.

Step-by-Step Solution

Establish the directional rules based on facing orientation.

For individuals facing the center (IN), left is clockwise and right is anti-clockwise. For individuals facing outward (OUT), left is anti-clockwise and right is clockwise.

Correctly translating left and right directions based on inward vs. outward facing is strictly required before placing anyone.

Assign positions 1 through 6 clockwise and place Priya to determine Mateo's seat.

Place Priya at position 1 (facing OUT). Because she faces outward, her left is anti-clockwise. The second position anti-clockwise from position 1 (sequence: 1 -> 6 -> 5) is position 5. Therefore, Mateo (facing IN) sits at position 5.

This establishes a starting anchor for the arrangement based on the first definite clue.

Place Lin based on Mateo's position.

Lin sits to the immediate right of Mateo. Mateo (at position 5) faces IN, so his right is anti-clockwise. The immediate position anti-clockwise from 5 is 4. Thus, Lin (facing OUT) sits at position 4.

Utilizes the second clue connected to a known position.

Place Kael based on Lin's position.

Kael sits second to the left of Lin. Lin (at position 4) faces OUT, so her left is anti-clockwise. The second position anti-clockwise from 4 (sequence: 4 -> 3 -> 2) is 2. Thus, Kael (facing IN) sits at position 2.

Maintains the chain of logic using established seats and specific facing directions.

Place the remaining two people, Nadia and Omar.

The empty seats are positions 3 and 6. Nadia cannot be an immediate neighbor of Priya (at position 1). Since position 6 is directly next to position 1, Nadia must take position 3 (facing OUT). Omar (facing IN) takes the final remaining seat at position 6.

Applies the final negative constraint to complete the deterministic seating layout.

Key Concept

Multi-variable seating arrangement requiring strict orientation rules (inward vs. outward) and sequential logical deduction.

Estimated Time:2m 0s

Question 4544Question →

A solid wooden cube is painted Black on two adjacent faces, while the remaining four faces are painted White. The large cube is then cut into $64$ identical smaller cubes. What is the total number of smaller cubes that have at least one Black face?

Show answer & explanation

Answer: 28

Answer

The large cube is divided into

64

smaller cubes, which means it forms a

4 \times 4 \times 4

grid. A single face of this large cube contains

4 \times 4 = 16

smaller cubes. Since two adjacent faces are painted Black, we count the cubes on both faces. The first face has

16

Black cubes. The second face also has

16

Black cubes, but it shares one edge with the first face. This shared edge consists of

4

smaller cubes. If we simply add

16 + 16 = 32

, we double-count the

4

cubes on the shared edge. Subtracting the overlap gives

32 - 4 = 28

unique smaller cubes that have at least one Black face.

Step-by-Step Solution

Find the edge length of the large cube in terms of smaller cubes.

n = 4

The total number of smaller cubes is

64

, and the total volume of a cube is

n^3

. Therefore,

n = \sqrt[3]{64} = 4

Count the smaller cubes on the first Black face.

16

cubes

Each face of an

n \times n \times n

cube contains

n^2

smaller cubes. Here,

4^2 = 16

cubes.

Count the smaller cubes on the second Black face and identify the overlap.

The second face adds

16

cubes, but the two faces share an edge of

4

cubes.

Since the two Black faces are adjacent, they intersect along one edge. The cubes on this edge are counted as part of both faces.

Calculate the total unique cubes with at least one Black face.

28

cubes

Using the inclusion-exclusion principle:

16

(first face) +

16

(second face) -

4

(shared edge) =

28

Key Concept

Calculation of painted smaller cubes from a larger cut cube, specifically handling adjacent face intersections.

Question 4545Question →

Consider the following policy initiative and evaluate which of the provided assumptions are logically implicit within it.

Statement: The State Agriculture Department has announced a new initiative providing a 75% subsidy on equipment costs for farmers who transition from traditional flood irrigation to drip irrigation systems.

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

Select all that apply

Providing financial incentives will encourage at least a portion of the farming community to adopt the new irrigation technology.

Drip irrigation systems possess operational benefits or water efficiencies that make them preferable to traditional flood irrigation.

Drip irrigation is currently the most widely used agricultural method in developed nations.

The government will eventually impose a complete ban on traditional flood irrigation for all crops.

Show answer & explanation

Answer: Providing financial incentives will encourage at least a portion of the farming community to adopt the new irrigation technology.; Drip irrigation systems possess operational benefits or water efficiencies that make them preferable to traditional flood irrigation.

Answer

The implicit assumptions are that financial incentives will encourage adoption, and that drip irrigation offers preferable benefits over traditional methods.

The correct selections represent the foundational beliefs necessary for the policy to exist. For the government to offer a subsidy for drip irrigation, it must inherently believe that the subsidy will work to motivate farmers and that drip irrigation is a preferable method worth funding.

Step-by-Step Solution

Analyze the core premise of the given statement.

The statement outlines a government policy offering a 75% subsidy to incentivize a shift from flood to drip irrigation.

Understanding the administrative action and its goal is the first step in identifying underlying assumptions.

Evaluate the candidate assumption regarding financial incentives.

The assumption is logically valid. The policy exists to drive a transition, implying the government believes the subsidy will successfully motivate farmers.

A proposed solution inherently assumes its own potential efficacy.

Evaluate the candidate assumption regarding the benefits of drip irrigation.

The assumption is valid. Promoting a transition to a new system implies the new system is recognized as superior or more efficient.

Governments do not sponsor transitions to alternative methods unless the alternative is deemed advantageous.

Evaluate the remaining candidate options for logical fallacies.

The options regarding developed nations (external fact) and future bans (future inference) are rejected.

Implicit assumptions must be foundational to the statement itself, avoiding outside knowledge or unstated future consequences.

Key Concept

Identifying Implicit Premises in Administrative Statements

Question 4546Question →

An architect is planning to pave a rectangular courtyard using exactly $240$ identical square stones. To ensure the space is functional, both the length and the width of the courtyard must consist of at least $5$ stones. Assuming the stones cannot be cut, how many different unordered pairs of dimensions (length and width) are possible for this courtyard?

Show answer & explanation

Answer: 6

Answer

The correct answer is derived by finding the total number of factor pairs for 240 (which is 10 pairs) and subtracting the 4 pairs that contain a factor smaller than 5 (1, 2, 3, and 4). This leaves exactly 6 valid unordered pairs.

Step-by-Step Solution

Find the prime factorization of 240.

240 = 2^4 \times 3^1 \times 5^1

Prime factorization is required to calculate the total number of factors (divisors) of the number.

Calculate the total number of factors for 240.

Total factors =

(4+1)(1+1)(1+1) = 5 \times 2 \times 2 = 20

Adding 1 to each exponent in the prime factorization and multiplying them yields the total number of divisors.

Calculate the total number of unordered pairs of dimensions.

Total pairs =

20 \div 2 = 10

Since 240 is not a perfect square, its 20 factors form exactly 10 distinct pairs

(a, b)

such that

a \times b = 240

Identify and exclude pairs where at least one dimension is less than 5.

The factors less than 5 are 1, 2, 3, and 4. These correspond to 4 invalid pairs: (1, 240), (2, 120), (3, 80), and (4, 60).

The problem specifically states that both dimensions must consist of at least 5 stones.

Calculate the final number of valid pairs.

10 - 4 = 6

valid pairs.

Subtracting the 4 invalid pairs from the total 10 pairs leaves the number of pairs that satisfy all conditions.

Key Concept

Factors, Multiples, and Prime Factorization

Alternative Method

Instead of finding all factors using prime factorization, one could systematically list the factor pairs starting from

1 \times 240

up to the square root of

240

(which is approximately

15.4

). The pairs are (1,240), (2,120), (3,80), (4,60), (5,48), (6,40), (8,30), (10,24), (12,20), and (15,16). Eliminating the first four pairs that contain a number smaller than 5 leaves exactly 6 valid options.

Estimated Time:1m 30s

Question 4547Question →

Five cargo ships—Apex, Beacon, Crest, Dawn, and Echo—arrived at a port one after another. Based on the following conditions, arrange the ships in the correct order of their arrival from first to last:

1. Crest arrived exactly two places after Beacon.
2. Dawn arrived before Echo but after Beacon.
3. Apex did not arrive first or last.
4. Echo arrived immediately after Apex.

Arrange the cargo ships in chronological order from the first to arrive to the last.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct chronological order of arrival is Beacon, Dawn, Crest, Apex, and Echo.

The conditions are uniquely satisfied when Beacon is 1st, Dawn is 2nd, Crest is 3rd, Apex is 4th, and Echo is 5th. Beacon must be first because placing it anywhere else either pushes Crest too far back or prevents Dawn and Echo from following their respective constraints.

Step-by-Step Solution

Determine the relative positions of Beacon and Crest.

Beacon is in position

x

, and Crest is in position

x+2

The condition states that Crest arrived exactly two places after Beacon.

Determine the possible positions for Apex and Echo.

Apex and Echo must occupy consecutive positions, with Apex directly before Echo. Since Apex cannot be first or last, they can only occupy positions 2 and 3, 3 and 4, or 4 and 5.

Echo arrived immediately after Apex, and Apex has boundary restrictions.

Test position assignments to satisfy all constraints.

If Beacon is 1st, Crest is 3rd. This leaves positions 4 and 5 for Apex and Echo. Dawn must arrive after Beacon and before Echo, so Dawn perfectly fits into the remaining 2nd position.

If Beacon were placed 2nd or later, it would create unresolvable conflicts with the spacing needed for Apex and Echo, or violate Dawn's placement.

Key Concept

Relative ordering and positional constraints

Question 4548Question →

You are the Chairperson of a State Selection Board overseeing the recruitment of police sub-inspectors. The physical endurance test is scheduled for tomorrow. Tonight, your close friend, who is also a senior cabinet minister, calls you and casually mentions that his nephew is appearing for the test. He does not explicitly ask for a favor, but remarks, 'I hope the boy's minor leg injury from last week doesn't unfairly disqualify him. Look out for him, he has great potential.' As an administrator committed to ethical conduct, what is the most appropriate course of action?

Publicly report the minister's phone call to the anti-corruption bureau and immediately disqualify the nephew from the recruitment process to set a strict precedent against nepotism.

Cancel the physical endurance test scheduled for tomorrow and order a complete restructuring of the evaluation committee to prevent any possibility of external political influence.

Ensure the selection process proceeds strictly according to established physical criteria for all candidates, relying on objective documentation like video recording, without giving any special instructions to the evaluating staff.

Direct the evaluation staff to be generally lenient regarding minor physical ailments for all candidates, ensuring the nephew benefits without demonstrating obvious, targeted favoritism.

Show answer & explanation

Answer: Ensure the selection process proceeds strictly according to established physical criteria for all candidates, relying on objective documentation like video recording, without giving any special instructions to the evaluating staff.

Answer

The correct course of action upholds the principles of impartiality and procedural integrity. By relying on established criteria and objective documentation (like video recording), the administrator neutralizes the subtle political pressure without overreacting or violating the rights of the candidate, who has not committed any wrongdoing.

Step-by-Step Solution

Analyze the core ethical conflict presented in the scenario.

The conflict is between personal relationship/subtle political pressure and the administrative duty of impartial recruitment.

Identifying the competing values is the first step in administrative decision-making.

Evaluate the nature of the pressure applied.

The minister made an implicit request rather than an overt illegal demand, meaning an immediate crisis response is not necessarily warranted.

Understanding the severity of the threat dictates the proportionality of the response.

Determine the course of action that upholds statutory rules without causing unnecessary administrative disruption.

Proceeding with standard, documented testing protocols ensures fairness to all candidates while protecting the process from manipulation.

Objective systems like standard video recording naturally counter subjective bias or subtle pressure.

Key Concept

Impartiality, Objective Evaluation, and Proportionality in Administration

Question 4549Question →

In a chemical manufacturing plant, three automated valves release specific additives into a continuous mixing tank. Valve A opens every $\frac{9}{2}$ minutes, Valve B opens every $\frac{15}{4}$ minutes, and Valve C opens every $\frac{45}{8}$ minutes. The system is programmed to record a 'synchronization event' whenever all three valves open at the exact same instant.

If the mixing process runs continuously for exactly $12$ hours, how many synchronization events will be recorded during this period? (Assume a synchronization event is recorded at the very beginning of the process, which counts as the first event).

129

32

271

33

Show answer & explanation

Answer: $33$

Answer

The correct answer is

33

synchronization events.

The synchronization interval is the LCM of the fractions

\frac{9}{2}

\frac{15}{4}

, and

\frac{45}{8}

. Using the formula

\frac{\text{LCM of numerators}}{\text{HCF of denominators}}

, we get

\frac{\text{LCM}(9, 15, 45)}{\text{HCF}(2, 4, 8)} = \frac{45}{2} = 22.5

minutes. Over a

12

-hour period (

720

minutes), there are precisely

\frac{720}{22.5} = 32

complete intervals. Since a synchronization event is recorded at the very beginning of the process, the total number of recorded events is

32 + 1 = 33

Step-by-Step Solution

Identify the mathematical concept required to find the synchronization interval.

The shortest time interval for all valves to open together is the Least Common Multiple (LCM) of their individual time cycles.

Synchronization occurs at common multiples of the cycle times, and the next event always occurs at the least common multiple.

Calculate the LCM of the fractions

\frac{9}{2}

\frac{15}{4}

, and

\frac{45}{8}

\text{LCM of fractions} = \frac{\text{LCM}(9, 15, 45)}{\text{HCF}(2, 4, 8)}

The standard formula for the LCM of fractions requires finding the LCM of the numerators divided by the HCF of the denominators.

Compute the LCM of numerators and HCF of denominators.

\text{LCM}(9, 15, 45) = 45

and

\text{HCF}(2, 4, 8) = 2

. Thus, the synchronization interval is

\frac{45}{2} = 22.5

minutes.

This establishes that every

22.5

minutes, all three valves will open simultaneously.

Calculate the total time in minutes for the

12

-hour shift.

12 \text{ hours} \times 60 \text{ minutes/hour} = 720 \text{ minutes}

Units must be consistent to accurately determine the number of intervals.

Determine the number of full intervals and total synchronization events.

\text{Number of intervals} = \frac{720}{22.5} = 32

. Total events

= 32 + 1 = 33

The division yields the number of periods between events. We add

1

to account for the initial synchronization event explicitly mentioned in the problem (the fencepost principle).

Key Concept

Calculating the Least Common Multiple (LCM) of fractions and applying it to a temporal boundary condition (fencepost counting).

Estimated Time:2m 0s

Question 4550Question →

Five researchers—Tariq, Uma, Victor, Wendy, and Xander—are seated evenly around a circular table. Tariq, Uma, and Victor are facing the center of the table, while Wendy and Xander are facing outwards (away from the center). The following seating conditions are observed:

1. Wendy is seated second to the left of Tariq.
2. Victor is seated to the immediate left of Wendy.
3. Uma is seated second to the right of Victor.

Based on these conditions, who is seated to the immediate right of Xander?

Uma

Wendy

Tariq

Victor

Show answer & explanation

Answer: Uma

Answer

Uma is seated to the immediate right of Xander.

Tracing the positions step-by-step: If Tariq is at seat 1, Wendy (second to his left, CW) is at seat 3. Wendy faces outward, so her immediate left (CCW) is seat 2, where Victor sits. Victor faces inward, so second to his right (CCW) is seat 5, where Uma sits. Xander takes the last seat, seat 4. Since Xander faces outward, his immediate right is in the CW direction, which is seat 5, occupied by Uma.

Step-by-Step Solution

Establish the directional rules for the circular table.

For inward-facing people (Tariq, Uma, Victor), Left is Clockwise (CW) and Right is Counter-Clockwise (CCW). For outward-facing people (Wendy, Xander), Left is CCW and Right is CW.

Correctly orienting relative directions based on facing is essential before placing anyone.

Place Tariq at an arbitrary starting position (let's call it Position 1) and find Wendy.

Tariq faces the center. His left is CW. Second to the left means moving 2 seats CW. Wendy is at Position 3.

Utilizes the first explicit seating condition.

Determine Victor's position relative to Wendy.

Wendy is at Position 3 and faces outwards. Her left is CCW. Immediate left means moving 1 seat CCW. Moving 1 seat CCW from Position 3 brings us to Position 2. Victor is at Position 2.

Applies the second condition, carefully applying the reversed directional rule for an outward-facing person.

Determine Uma's position relative to Victor.

Victor is at Position 2 and faces the center. His right is CCW. Second to the right means moving 2 seats CCW. Moving 2 seats CCW from Position 2 (past Position 1) lands on Position 5. Uma is at Position 5.

Applies the third condition.

Place Xander in the remaining seat and answer the final question.

Xander must be in the only empty seat, Position 4. Xander faces outwards, so his right is CW. Moving 1 seat CW from Position 4 gives Position 5, which is occupied by Uma.

Completes the logical deduction to identify the requested relationship.

Key Concept

Positional mapping with mixed facing orientations in circular seating arrangements.

Question 4551Question →

Eight delegates are attending a bilateral summit. They are seated in two parallel rows, with four delegates in each row, such that there is an equal distance between adjacent persons.

In Row 1: P, Q, R, and S are seated, and all of them are facing South.
In Row 2: W, X, Y, and Z are seated, and all of them are facing North.
Therefore, each delegate in a row faces exactly one delegate from the other row.

Read the following information carefully:
- R sits second to the right of P.
- The delegate who faces P sits to the immediate left of Y.
- W sits at an extreme end of the row.
- Only one person sits between W and X.
- Q does not face Z.

Who sits to the immediate right of the delegate who faces X?

Show answer & explanation

Answer: S

Answer

Based on the logical deductions from the clues, the final seating arrangement from left to right is R, S, P, Q in the South-facing row, and W, Z, X, Y in the North-facing row. The delegate facing X is P. Since P is facing South, their immediate right is the seat to their left from the viewer's perspective, which is occupied by S.

Step-by-Step Solution

Determine the relative positions in Row 1 based on facing direction.

Since Row 1 faces South, their 'right' translates to the viewer's 'left'. Placing R second to the right of P means R is positioned two seats to the left of P from our perspective. Possible positions (from Left to Right) are either (Seat 1: R, Seat 3: P) or (Seat 2: R, Seat 4: P).

To establish the initial positional anchors for the South-facing row.

Use the cross-row relationships to fix exact seats for P, R, and Y.

If P is at Seat 4, the person facing P (Seat 4 in Row 2) must be to the immediate left of Y. Since Row 2 faces North, this forces Y to an impossible 5th seat. Therefore, P must be at Seat 3 and R at Seat 1. The delegate facing P is at Seat 3 in Row 2, which makes Y sit at Seat 4.

To eliminate the invalid configuration and lock in the definitive locations of P, R, and Y.

Complete Row 2 using the constraints for W and X.

In Row 2, Seat 4 is occupied by Y, so W must be at the other extreme end (Seat 1). With exactly one person sitting between W and X, X must be seated at Seat 3. The remaining Seat 2 is occupied by Z.

To finalize the entire North-facing row layout.

Finalize Row 1 using Q's constraint.

Row 2 has Z at Seat 2. Since Q does not face Z, Q cannot be at Seat 2 in Row 1. Thus, Q takes Seat 4, leaving S at Seat 2. The arrangement is complete: Row 1 (R, S, P, Q) and Row 2 (W, Z, X, Y).

To complete the full seating matrix and ensure all conditions are met.

Answer the specific question asked in the stem.

X is at Seat 3. The delegate facing X is P. P is facing South, so P's immediate right is the seat immediately to their left from the viewer's perspective (Seat 2), which is occupied by S.

To accurately derive the final answer from the completed seating arrangement.

Key Concept

Linear and Parallel Seating Arrangements

Question 4552Question →

An industrial control system tracks part serial numbers using a cyclic validation algorithm. A batch identifier $N$ is generated using the formula $N = 65^{202} - 32^{145} - 45$ . The system validates this batch by computing the positive remainder when $N$ is divided by the base modulus $33$ . What is the expected positive remainder for this batch?

Show answer & explanation

Answer: 23

Answer

The expected positive remainder is 23.

By applying the principles of modular arithmetic, we can simplify each term of the expression

65^{202} - 32^{145} - 45

modulo

33

. First,

65 \equiv -1 \pmod{33}

, so

65^{202} \equiv (-1)^{202} = 1

. Next,

32 \equiv -1 \pmod{33}

, so

32^{145} \equiv (-1)^{145} = -1

. Substituting these simplified values back into the expression yields

1 - (-1) - 45 = 2 - 45 = -43

. To find the positive remainder when dividing by

33

, we add a multiple of

33

(in this case,

66

) to

-43

, which results in

23

Step-by-Step Solution

Express the base numbers

65

and

32

modulo

33

65 \equiv -1 \pmod{33}

and

32 \equiv -1 \pmod{33}

Finding remainders close to

1

-1

significantly simplifies the evaluation of large exponents.

Evaluate the first term,

65^{202}

modulo

33

(-1)^{202} = 1

An even power of a negative one results in positive one.

Evaluate the second term,

32^{145}

modulo

33

(-1)^{145} = -1

An odd power of a negative one results in negative one.

Substitute the simplified terms back into the original expression modulo

33

N \equiv 1 - (-1) - 45 \equiv 2 - 45 \equiv -43 \pmod{33}

Combines all parts of the expression to find the overall remainder before final normalization.

Convert the negative remainder into a valid positive remainder.

-43 \equiv -43 + 66 = 23 \pmod{33}

Remainders must be positive integers between

0

and the modulus minus one. Adding the smallest multiple of

33

that is greater than

43

(which is

66

) gives the valid positive remainder.

Key Concept

Modular Arithmetic and Remainder Theorem with Negative Remainders

Alternative Method

You could also simplify the constant term first:

-45 \equiv -12 \equiv 21 \pmod{33}

. Then the expression becomes

1 - (-1) + 21 = 23

Estimated Time:1m 30s

Question 4553Question →

An official press release from the Regional Transport Office (RTO) states the following:

"In a bid to enhance passenger safety and bring down highway collision rates, it is now mandatory for all inter-state commercial buses to be equipped with electronic speed limiters capped at 80 km/h."

Two candidate assumptions are proposed based on the text above:
I. The mandatory installation of electronic speed limiters will successfully prevent these buses from traveling faster than 80 km/h.
II. Driving at speeds exceeding 80 km/h is the sole cause of highway collisions involving inter-state commercial buses.

Which of the following options correctly identifies the assumption(s) implicit in the RTO's press release?

Only assumption I is implicit

Only assumption II is implicit

Both assumptions I and II are implicit

Neither assumption I nor II is implicit

Show answer & explanation

Answer: Only assumption I is implicit

Answer

Only assumption I is implicit

Assumption I is logically implicit because the RTO's entire strategy to reduce collisions relies on the unstated premise that the electronic speed limiters will successfully perform their intended function. Assumption II is invalid because, although the text implies high speed is a contributing factor (hence the mandate to 'bring down' collision rates), it does not assume that speeding is the 'sole' cause of all such incidents.

Step-by-Step Solution

Analyze the core objective and the mandated action presented in the statement.

The RTO's objective is to 'enhance passenger safety and bring down highway collision rates'. To achieve this, the action is mandating 'electronic speed limiters capped at 80 km/h'.

Identifying the cause-and-effect relationship intended by the author is strictly necessary to deduce their underlying assumptions.

Evaluate Assumption I strictly against the logic of the original statement.

Assumption I suggests the limiters will successfully cap the speed at 80 km/h. This is a required, unstated premise. If the RTO did not assume the devices would function as intended, mandating them would not achieve their stated safety objective.

An assumption is a premise that must be taken for granted for the statement's logic to hold true.

Evaluate Assumption II, paying close attention to qualitative modifiers and absolute language.

Assumption II claims that speeding over 80 km/h is the 'sole cause' of collisions. While the statement aims to 'bring down' (reduce) collisions, which assumes speed is a contributing factor, it does not logically assume speed is the only factor.

Extreme language (such as 'sole', 'only', 'always') in an assumption usually makes it logically invalid unless the original text explicitly supports such an absolute condition.

Key Concept

Evaluating Implicit Premises and Identifying Extreme Language Traps in Logical Reasoning

Question 4554Question →

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

Statement:
"The National Cyber Security Agency advises all smartphone users to install the latest OS update immediately to patch a critical vulnerability that could allow hackers to access personal data."

Assumptions:
I. At least some smartphone users will pay attention to this advisory and update their operating systems.
II. Hackers have already successfully stolen personal data from users who have not yet installed this update.

Based on the given statement, which of the following options is correct?

Only assumption I is implicit

Only assumption II is implicit

Both assumptions I and II are implicit

Neither assumption I nor II is implicit

Show answer & explanation

Answer: Only assumption I is implicit

Answer

The option stating that only assumption I is implicit is the correct answer. Issuing a public advisory inherently assumes that people will heed it, whereas the statement does not assume a hack has already occurred.

Assumption I is valid because the very act of issuing a public advisory fundamentally relies on the premise that the target audience will pay attention and take the suggested action. Assumption II is invalid because the statement only identifies a potential vulnerability ('could allow' access) and does not assume that hackers have already successfully exploited it to steal data.

Step-by-Step Solution

Analyze the nature and purpose of the given statement.

The statement is a public advisory issued by a government agency urging action (installing an update) to prevent a negative outcome (data theft).

Identifying the type of statement helps in determining its underlying logical premises.

Evaluate Assumption I against the statement.

Assumption I states that some users will follow the advisory. This is implicitly assumed.

Any public notice, warning, or advisory is issued under the fundamental assumption that it will be read and acted upon by at least a portion of the target audience. Otherwise, issuing it would be pointless.

Evaluate Assumption II against the statement.

Assumption II claims that hackers have already stolen data. This is not implicitly assumed.

The statement uses the phrase 'could allow hackers to access', indicating a potential future risk or vulnerability. It does not state or assume that the vulnerability has already been actively exploited.

Key Concept

Identifying implicit premises in public advisories and distinguishing them from unstated inferences or factual exaggerations.

Question 4555Question →

During an international climate summit, a group of delegates is seated in a single straight row facing the main stage. Julian is seated $17^{\text{th}}$ from the left end of the row, while Nadia is seated $22^{\text{nd}}$ from the right end. If Julian and Nadia interchange their seats, Julian's new position becomes $33^{\text{rd}}$ from the left end. Based on this information, what is the total number of delegates seated in the row?

49

54

55

38

Show answer & explanation

Answer: $54$

Answer

The correct answer is

54

The total number of individuals in a line can be determined by adding a single position's rank from the left and its rank from the right, then subtracting 1 to account for double-counting. After interchanging seats, Julian moves to Nadia's original seat. This seat is

33^{\text{rd}}

from the left (Julian's new position) and

22^{\text{nd}}

from the right (Nadia's original position). Therefore, the total number of delegates is

33 + 22 - 1 = 54

Step-by-Step Solution

Determine the position of a single specific seat from both ends of the row.

Nadia's original seat is given as

22^{\text{nd}}

from the right end. After the interchange, Julian moves into this seat, and his new position is given as

33^{\text{rd}}

from the left end. Thus, this single seat is

33^{\text{rd}}

from the left and

22^{\text{nd}}

from the right.

To calculate the total number of people in a row, we must know the exact position of one distinct point measured from both the left and right extremes.

Apply the total count formula:

\text{Total} = (\text{Position from Left}) + (\text{Position from Right}) - 1

\text{Total} = 33 + 22 - 1 = 54

Adding the left and right positions counts the specific intersection seat twice. We must subtract 1 to ensure a correct, un-duplicated headcount.

Key Concept

Order and Ranking

Estimated Time:1m 0s

Question 4556Question →

You are the Chief Medical Officer (CMO) of a district hospital during a severe outbreak of a novel viral fever. The hospital is operating at full capacity. A highly influential local politician demands that you clear a general ward—currently occupied by recovering, but still vulnerable, patients—to create an exclusive isolation unit for his extended family, who are exhibiting mild symptoms. The politician subtly threatens to block the hospital's discretionary emergency medicine funding if you do not comply. Arrange the following courses of action from the most ethically sound and administratively appropriate to the least appropriate.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence begins with adhering strictly to triage and reporting the threat, followed by offering standard admission without reporting, then abandoning the post, and ends with the most unethical act of evicting vulnerable patients.

The most appropriate action directly addresses both the medical reality (triage) and the administrative threat (reporting to superiors). The next best action maintains medical ethics but ignores the threat. Dereliction of duty is severe, but actively harming existing vulnerable patients for political favor is the ultimate breach of administrative and medical ethics, placing it last.

Step-by-Step Solution

Evaluate the primary administrative and medical duty in the scenario.

The paramount duty of a CMO is non-maleficence (do no harm) and fair, objective triage of patients regardless of political influence.

Medical ethics and administrative impartiality dictate that healthcare resources must be allocated based strictly on medical need.

Assess the ethical standing of each proposed action.

Reporting the threat shows highest integrity. Offering standard admission is ethical but passive. Abandoning the post is a passive failure. Evicting patients is an active, corrupt harm.

Administrative actions must be judged on adherence to statutory rules, the courage to face coercion, and the protection of the public interest.

Key Concept

Administrative Objectivity, Medical Ethics, and Resistance to Undue Influence

Question 4557Question →

A municipal election committee is distributing ballots to various polling stations. When they pack the ballots in bundles of $45$ , $54$ , or $72$ , they find that they are always left with $38$ , $47$ , and $65$ unbundled ballots, respectively. If the total number of ballots printed is the largest possible 4-digit number satisfying these conditions, what is the exact total number of ballots?

Show answer & explanation

Answer: 9713

Answer

9713

The problem describes a scenario where the difference between each divisor (

45

54

72

) and its respective remainder (

38

47

65

) is exactly

7

. This means that if

7

more ballots were added, the total would be perfectly divisible by all three numbers. Therefore, the required total is exactly

7

less than a common multiple of these divisors. The LCM of

45

54

, and

72

1080

. The largest

4

-digit multiple of

1080

9720

(

1080 \times 9

). Subtracting the constant difference of

7

from

9720

gives the final answer of

9713

Step-by-Step Solution

Calculate the difference between each bundle size and its corresponding remainder.

45 - 38 = 7

54 - 47 = 7

, and

72 - 65 = 7

To identify if there is a constant difference, which allows the use of the LCM minus constant method.

Determine the Least Common Multiple (LCM) of the bundle sizes

45

54

, and

72

\text{LCM}(45, 54, 72) = 1080

The LCM represents the smallest bundle size that perfectly divides by all three numbers. Prime factorizations:

45 = 3^2 \times 5

54 = 2 \times 3^3

72 = 2^3 \times 3^2

. LCM

= 2^3 \times 3^3 \times 5 = 1080

Find the largest 4-digit multiple of the LCM.

1080 \times 9 = 9720

The problem asks for the largest 4-digit number. Dividing

9999

1080

yields

9.258...

, so the largest integer multiplier is

9

Subtract the constant difference from this largest multiple.

9720 - 7 = 9713

Since each division left a remainder that was

7

short of a full bundle, subtracting

7

from a perfect multiple satisfies all three remainder conditions.

Key Concept

Solving simultaneous remainder problems where the difference between divisors and remainders is constant, by utilizing the LCM and scaling to a specific boundary range.

Question 4558Question →

Six delegates (Aisha, Bilal, Chloe, David, Elena, and Farid) are seated in a single row of chairs facing North. The chairs are numbered consecutively from Seat 1 (leftmost) to Seat 6 (rightmost).

The seating arrangement is determined by the following conditions:
1. Farid is seated immediately to the right of Elena.
2. Chloe is seated immediately to the right of David.
3. There are exactly two delegates seated between Aisha and Bilal.
4. Elena is seated at Seat 5.
5. Aisha is seated to the left of Elena, but to the right of Chloe.

Match the following delegates to their correct seat numbers.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Bilal sits at Seat 1, David at Seat 2, Aisha at Seat 4, and Farid at Seat 6.

By anchoring Elena at Seat 5 and extending Farid to Seat 6, the remaining constraints deterministically force David and Chloe into Seats 2 and 3. This leaves only Seats 1 and 4 for Bilal and Aisha, correctly separated by two seats.

Step-by-Step Solution

Place Elena and Farid based on direct statements.

Elena is at Seat 5, and Farid is at Seat 6.

Rule 4 explicitly places Elena at Seat 5. Rule 1 states Farid is to her immediate right. Since they are facing North, the immediate right is the next higher seat number.

Determine the possible positions for David and Chloe.

David must be at Seat 2 and Chloe at Seat 3.

Rule 5 states Aisha is to the right of Chloe and left of Elena, meaning Chloe must be at Seat 3 or lower. Rule 2 states Chloe is immediately right of David. If David were at Seat 1 and Chloe at Seat 2, Aisha would be at Seat 3 or 4, which creates a conflict when trying to place Bilal with exactly two seats apart. Thus, David takes Seat 2 and Chloe takes Seat 3.

Place Aisha and Bilal in the remaining seats.

Aisha is at Seat 4, and Bilal is at Seat 1.

With Seats 2, 3, 5, and 6 occupied, only Seats 1 and 4 remain. Rule 5 states Aisha is right of Chloe (Seat 3), so Aisha must be at Seat 4. Rule 3 states there are two seats between Aisha and Bilal. Seat 4 and Seat 1 have exactly two seats (Seats 2 and 3) between them, confirming Bilal is at Seat 1.

Key Concept

Linear Seating Arrangement with Position and Neighbor Constraints

Question 4559Question →

Consider the following set of 8 numbers:

S = \left\{ -7, 0, \pi, \frac{22}{7}, \sqrt{169}, \sqrt{50}, 2.5, 0.\overline{45} \right\}

Let:
$P =$ the number of integers in $S$
$Q =$ the number of rational numbers in $S$
$R =$ the number of irrational numbers in $S$
$T =$ the number of whole numbers in $S$

Calculate the exact value of $(Q \times R) + (P \times T)$ .

Show answer & explanation

Answer: 18

Answer

Based on mathematical definitions: P (integers) = 3 because -7, 0, and 13 are integers. Q (rationals) = 6 because it includes the 3 integers plus 22/7, 2.5, and 0.4545... R (irrationals) = 2 because it includes pi and sqrt(50). T (whole numbers) = 2 because it includes the non-negative integers 0 and 13. Plugging these into (Q * R) + (P * T) yields (6 * 2) + (3 * 2) = 18.

Step-by-Step Solution

Evaluate and simplify each number in the set to determine its properties.

\sqrt{169} simplifies to exactly 13. The repeating decimal 0.\overline{45} can be written as a fraction (45/99).

Numbers must be evaluated in their simplest form to avoid misclassification.

Determine the value of P by counting the integers.

The integers are -7, 0, and 13. Therefore, P = 3.

Integers include positive numbers, negative numbers, and zero, with no decimal or fractional parts.

Determine the value of Q by counting the rational numbers.

The rational numbers are -7, 0, 22/7, 13, 2.5, and 0.\overline{45}. Therefore, Q = 6.

Rational numbers are any numbers that can be expressed as a ratio of two integers.

Determine the value of R by counting the irrational numbers.

The irrational numbers are \pi and \sqrt{50}. Therefore, R = 2.

Irrational numbers have non-terminating, non-repeating decimal expansions.

Determine the value of T by counting the whole numbers.

The whole numbers are 0 and 13. Therefore, T = 2.

Whole numbers consist only of the non-negative integers.

Substitute the counts into the requested mathematical expression and calculate the final result.

(6 \times 2) + (3 \times 2) = 12 + 6 = 18.

To answer the specific question given in the stem.

Key Concept

Classification of real numbers into integers, rational numbers, irrational numbers, and whole numbers.

Question 4560Question →

A manufacturing plant produces a total of $10 + 35 \times 41^{43}$ small electronic components. These components are to be distributed equally into shipping containers that hold exactly $42$ units each. How many components will be left unpacked after filling as many containers as possible?

Show answer & explanation

Answer: 17

Answer

To find the leftover components, we must calculate the remainder of the total amount modulo 42. Using modular arithmetic, 41 is congruent to -1 modulo 42. Raising -1 to the odd power of 43 yields -1. Multiplying this by 35 gives -35. Adding 10 results in -25. Because a physical remainder must be positive, we add the divisor (42) to the intermediate value of -25 to find the equivalent positive remainder, which is 17.

Step-by-Step Solution

Set up the modulo equation to find the remainder of the total components divided by 42.

(10 + 35 \times 41^{43}) \pmod{42}

The number of unpacked components represents the mathematical remainder when the total quantity is divided by the container capacity.

Simplify the large base (41) with respect to the divisor (42).

41 \equiv -1 \pmod{42}

Replacing a large base with its smallest absolute congruent value simplifies large exponent calculations.

Substitute -1 into the expression and evaluate the odd exponent.

10 + 35 \times (-1)^{43} \equiv 10 + 35 \times (-1) \pmod{42}

A negative base raised to an odd power evaluates to a negative result.

Perform the multiplication and addition following standard order of operations.

10 - 35 = -25 \pmod{42}

Multiplication must be completed strictly before addition.

Convert the negative remainder into a valid strictly positive remainder.

-25 + 42 = 17

Physical remainders cannot be negative. To find the equivalent positive remainder in modular arithmetic, add the divisor to the negative result.

Key Concept

Modular Arithmetic and Negative Remainders

Estimated Time:1m 30s

PreviousPage 228 / 230Next

All practice questions

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer