Examkin

All practice questions

4581 questions

All types4,581Multiple choice2,372Multiple select712True / False174Numeric188Matching670Ordering415Fill in the blank20Short answer30
Question 561Question →

A botanist leaves her base camp to collect soil samples in a dense forest. She first walks 35 m35\text{ m} straight East. She then turns 135∘135^{\circ} to her left and walks 202 m20\sqrt{2}\text{ m}. Finally, she turns 135∘135^{\circ} to her left again and walks 12 m12\text{ m}. What is the shortest straight-line distance, in meters, between her current position and the base camp?

Show answer & explanation

Answer: 17

Answer

The shortest straight-line distance between the botanist's current position and the base camp is 17 meters.
By breaking each movement into its horizontal and vertical components on a Cartesian plane, the final position is found to be 15 meters East and 8 meters North of the origin. The shortest direct distance is the hypotenuse of the triangle formed by these coordinates: √(15² + 8²) = 17.

Step-by-Step Solution

1
Plot the first movement on a coordinate plane.
Position 1 is at (35,0)(35, 0).
Setting the starting base camp at the origin (0,0)(0, 0), moving East means traveling along the positive x-axis.
2
Calculate the vector and position for the second movement.
Position 2 is at (15,20)(15, 20).
Facing East (0°), a left turn of 135° results in facing North-West. Moving 202 m20\sqrt{2}\text{ m} diagonally at a 45° angle to the axes translates to Δx=−20 m\Delta x = -20\text{ m} (West) and Δy=+20 m\Delta y = +20\text{ m} (North). New coordinate: (35−20,0+20)(35 - 20, 0 + 20).
3
Calculate the vector and position for the third movement.
Position 3 is at (15,8)(15, 8).
Facing North-West, another left turn of 135° places her facing directly South. Walking 12 m12\text{ m} South subtracts 12 from the y-coordinate. New coordinate: (15,20−12)(15, 20 - 12).
4
Calculate the shortest straight-line distance to the origin.
The final distance is 17 m17\text{ m}.
The straight-line distance from (0,0)(0, 0) to (15,8)(15, 8) forms a right-angled triangle. Using Pythagoras: d=152+82=225+64=289=17d = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17.

Key Concept

Calculating continuous displacement using angular turns, vector breakdown, and the Pythagorean theorem.
Estimated Time:1m 30s
Question 562Question →

Consider the following statements regarding a new national cybersecurity directive:

Statement 1: All government departments handling citizen financial records must migrate to quantum-resistant encryption by 2028.
Statement 2: Some municipal water authorities handle citizen financial records for billing purposes.
Statement 3: No department utilizing quantum-resistant encryption is vulnerable to standard brute-force attacks.

Which of the following conclusions can be logically drawn from the statements provided? (Select all valid inferences)

Select all that apply

Show answer & explanation

Answer: Some municipal water authorities are required to migrate to quantum-resistant encryption by 2028.; At least some municipal water authorities will not be vulnerable to standard brute-force attacks once they implement the mandated migration.

Answer

The logically valid inferences are that some municipal water authorities must migrate to quantum-resistant encryption, and that those specific authorities will not be vulnerable to standard brute-force attacks upon migration.
The valid conclusions are derived through direct categorical syllogism. Since all departments handling financial records must migrate, and some water authorities fall into this category, it is a logical certainty that some water authorities must migrate. Consequently, because no department using this encryption is vulnerable to standard brute-force attacks, the water authorities that execute this migration will inherently gain this invulnerability.

Step-by-Step Solution

1
Analyze the relationship between Statement 1 and Statement 2.
Identify that 'some municipal water authorities' fall into the category of 'government departments handling citizen financial records'.
To establish whether the encryption mandate applies to the municipal water authorities.
2
Apply the mandate rule from Statement 1 to the findings from Step 1.
Deduce that the specific municipal water authorities handling financial records are legally required to migrate to quantum-resistant encryption by 2028.
To test the validity of the conclusion regarding mandatory migration.
3
Evaluate Statement 3 in relation to the established facts.
Deduce that since these specific water authorities must utilize quantum-resistant encryption, they will consequently not be vulnerable to standard brute-force attacks.
To test the validity of the conclusion regarding future vulnerability.
4
Examine the remaining proposed conclusions for logical fallacies or external assumptions.
Reject the conclusions claiming financial burden and departmental exemptions.
Valid conclusions must be derived exclusively from the provided text without relying on outside knowledge or inverse fallacies.

Key Concept

Categorical Syllogism and Direct Logical Deduction
Question 563Question →

Read the following information carefully about a family of six members: P, Q, R, S, T, and U.

• There are two married couples in the family across three generations.
• Q is a doctor and the father of T.
• U is a contractor and the paternal grandfather of R.
• S is a housewife and the paternal grandmother of T.
• The professions in the family are: one doctor, one contractor, one nurse, one housewife, and two students.
• The nurse is the wife of the doctor.
• R is the sister of T.

Determine the identity and position of each family member, then match the individuals on the left with their exact relationship to P.

Click a left item, then click its matching right item

Items

Q
U
S
R

Matches

Show answer & explanation

Answer

Q is the Husband, U is the Father-in-law, S is the Mother-in-law, and R is the Daughter of P.
Based on the generational breakdown, P is the mother in the second generation. Therefore, her spouse Q is her Husband, her spouse's father U is her Father-in-law, her spouse's mother S is her Mother-in-law, and her female child R is her Daughter.

Step-by-Step Solution

1
Identify the first-generation couple based on the grandparents' clues.
U (paternal grandfather) and S (paternal grandmother) are married to each other.
Since there are two married couples and they are described as paternal grandparents of the children, they form the oldest generation couple.
2
Identify the second-generation parents and their professions.
Q is the doctor and the father. He is married to P, who must be the nurse.
Q is explicitly stated as the father of T. The nurse is the wife of the doctor. Since P is the only unassigned adult, P must be the nurse and Q's wife.
3
Identify the third-generation children and their genders.
R and T are the children of Q and P. R is female (sister of T).
Both are grandchildren of U and S, and children of Q. R is stated to be the sister, establishing her gender as female.
4
Determine the exact relationship of Q, U, S, and R to P.
Q is P's husband; U is Q's father (P's father-in-law); S is Q's mother (P's mother-in-law); R is P's daughter.
Mapping the family tree directly to P yields these specific relationships.

Key Concept

Multi-generational family tree deduction and mapping relationships to a specific unstated family member.
Question 564Question →

Read the following premises carefully and determine which of the given conclusions logically follow(s), assuming the premises to be absolutely true even if they seem to be at variance with commonly known facts.

Premises:
1. All encryption algorithms are security protocols.
2. Some security protocols are vulnerable systems.
3. No vulnerable system is approved for military use.

Conclusions:
I. Some encryption algorithms are vulnerable systems.
II. Some security protocols are not approved for military use.
III. No encryption algorithm is approved for military use.

Which of the following options is correct?

Show answer & explanation

Answer: Only conclusion II follows

Answer

Only conclusion II logically follows from the given premises.
Based on the rules of categorical syllogism, Conclusion II correctly links a particular affirmative (Some S are V) with a universal negative (No V are A) to yield a valid particular negative deduction (Some S are not A). The other conclusions attempt to force connections that are not strictly guaranteed by the premises.

Step-by-Step Solution

1
Analyze Conclusion I using the premises.
Conclusion I is invalid.
Premise 1 states 'All encryption algorithms are security protocols' and Premise 2 states 'Some security protocols are vulnerable systems'. The middle term 'security protocols' is not distributed in a way that connects 'encryption algorithms' to 'vulnerable systems'. The vulnerable systems might be a completely different subset.
2
Analyze Conclusion II using the premises.
Conclusion II is valid.
Premise 2 states 'Some security protocols are vulnerable systems' and Premise 3 states 'No vulnerable system is approved for military use'. The specific subset of security protocols that are vulnerable cannot be approved for military use. Therefore, it is definitively true that some security protocols are not approved for military use.
3
Analyze Conclusion III using the premises.
Conclusion III is invalid.
Since we could not logically establish that any encryption algorithms are vulnerable systems (from Step 1), we cannot conclude that they share the negative condition of being excluded from military use. They might fall entirely into the non-vulnerable, safe category of security protocols.

Key Concept

Syllogistic logic involving universal affirmative, particular affirmative, and universal negative propositions, emphasizing the rules of the undistributed middle.
Estimated Time:1m 30s
Question 565Question →

When the mathematical expression 4463−17244^{63} - 17^2 is divided by 4545, what is the final positive remainder?

Show answer & explanation

Answer: 25

Answer

25
Applying modular arithmetic rules, the base 4444 is congruent to −1(mod45)-1 \pmod{45}. Raising −1-1 to an odd power (6363) keeps the value as −1-1. For the second term, 17217^2 equals 289289, which leaves a remainder of 1919 when divided by 4545. Subtracting the second remainder from the first gives −1−19=−20-1 - 19 = -20. Because a standard remainder must be positive, adding the divisor (4545) to −20-20 yields the final correct answer of 2525.

Step-by-Step Solution

1
Apply modular arithmetic to the first term, 446344^{63}, relative to the divisor 4545.
44≡−1(mod45)44 \equiv -1 \pmod{45}, so 4463≡(−1)63=−1(mod45)44^{63} \equiv (-1)^{63} = -1 \pmod{45}
Using negative remainders for bases close to the divisor significantly simplifies large power calculations.
2
Evaluate the second term, 17217^2, and find its remainder when divided by 4545.
172=28917^2 = 289. Dividing 289289 by 4545 yields a quotient of 66 (270270) with a remainder of 1919.
The constant term must be reduced modulo 45 to properly combine it with the first term.
3
Combine the simplified terms according to the original expression structure.
−1−19=−20(mod45)-1 - 19 = -20 \pmod{45}
The remainder of a difference is equivalent to the difference of the individual remainders.
4
Convert the resulting negative remainder into an equivalent positive remainder.
−20+45=25-20 + 45 = 25
Standard remainders must be non-negative. Adding the divisor to a negative remainder provides the mathematically correct positive value.

Key Concept

Modular Arithmetic and Negative Remainders
Question 566Question →

A State Revenue Secretary is rolling out a new digital land registry policy to modernise district-level administration. Arrange the following events in the correct chronological order to accurately represent the standard communication process cycle, from initial conceptualisation to the final feedback loop.

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct chronological sequence following the communication cycle is: Intent formulation by the Secretary, Drafting the operational circular, Email dispatch of the circular, Reading and interpretation by the Sub-Registrars, and Submitting the compliance report.
The standard organisational communication process follows a linear but cyclical model. It begins with the Sender forming an idea (the Secretary conceptualising the policy rollout). This is followed by Encoding, where the idea is structured into a message (drafting the circular). The encoded message is then Transmitted via a specific Channel (the secure administrative email). The Receiver then Decodes the message (the Sub-Registrars reading and interpreting the new workflow). Finally, the Receiver provides Feedback (submitting a compliance report) to the Sender, acknowledging receipt and understanding, which closes the communication loop.

Step-by-Step Solution

1
Identify the initial source and formulation of the message (Sender/Ideation).
The Secretary recognising the need to standardise the protocol represents the beginning of the communication process.
All communication originates with a sender who has a distinct purpose or idea to convey.
2
Identify the encoding process.
Drafting the comprehensive operational circular is the second step.
The sender must translate their abstract intent into a tangible format (words, formal document) before it can be transmitted.
3
Identify the transmission through a medium (Channel).
Dispatching the circular via the secure internal email network is the third step.
The encoded message requires a medium or channel to travel from the sender to the targeted receivers.
4
Identify the receiver's decoding process.
The Sub-Registrars reading and interpreting the workflow changes is the fourth step.
Once the message arrives, the receivers must decode and extract meaning from the transmitted text to understand the instructions.
5
Identify the feedback loop.
Sending a compliance report and technical queries back to the Secretary is the final step.
Feedback confirms successful delivery and understanding, thereby completing the two-way communication cycle.

Key Concept

The sequential stages of the standard communication process model (Sender, Encoding, Channel, Decoding, Receiver, Feedback).
Question 567Question →

Which of the following statements regarding the classification and fundamental properties of numbers are mathematically correct? (Select all that apply)

Select all that apply

Show answer & explanation

Answer: The sum of any rational number and any irrational number always results in an irrational number.; The number 22 is the only even prime number, while all even integers greater than 22 are composite numbers.

Answer

The correct statements are that the sum of a rational and an irrational number is always irrational, and that the number 2 is the only even prime number.
The correct statements highlight foundational number properties: a rational number added to an irrational number always forms a new irrational number, and the number 2 is unique as the only even prime number in mathematics, with all other even numbers above it containing multiple divisors.

Step-by-Step Solution

1
Evaluate the closure property of rational and irrational numbers described in the first statement.
The statement is determined to be mathematically true, as combining a rational with an irrational always yields an irrational result.
To verify if the first classification rule provided is accurate.
2
Analyze the sign classification of the integer 0 as presented in the second statement.
The statement is found to be false because 0 is a neutral integer, neither positive nor negative.
To test the common misconception that 0 holds a positive value.
3
Differentiate between the exact value of the constant pi and its fractional approximations in the third statement.
The statement is deemed false because pi is fundamentally irrational, and 22/7 is only an approximation.
To address the frequent conflation of pi with its common rational substitute.
4
Determine the prime and composite nature of even integers as claimed in the fourth statement.
The statement is recognized as true since 2 is the solely existing even prime number, with all larger even integers being composite.
To confirm the parity properties of prime numbers.

Key Concept

Classification of rational vs irrational numbers, neutral integers, and prime properties.
Question 568Question →

The State Environmental Protection Agency (SEPA) evaluates industrial facilities for regulatory compliance based on their operations up to August 2026. The criteria for granting a 'Standard Compliance Clearance' are as follows:

1. Monthly particulate emissions must not exceed 50 tons.
2. An active carbon filtration unit must have been installed and operational before January 1, 2026.
3. The annual environmental audit must have been submitted on or before June 30, 2026.
4. There must be zero incidents of untreated wastewater discharge.

Administrative Actions & Exceptions:
* (a) If a facility meets all criteria EXCEPT (2), but has documentary proof of placing an advance equipment order, refer the case to the 'Compliance Extension Board'.
* (b) If a facility meets all criteria EXCEPT (3), and the delay is verified to be due to a documented natural disaster, grant a '30-day Grace Period'.
* (c) If a facility violates criterion (4), immediately issue a 'Show Cause Notice'. However, if the wastewater discharge was explicitly authorized by the municipal emergency protocol, issue a 'Standard Warning' instead.
* (d) Any facility whose monthly particulate emissions exceed 100 tons must be issued an immediate 'Operations Shutdown' order, bypassing all other administrative procedures.

Consider the following facilities:
* Facility M: Monthly emissions are 45 tons. Filtration unit not installed, but proof of an advance order placed in March 2026 was provided. Audit submitted in May 2026. Zero wastewater discharge incidents.
* Facility N: Monthly emissions are 30 tons. Filtration unit installed in 2025. Audit was not submitted by June 30, 2026, due to a severe regional earthquake in June. Had one incident of untreated wastewater discharge, which was explicitly authorized under the municipal emergency protocol.
* Facility P: Monthly emissions are 120 tons. Filtration unit installed in 2025. Audit submitted on time. Zero wastewater discharge incidents.

Based on the criteria provided, which of the following statements regarding the administrative decisions for these facilities are correct?

Select all that apply

Show answer & explanation

Answer: Facility M's case should be referred to the Compliance Extension Board.; Facility N should be granted a 30-day Grace Period and be issued a Standard Warning.

Answer

The correct statements are that Facility M should be referred to the Compliance Extension Board, and Facility N should be granted a 30-day Grace Period and issued a Standard Warning.
Facility M triggers the advance order exception, correctly resulting in a referral to the Compliance Extension Board. Facility N correctly receives a compound decision: a 30-day grace period for its disaster-delayed audit and a Standard Warning for its explicitly authorized wastewater discharge.

Step-by-Step Solution

1
Evaluate Facility M against the criteria and exceptions.
Facility M meets criteria 1, 3, and 4. It fails criterion 2 (no filtration unit) but has proof of an advance order.
Exception (a) explicitly states that failing only criterion 2, given proof of an advance order, results in a referral to the Compliance Extension Board. Therefore, the statement regarding Facility M is correct.
2
Evaluate Facility N against the criteria and exceptions.
Facility N meets criteria 1 and 2. It fails criterion 3 due to a natural disaster and fails criterion 4 due to an authorized discharge.
Exception (b) applies to the delayed audit, granting a 30-day Grace Period. Exception (c) applies to the authorized discharge, modifying the penalty to a Standard Warning. Therefore, the statement outlining both the Grace Period and Standard Warning for Facility N is correct.
3
Evaluate Facility P against the criteria and exceptions.
Facility P has monthly emissions of 120 tons, failing criterion 1.
Because emissions exceed 100 tons, Exception (d) takes precedence over all other protocols, demanding an immediate Operations Shutdown. Therefore, suggesting a Show Cause Notice for Facility P is incorrect.

Key Concept

Analytical Decision Making and Criteria Selection
Estimated Time:1m 45s
Question 569Question →

A solid wooden block has dimensions 12 cm×15 cm×18 cm12 \text{ cm} \times 15 \text{ cm} \times 18 \text{ cm}. Three of its mutually adjacent faces (which meet at a single corner) are painted Red, and the remaining three faces are painted Blue. The block is then cut into smaller, identical cubes of size 1 cm×1 cm×1 cm1 \text{ cm} \times 1 \text{ cm} \times 1 \text{ cm}. How many of these smaller cubes have AT LEAST one Red face AND at least one Blue face?

Show answer & explanation

Answer: 84

Answer

84
The unit cubes with both colors lie exactly on the 6 boundary edges separating the Red and Blue faces. By summing the lengths of these 6 edges (12+12+15+15+18+18=9012+12+15+15+18+18 = 90) and subtracting the 6 overlapping corner cubes to correct for double-counting, we arrive at exactly 84 cubes.

Step-by-Step Solution

1
Identify the configuration of the painted faces on the block.
The 3 Red faces share one corner, and the 3 Blue faces share the diagonally opposite corner.
To accurately locate where the Red and Blue paints meet on the surface.
2
Determine which unit cubes satisfy the condition of having both Red and Blue faces.
Only the cubes lying exactly on the boundary edges between a Red face and a Blue face will have both colors.
Cubes fully within a Red face have no Blue paint, internal cubes have no paint, and purely corner cubes not on the boundary have only one color.
3
Count the number of boundary edges.
There are exactly 6 boundary edges forming a continuous zig-zag hexagonal ring around the block.
Each of the 3 Red faces shares an edge with exactly 2 Blue faces.
4
Determine the lengths of these 6 boundary edges.
The ring wraps around all three dimensions symmetrically, covering two edges of 12 cm, two of 15 cm, and two of 18 cm.
A cuboid has 4 edges of each dimension; the boundary ring traverses exactly half of them.
5
Calculate the total number of unit cubes on these edges using the inclusion-exclusion principle.
2×(12+15+18)−6=90−6=842 \times (12 + 15 + 18) - 6 = 90 - 6 = 84 cubes.
Summing the edge lengths gives 90 cubes, but the 6 corners connecting these edges in the ring are counted twice, so we must subtract 6 to prevent double-counting.

Key Concept

Spatial reasoning and painted cube boundary edge calculation.
Question 570Question →

What is the Highest Common Factor (HCF) of the fractions 23\frac{2}{3}, 49\frac{4}{9}, and 815\frac{8}{15}?

Show answer & explanation

Answer: 245\frac{2}{45}

Answer

The correct answer is 245\frac{2}{45}.
The HCF of a set of fractions is found by dividing the HCF of the numerators by the LCM of the denominators. The numerators are 2, 4, and 8, and their HCF is 2. The denominators are 3, 9, and 15, and their LCM is 45. Combining these yields the correct value of 245\frac{2}{45}.

Step-by-Step Solution

1
Identify the formula for finding the HCF of fractions.
HCF of fractions = (HCF of numerators) / (LCM of denominators).
This standard formula is required to correctly solve for the highest common factor of any set of fractions.
2
Extract the numerators and find their Highest Common Factor (HCF).
The numerators are 2, 4, and 8. Their HCF is 2.
2 is the largest integer that divides 2, 4, and 8 without leaving a remainder.
3
Extract the denominators and find their Least Common Multiple (LCM).
The denominators are 3, 9, and 15. Their LCM is 45.
45 is the smallest positive integer that is a multiple of 3, 9, and 15.
4
Apply the calculated values to the fraction HCF formula.
The final fraction is 245\frac{2}{45}.
Substituting the HCF of the numerators (2) and the LCM of the denominators (45) into the formula gives the final answer.

Key Concept

The HCF of two or more fractions is calculated by dividing the HCF of their numerators by the LCM of their denominators.
Question 571Question →

Analyze the two statements below and the subsequent conclusions to determine logical validity.

Statement I: The municipal zoning authority has ruled that any newly constructed commercial complex with a footprint exceeding 50,00050,000 square feet must allocate a minimum of 15%15\% of its total ground area to public recreational spaces.
Statement II: A consortium of major real estate developers recently suspended the construction of their planned 80,00080,000 square feet commercial projects.

Conclusion I: The developers suspended their projects specifically because they were unwilling to comply with the 15%15\% public recreational space mandate.
Conclusion II: Any newly constructed commercial complex that legally allocates only 10%10\% of its ground area to public recreational spaces must have a footprint of 50,00050,000 square feet or less.

Which of the given conclusions logically follows from the statements?

Show answer & explanation

Answer: Only Conclusion II follows

Answer

The correct option is that Only Conclusion II follows.
Conclusion II is a valid deduction because it represents the contrapositive of Statement I. Statement I creates a rule: If Area > 50,00050,000, then Recreation ≥15%\ge 15\%. Conclusion II tests the inverse: if Recreation is only 10%10\% (which is <15%< 15\%), then the Area cannot be >50,000> 50,000. Conclusion I is invalid because the text never explicitly states why the developers suspended their projects; inferring their motivation is an assumption, not a logical deduction.

Step-by-Step Solution

1
Analyze Statement I to extract logical rules and boundaries.
Identified the conditional rule: If a complex is greater than 50,00050,000 square feet, then it must allocate at least 15%15\% to recreational spaces.
To establish the primary logical parameters that govern the scenario.
2
Evaluate Conclusion I against the provided statements.
Conclusion I is invalid. While Statement II confirms that large projects were suspended, neither statement provides the reason for this suspension.
To verify if the first conclusion relies on unstated assumptions or external knowledge.
3
Evaluate Conclusion II using formal logic against Statement I.
Conclusion II is valid. Using the contrapositive (modus tollens): if the recreational space is less than 15%15\% (e.g., 10%10\%), then the condition of being greater than 50,00050,000 square feet must be false.
To check whether the second conclusion represents a mathematically and logically sound deduction from the given rules.

Key Concept

Logical Deduction, Modus Tollens, and Distinguishing Inference from Assumption
Question 572Question →

Arrange the following geometric figures in a logical classification sequence based on the sum of their interior angles, in ascending order (from the smallest sum to the largest sum).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct logical sequence from smallest to largest sum of interior angles is: Triangle, Pentagon, Hexagon, Heptagon, and Octagon.
The sum of the interior angles of a polygon increases as the number of its sides increases, defined by the formula S=(n−2)×180∘S = (n-2) \times 180^\circ. Therefore, ordering the shapes by their interior angle sums is mathematically identical to sorting them by their side count. The correct progression is Triangle (3 sides), Pentagon (5 sides), Hexagon (6 sides), Heptagon (7 sides), and Octagon (8 sides).

Step-by-Step Solution

1
Identify the relationship between a polygon and the sum of its interior angles.
The formula for the sum of interior angles is S=(n−2)×180∘S = (n-2) \times 180^\circ, where nn is the number of sides.
Establishing the mathematical rule is required to find the underlying numerical logic for the sequence.
2
Determine the number of sides (nn) for each geometric figure.
Triangle (n=3n=3), Pentagon (n=5n=5), Hexagon (n=6n=6), Heptagon (n=7n=7), Octagon (n=8n=8).
The number of sides directly dictates the magnitude of the interior angle sum.
3
Arrange the figures based on the side count in ascending order.
Since a larger number of sides always results in a larger sum of interior angles, ordering by nn gives: Triangle (3) < Pentagon (5) < Hexagon (6) < Heptagon (7) < Octagon (8).
This establishes the correct classification sequence from smallest to largest without needing to manually calculate every angle sum.

Key Concept

Classification based on numerical and geometric properties
Question 573Question →

Calculate the true positive remainder obtained upon dividing the numerical expression 6795−3×534267^{95} - 3 \times 53^{42} by 1717.

Show answer & explanation

Answer: 4

Answer

The correct positive remainder is 4.
By evaluating the expression under modulo 1717, we first simplify the base numbers: 67≡−1(mod17)67 \equiv -1 \pmod{17} and 53≡2(mod17)53 \equiv 2 \pmod{17}. This reduces the overarching expression to (−1)95−3×242(-1)^{95} - 3 \times 2^{42}. The first term trivially becomes −1-1. For the second term, we can utilize the fact that 24=16≡−1(mod17)2^4 = 16 \equiv -1 \pmod{17}. Therefore, 2422^{42} can be broken down into (24)10×22≡(−1)10×4≡4(mod17)(2^4)^{10} \times 2^2 \equiv (-1)^{10} \times 4 \equiv 4 \pmod{17}. Substituting these simplified values back into the expression yields −1−3(4)=−13-1 - 3(4) = -13. Because standard remainders must be positive, we adjust the negative result by adding the divisor 1717 to −13-13, which gives the true positive remainder of 44.

Step-by-Step Solution

1
Reduce the base numbers 6767 and 5353 to smaller equivalent values modulo 1717.
67=17×4−1  ⟹  67≡−1(mod17)67 = 17 \times 4 - 1 \implies 67 \equiv -1 \pmod{17}. And 53=17×3+2  ⟹  53≡2(mod17)53 = 17 \times 3 + 2 \implies 53 \equiv 2 \pmod{17}.
To drastically simplify large exponentiations by substituting smaller, manageable equivalent bases.
2
Evaluate the remainder of the first term, 6795(mod17)67^{95} \pmod{17}.
(−1)95=−1(mod17)(-1)^{95} = -1 \pmod{17}.
An odd exponent applied to a base of −1-1 preserves the negative sign.
3
Simplify the second term's exponentiation, 242(mod17)2^{42} \pmod{17}, by identifying a nearby power of 22 that relates to 1717.
Observe that 24=16≡−1(mod17)2^4 = 16 \equiv -1 \pmod{17}.
Finding a power that equals 11 or −1-1 modulo 1717 creates a highly efficient shortcut for reducing massive exponents.
4
Break down 2422^{42} using the established property of 242^4.
242=(24)10×22≡(−1)10×4≡1×4=4(mod17)2^{42} = (2^4)^{10} \times 2^2 \equiv (-1)^{10} \times 4 \equiv 1 \times 4 = 4 \pmod{17}.
To substitute the −1-1 equivalence and systematically compute the modular value of the second term.
5
Combine both simplified terms into the original arithmetic expression.
The expression evaluates to −1−3×4=−1−12=−13(mod17)-1 - 3 \times 4 = -1 - 12 = -13 \pmod{17}.
To find the overall aggregate modular value of the complete mathematical expression.
6
Convert the negative result into the equivalent true positive remainder.
−13+17=4-13 + 17 = 4.
By definition, a valid remainder must be a non-negative integer strictly less than the divisor.

Key Concept

Modular Arithmetic, Exponent Rules, and Negative Remainders
Estimated Time:3m 0s
Question 574Question →

In a cryptographic family ledger, relational links between individuals are denoted by specific symbols:

P@QP @ Q means PP is the father of QQ.
P#QP \# Q means PP is the wife of QQ.
P$QP \$ Q means PP is the brother of QQ.
P%QP \% Q means PP is the daughter of QQ.

Based on the coded expression Aditi%Meera#Dev$Shashi\text{Aditi} \% \text{Meera} \# \text{Dev} \$ \text{Shashi}, what is the exact relationship of Shashi to Aditi?

Show answer & explanation

Answer: Data is insufficient to determine

Answer

Data is insufficient to determine
The relationship chain establishes that Shashi is the sibling of Aditi's father. However, the expression does not provide a relational symbol that defines Shashi's gender. Without knowing if Shashi is a man or a woman, the exact relationship (Paternal Uncle or Paternal Aunt) cannot be determined. Therefore, the data is insufficient.

Step-by-Step Solution

1
Analyze the first part of the expression: Aditi%Meera\text{Aditi} \% \text{Meera}
Aditi is the daughter of Meera. (Aditi is female).
Applies the defined rule for the %\% symbol.
2
Analyze the next part: Meera#Dev\text{Meera} \# \text{Dev}
Meera is the wife of Dev. Therefore, Dev is male and is the father of Aditi.
Applies the rule for the #\# symbol and establishes the parental link.
3
Analyze the final part: Dev$Shashi\text{Dev} \$ \text{Shashi}
Dev is the brother of Shashi. This makes Shashi the sibling of Aditi's father.
Applies the rule for the $\$ symbol.
4
Evaluate Shashi's gender to determine the precise relationship.
The symbol $\$ defines Dev as a male brother, but it does not specify whether the sibling (Shashi) is male or female.
Logical constraints require explicit proof of gender. Proper names cannot be used to deduce gender in blood relation puzzles.

Key Concept

Coded Blood Relations and Logical Gender Ambiguity
Estimated Time:1m 0s
Question 575Question →

In the real number system, numbers are broadly classified into rational and irrational categories. Based on strict mathematical definitions, which of the following is an irrational number?

Show answer & explanation

Answer: π\pi

Answer

The mathematical constant π\pi is an irrational number.
The constant π\pi cannot be expressed exactly as a fraction of two integers. Its decimal representation is non-terminating and non-repeating, which is the defining characteristic of an irrational number.

Step-by-Step Solution

1
Recall the definition of a rational number.
A rational number can be written exactly as a fraction pq\frac{p}{q} where pp and qq are integers and q≠0q \neq 0.
Establishing the rule allows us to eliminate options that fit this criteria.
2
Evaluate the fractional and decimal options.
227\frac{22}{7} is explicitly a ratio of integers. 3.141593.14159 is a terminating decimal, which can also be written as a fraction.
Both perfectly fit the definition of a rational number and must be eliminated.
3
Evaluate the square root option.
25\sqrt{25} simplifies to the integer 55.
Since 5=515 = \frac{5}{1}, it is also a rational number.
4
Evaluate the mathematical constant.
π\pi represents a non-terminating, non-repeating decimal that cannot be written exactly as a fraction.
Numbers with non-terminating and non-repeating decimal expansions are classified as irrational numbers.

Key Concept

Distinguishing between rational numbers (exact fractions, terminating decimals, perfect roots) and irrational numbers (non-terminating, non-repeating decimals like the constant pi).
Question 576Question →

Six software engineers—Arthur, Bianca, Caleb, Daphne, Elias, and Fiona—are seated in a single straight row facing their monitors (North). Read the conditions below and determine their exact seating arrangement from left to right.

1. Elias sits exactly fourth to the right of Arthur.
2. Bianca sits immediately to the left of Daphne, but neither of them sits at an extreme end of the row.
3. Caleb sits somewhere to the right of Bianca but is not an immediate neighbor of Daphne.
4. Fiona does not sit next to Arthur.

Arrange the engineers in the correct seating order from extreme left (position 11) to extreme right (position 66).

Drag items to arrange them in the correct order

Show answer & explanation

Answer

The correct sequence from left to right is Arthur, Bianca, Daphne, Fiona, Elias, and Caleb.
By analyzing the widest gap constraint, Elias must be at position 55 and Arthur at position 11, or Elias at position 66 and Arthur at position 22. Placing Arthur at 22 forces Bianca and Daphne into internal slots that inevitably eliminate any valid position for Caleb (who must be right of Bianca but not next to Daphne). Therefore, Arthur must be at position 11 and Elias at position 55. Bianca and Daphne must sit at positions 22 and 33. Caleb must be placed at position 66 to be right of Bianca and avoid neighboring Daphne at 33. Finally, Fiona takes the remaining position 44, successfully ensuring she is not adjacent to Arthur at 11.

Step-by-Step Solution

1
Analyze the widest positional gap between Elias and Arthur.
Elias is fourth to the right of Arthur. This requires a block of five seats (Arthur, _, _, _, Elias). They can only occupy positions (1 and 5) or (2 and 6).
Starting with the largest spatial footprint limits the overarching frameworks available for the remaining conditions.
2
Place the Bianca-Daphne pair into the possible frameworks.
Bianca is immediately left of Daphne. In framework (2 and 6), they must occupy (3,4) or (4,5). In framework (1 and 5), they must occupy (2,3) or (3,4). Since neither can be at the extreme ends, all these pairs initially seem viable.
Contiguous blocks must fit into unbroken sequences of empty seats without touching the ends.
3
Test Caleb's placement based on Bianca and Daphne's positions.
If Arthur is at 2 and Elias at 6, Bianca and Daphne at (3,4) forces Caleb to 5, which makes Caleb and Daphne immediate neighbors (violating condition 3). Thus, Arthur must be at position 1 and Elias at 5.
Eliminating invalid frameworks ensures a singular logical path forward.
4
Finalize positions within the confirmed (Arthur at 1, Elias at 5) framework.
Bianca and Daphne must be at (2,3). If they were at (3,4), Caleb (who must be right of Bianca) would have to be at 6, but Fiona would be forced to position 2 (next to Arthur, violating condition 4). With Bianca at 2 and Daphne at 3, Caleb must be at 6 to avoid neighboring Daphne. Fiona takes the remaining position 4.
Iteratively applying the negative constraints (not neighboring) eliminates the final false sub-cases.

Key Concept

Linear seating arrangement with overlapping positional and negative constraints
Estimated Time:2m 30s
Question 577Question →

Carefully analyze the following premises regarding hypothetical physics concepts. Based exclusively on these premises, which of the following conclusions logically and definitively follow? Select all correct conclusions.

Premises:
1. All perpetual motion machines are infinite energy sources.
2. No infinite energy source is compatible with thermodynamics.
3. Some theoretical frameworks are compatible with thermodynamics.

Select all that apply

Show answer & explanation

Answer: No perpetual motion machine is compatible with thermodynamics.; Some theoretical frameworks are not infinite energy sources.

Answer

The logically valid conclusions are that no perpetual motion machine is compatible with thermodynamics, and some theoretical frameworks are not infinite energy sources.
The correct conclusions rely purely on the structural rules of deduction. The conclusion regarding perpetual motion machines correctly chains the first two premises to form a universal negative. The conclusion regarding theoretical frameworks correctly combines the particular affirmative of premise 3 with the universal negative of premise 2.

Step-by-Step Solution

1
Analyze the relationship between perpetual motion machines (P) and compatibility with thermodynamics (C) using the first two premises.
Since all P are infinite energy sources (I), and no I are C, it directly and universally follows that no P are C.
This is a standard universal negative deduction (Celarent valid mood in classical syllogism).
2
Analyze the relationship between theoretical frameworks (T) and infinite energy sources (I) using the second and third premises.
Some T are C. Since no I are C (which equivalently means no C are I), the specific T that are C cannot be I. Therefore, some T are not I.
This is a valid particular negative deduction (Festino/Ferio mood logic).
3
Evaluate the remaining candidate conclusions for logical fallacies or structural invalidity.
Concluding 'Some P are I' from 'All P are I' is invalid in modern Boolean logic because P (perpetual motion machines) is mathematically an empty set (existential fallacy). Concluding 'No T is P' is invalid because the premises only allow us to prove 'Some T are not P', not 'All T are not P'.
Conclusions must necessarily follow from the premises without assuming unstated facts, illicit conversions, or the physical existence of hypothetical entities.

Key Concept

Evaluating categorical syllogisms using modern Boolean logic rules (avoiding existential import) and identifying valid deductive inferences.
Question 578Question →

You are the District Magistrate overseeing the land acquisition process for a crucial state highway. During the final verification stage, you discover that the head of the local village council (Panchayat) has forged documents to claim compensation for a parcel of public land, falsely presenting it as his private property. The highway project is already significantly delayed, and there is intense pressure from the state government to commence construction immediately. Upon being confronted, the Panchayat head implicitly threatens that if you initiate an investigation, he will instigate the villagers to violently protest and halt the construction indefinitely. Which of the following is the most appropriate course of action?

Show answer & explanation

Answer: Suspend the specific compensation payment to the Panchayat head, initiate a formal legal inquiry into the forgery, and proactively engage with the villagers through community meetings to explain the facts.

Answer

The most appropriate course of action is to suspend the fraudulent payment, initiate a legal inquiry into the forgery, and proactively engage the community to explain the facts and prevent manipulated protests.
The correct answer balances the strict enforcement of the law with pragmatic administrative foresight. By stopping the fraudulent payment and initiating an inquiry, the officer upholds integrity and statutory duty. By proactively engaging with the community, the officer neutralizes the threat of manufactured protests through transparency, ensuring the project can continue without resorting to disproportionate force or capitulating to blackmail.

Step-by-Step Solution

1
Analyze the competing interests and core conflict in the scenario.
The conflict lies between upholding the rule of law (preventing fraud) and maintaining public order to ensure timely project execution.
Identifying the competing values is the first step in resolving any administrative dilemma.
2
Evaluate the ethical and legal implications of yielding to the threat.
Allowing the fraudulent payment violates statutory duty and administrative integrity. Expediency cannot justify complicity in corruption.
Civil servants must strictly adhere to statutory protocols, especially regarding public funds.
3
Assess the nature of the threatened public protest.
The threat relies on the villagers being misinformed or manipulated by the Panchayat head to protect his personal illegal gains.
Understanding the root cause of potential unrest helps in formulating a preventive strategy.
4
Determine the most balanced, legally sound, and practical action.
Halting the specific fraudulent payment, starting a formal inquiry, and communicating directly with the villagers addresses the legal violation while neutralizing the threat of manipulated unrest.
This upholds administrative propriety, due process, and public interest without unnecessarily halting the entire project or using disproportionate force.

Key Concept

Balancing rule of law with public order management through transparency
Question 579Question →

During a grievance redressal hearing, a Chief Settlement Officer verbally assures a group of displaced residents, "We are prioritizing your rehabilitation." However, the officer delivers this statement with a rapid, impatient speaking pace, frequently sighs audibly, and maintains a flat, monotonous pitch. Consequently, the residents feel dismissed and escalate their issue to a higher authority. Which non-verbal element of the communication process directly undermined the officer's verbal message in this scenario?

Show answer & explanation

Answer: Paralinguistics, because the vocal characteristics and delivery style contradicted the reassuring words.

Answer

Paralinguistics, because the vocal characteristics and delivery style contradicted the reassuring words.
The correct answer accurately identifies that speaking pace, pitch, and audible sighs are paralinguistic elements. In professional communication, when paralinguistic cues (how something is said) contradict the verbal message (what is said), the receiver almost always trusts the non-verbal cues. This misalignment directly caused the residents to feel dismissed.

Step-by-Step Solution

1
Analyze the source of the communication breakdown described in the scenario.
The breakdown was actively caused by the officer's rapid pace, audible sighs, and monotonous pitch contradicting the reassuring spoken words.
Identifying the specific disruptive behaviors is the first step in mapping them to formal communication models.
2
Classify the identified behaviors into the correct category of non-verbal communication.
Pace, pitch, and sighs are vocal cues that accompany speech, which are formally classified under paralinguistics (or vocalics).
Differentiating between types of non-verbal channels ensures the correct theoretical element is chosen.

Key Concept

Paralinguistics and Non-Verbal Communication
Question 580Question →

You are a civil supplies officer managing grain transportation to three drought-affected districts. Your available logistics budget is $50,000. Standard operating protocol dictates the following rules for resource allocation:

1. Districts must receive funding in strict accordance with their severity priority: District Alpha (Critical, requires 20,000),DistrictBeta(Severe,requires20,000), District Beta (Severe, requires 25,000), and District Gamma (Moderate, requires 15,000).2.Higher−prioritydistrictsmustbefullyfundedbeforeanyremainingfundscanbeallocatedtolower−prioritydistricts.3.Amandatorystatutoryreserveof15,000). 2. Higher-priority districts must be fully funded before any remaining funds can be allocated to lower-priority districts. 3. A mandatory statutory reserve of 5,000 must be retained at all times unless a formal state-level emergency is declared, which has not occurred.

Based on the stated procedural rules and resource constraints, which of the following administrative actions are correct? (Select all that apply)

Select all that apply

Show answer & explanation

Answer: Fully funding the logistical needs of District Alpha and District Beta while strictly maintaining the $5,000 statutory reserve.; Withholding transport logistics funding for District Gamma during the current allocation cycle until a supplementary budget is formally sanctioned.

Answer

The correct actions are to fully fund District Alpha and District Beta while maintaining the statutory reserve, and to withhold funding for District Gamma until additional budget is sanctioned.
The correct administrative actions demonstrate strict adherence to the priority ranking by fully funding the Critical and Severe districts, maintaining the mandated reserve, and procedurally withholding funds from the Moderate district due to exhausted resources, without resorting to illegal or unauthorized measures.

Step-by-Step Solution

1
Calculate the available funds for actual district allocation.
50,000(Total)−50,000 (Total) - 5,000 (Reserve) = $45,000 available.
Protocol strictly requires retaining the $5,000 reserve because no state-level emergency has been officially declared.
2
Allocate the available funds sequentially according to the priority rules.
District Alpha receives its full 20,000(leaving20,000 (leaving 25,000). District Beta then receives its full 25,000(leaving25,000 (leaving 0).
Rule 2 dictates that higher-priority districts must be fully funded before lower-priority ones receive any allocation.
3
Determine the administrative outcome for the remaining district.
District Gamma receives $0 in this allocation cycle and must wait for formal supplementary funds.
All available funds were exhausted by higher-priority obligations, and statutory limits prevent bypassing rules or utilizing illegal measures to compensate for the shortfall.

Key Concept

Resource allocation under strict budgetary constraints requires sequential prioritization and absolute adherence to statutory limits, prioritizing procedural integrity over informal or extreme problem-solving.
PreviousPage 29 / 230Next
All practice questions — State PSC Exam | Examkin