UKPSC Prelims 2024 CSAT (Paper II) with Answers & Explanations

Explanation

Substituting the symbols >, ≥, and = from option c) into the blanks of the given expression X _ N _ O ≥ S _ P yields:

X > N ≥ O ≥ S = P

To verify if S < X (or X > S) holds true: The relationship path from X to S is X > N ≥ O ≥ S. Since the strict inequality symbol > is present in the path, it is established that X > S, which means S < X definitely holds true.

To verify if N ≥ P holds true: The relationship path from N to P is N ≥ O ≥ S = P. Combining these signs gives N ≥ P, which definitely holds true.

Explanation

To make the equation correct, the numbers 12 and 24 should be interchanged.

The original equation with the formatted exponent is: 6³ ÷ 12 + 4 × √(81) - 28 ÷ 2 + 24 = 43

After interchanging 12 and 24, the equation becomes: 6³ ÷ 24 + 4 × √(81) - 28 ÷ 2 + 12 = 43

Applying the BODMAS rules, we first simplify the power and the square root: 6³ = 216 √(81) = 9

Substituting these values back into the equation: 216 ÷ 24 + 4 × 9 - 28 ÷ 2 + 12 = 43

Next, perform the division and multiplication operations: 216 ÷ 24 = 9 4 × 9 = 36 28 ÷ 2 = 14

Now, substitute these results back to perform addition and subtraction: 9 + 36 - 14 + 12 = 43 45 - 14 + 12 = 43 31 + 12 = 43 43 = 43

Since the Left Hand Side equals the Right Hand Side, the equation is correct.

Explanation

The statement presents a purely negative consequence of excessive social media use, claiming it reduces face-to-face interaction. The argument introduces a counter-perspective by highlighting positive outcomes of social media, such as enabling new forms of social connection and community building. By pointing out these overlooked benefits, the argument challenges the completeness and absolute nature of the initial statement. It does not fully disprove (refute) the claim about decreased face-to-face interaction, but it provides a balanced perspective that undermines the force of the negative generalization, thereby weakening the statement.

Explanation

The statement asserts that regular exercise contributes positively to overall health and well-being. The argument supports this assertion by referencing numerous scientific studies that demonstrate the positive effects of exercise on both physical and mental health. By citing empirical scientific evidence that aligns with and provides a factual basis for the initial claim, the argument directly reinforces and strengthens the validity of the statement.

Explanation

The statement asserts that a shorter, four-day workweek will lead to increased employee productivity. For this outcome to occur, employees must be capable of completing and managing their tasks effectively within the newly constrained timeframe. The assumption provides this critical, necessary condition for the success of the policy. Because the assumption supports the feasibility and practical success of the four-day workweek, it directly strengthens the statement.

Explanation

The statement posits that promoting gender equality in education results in greater economic prosperity. The assumption supports this claim by providing the underlying mechanism, explaining that equal access to education across genders fosters innovation and productivity, both of which are direct drivers of economic growth. By establishing this logical link between educational equality and economic development, the assumption directly validates and strengthens the statement.

Explanation

Interchanging + and × in the given equation makes the expression mathematically correct. Applying the BODMAS rule to the left-hand side of the modified equation:

12 × 30 ÷ 6 - 15 + 23

First, perform the division operation:

30 ÷ 6 = 5

The equation becomes:

12 × 5 - 15 + 23

Next, perform the multiplication operation:

12 × 5 = 60

The equation becomes:

60 - 15 + 23

Finally, perform the addition and subtraction:

60 - 15 + 23 = 45 + 23 = 68

This satisfies the equation as the left-hand side equals the right-hand side.

Explanation

Implementing stricter regulations on industrial emissions is a highly effective, direct administrative action that addresses one of the primary root causes of pollution. By curbing hazardous gases and particulate matter at the source, industrial emissions are significantly reduced, leading to immediate and sustainable environmental improvements.

Launching public awareness campaigns about pollution is a vital complementary strategy that addresses pollution from a social and behavioral standpoint. Raising awareness fosters citizen responsibility, promotes eco-friendly practices among the general public, and builds widespread community support for administrative measures. Combining regulatory enforcement on industries with public sensitization provides a comprehensive, sustainable, and highly effective approach to resolving the pollution crisis.

Explanation

The given statement highlights a growing imbalance where the domestic demand for crude oil is rising at a rate faster than its indigenous production, leading to a supply-demand deficit.

To address this deficit, two logical pathways or solutions emerge directly from this scenario. First, because domestic production is insufficient to meet the rising demand, crude oil must be imported from other countries to fill the gap and satisfy domestic requirements. Therefore, conclusion (1) logically follows. Second, to bring the supply and demand back into balance and reduce dependency on external sources, measures must be taken to reduce or optimize domestic demand. Thus, conclusion (2) also logically follows as a necessary action to manage the shortfall.

Explanation

The statement means that ability acts as a form of wealth for a poor person. Since wealth is a resource that provides security and the means to acquire what one needs, equating ability to wealth for a poor man implies that his ability is the primary asset that enables him to earn livelihood and wealth. Therefore, a poor man can earn wealth if he has ability, making conclusion (4) logically sound.

Explanation

Enhancing vocabulary and language skills is a direct and fundamental component of cognitive development and brain function. Regular reading stimulates neural pathways involved in comprehension, speech production, and memory, thereby strengthening overall cognitive abilities. Therefore, the statement that reading enhances vocabulary and language skills serves as a direct justification for the claim that reading regularly improves cognitive function.

Explanation

Using the laws of exponents, the given product can be simplified as: 3ᵃ× 3ᵇ× 3ᶜ = 3ᵃ⁺ᵇ⁺ᶜ

Expressing 729 as a power of 3 gives: 729 = 3⁶

Equating the powers, we get: a + b + c = 6

Since a, b, and c are different positive integers, the only possible combination of three distinct positive integers that sum to 6 is 1, 2, and 3.

Substituting these values into the given expression yields: 3¹ + 3² + 3³ = 3 + 9 + 27 = 39

Explanation

The given ratio is: (a + b)/(a - b) = ⅕

Cross-multiplying the terms gives: 5(a + b) = a - b 5a + 5b = a - b 4a = -6b (a)/(b) = -³⁄₂

Squaring both sides of the equation yields: a²/b² = ⁹⁄₄

We want to find the value of the ratio: (a² - b²)/(a² + b²)

Dividing both the numerator and the denominator by b²: (a²/b² - 1)/(a²/b² + 1) = (9/4 - 1)/(9/4 + 1) = (5/4)/(13/4) = ⁵⁄₁₃

Explanation

To find the largest integer n for which 40! is divisible by 3ⁿ, Legendre's formula is used to calculate the exponent of the prime 3 in the prime factorization of 40!.

Legendre's formula states that the exponent of a prime p in the prime factorization of N! is given by:

Eₚ(N!) = ₖ₌₁^∞ ⌊ (N)/(pᵏ) ⌋

Substituting N = 40 and p = 3 into the formula:

E₃(40!) = ⌊ ⁴⁰⁄₃ ⌋ + ⌊ (40)/(3²) ⌋ + ⌊ (40)/(3³) ⌋ + ⌊ (40)/(3⁴) ⌋ + …

E₃(40!) = ⌊ ⁴⁰⁄₃ ⌋ + ⌊ ⁴⁰⁄₉ ⌋ + ⌊ ⁴⁰⁄₂₇ ⌋ + ⌊ ⁴⁰⁄₈₁ ⌋

Evaluating each term:

⌊ ⁴⁰⁄₃ ⌋ = 13

⌊ ⁴⁰⁄₉ ⌋ = 4

⌊ ⁴⁰⁄₂₇ ⌋ = 1

⌊ ⁴⁰⁄₈₁ ⌋ = 0

Adding these values together:

E₃(40!) = 13 + 4 + 1 + 0 = 18

Explanation

Deploying additional personnel and equipment is an immediate, practical measure to clear the water and mitigate the flooding. Issuing public advisories to avoid non-essential travel helps manage the immediate traffic congestion and prevents further public inconvenience. Both of these actions directly address the acute problem described in the statement.

Explanation

Argument II is a strong and logically sound argument because cooking gas is a primary utility for households. A reduction in the subsidy on cooking gas leads to an immediate increase in its price. Since fuel and cooking costs are fundamental components of the household budget, higher cooking gas prices directly increase the cost of food preparation and other daily commodities. This upward pressure on costs contributes to overall inflation in the economy, adversely affecting the general population. This provides a valid and significant economic reason against reducing the subsidy.

Explanation

The passage emphasizes that the value of an artwork is completely independent of external utility, moral purposes, or didactic instruction, and is instead determined solely by the artwork itself. By stating that art is self-attained and brought into being solely by the spirit of art, the text underscores the intrinsic value of creative work. This concept is the core tenet of the aesthetic philosophy known as art for art's sake, which asserts that art is self-sufficient and exists entirely for its own sake.

Explanation

Let the incomes of X, Y and Z be 12x, 9x and 7x respectively, and let their expenditures be 15y, 9y and 8y respectively.

Since X saves 25% of his income, his expenditure must be 75% of his income. This relation can be represented as: 15y = ⁷⁵⁄₁₀₀ × 12x 15y = ¾ × 12x 15y = 9x (x)/(y) = ¹⁵⁄₉ = ⁵⁄₃

By setting x = 5 and y = 3, we can determine the individual values of income and expenditure: Income of X = 12 × 5 = 60 Income of Y = 9 × 5 = 45 Income of Z = 7 × 5 = 35

Expenditure of X = 15 × 3 = 45 Expenditure of Y = 9 × 3 = 27 Expenditure of Z = 8 × 3 = 24

Savings are calculated as Income minus Expenditure: Savings of X = 60 - 45 = 15 Savings of Y = 45 - 27 = 18 Savings of Z = 35 - 24 = 11

The ratio of their savings is 15 : 18 : 11.

Explanation

By definition, a prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. Because 1 has only one positive divisor, it does not meet this definition and is classified as neither prime nor composite. The first and smallest prime number is 2. Therefore, statement (iv) is incorrect.

Explanation

Assertion (A) is correct because crude oil exists in vast quantities in natural subterranean reservoirs across various regions of the globe, making it abundantly available in nature.

Reason (R) is also correct because crude oil is the primary raw material processed to make petrol, diesel, synthetic rubber, plastics, and lubricating oils, which are the essential fuels and manufacturing inputs for automobiles.

The reason, however, is not the correct explanation of the assertion. The geological abundance of crude oil is a consequence of prehistoric organic matter undergoing chemical transformations over millions of years under high pressure and temperature, which is entirely independent of its human consumption or utility in the automobile industry.

Explanation

Two circles with radii r₁ and r₂ have exactly 3 common tangents if and only if they touch externally. For two circles to touch externally, the distance d between their centres must be equal to the sum of their radii, which is expressed by the formula:

d = r₁ + r₂

Given the radii are r₁ = 4 cm and r₂ = 5 cm, substituting these values into the formula yields:

d = 4 + 5 = 9 cm

Explanation

Vaccines work by introducing a weakened or inactive part of a pathogen into the body, prompting the immune system to produce antibodies and develop immunological memory to prevent future infections, which makes Assertion (A) true. It is also true that vaccines are essential for pediatric health and must be administered to children to protect them from infectious diseases at an early age, making Reason (R) true. However, the assertion describes the medical function of vaccines, whereas the reason states a demographic policy and healthcare requirement for children, which does not explain the biological mechanism of how vaccines prevent diseases.

Explanation

The group consists of seven members: A, B, C, D, E, F, and G travelling in three different vehicles (Maruti, Santro, and Opel).

The condition states that there are at least two passengers in each vehicle, and only one of them is a male in each vehicle. Since there are three vehicles and each has exactly one male passenger, the total number of male passengers in the group is exactly 3.

Subtracting the number of male passengers from the total number of travellers gives the number of female (lady) passengers:

7 - 3 = 4

Thus, there are exactly four lady members in the group.

Explanation

To find which statement is incorrect, we can determine the gender of each person. Each of the three vehicles has only one male passenger, meaning there are exactly three male passengers and four female passengers among the seven individuals.

From the given information: B is a male engineer. D is a male doctor. A and F are sisters, so both are female. C is a lady doctor, so C is female. B travels only with G in Maruti. Since B is male and each vehicle has only one male, G must be female.

Thus, the four females are A, F, C, and G. This leaves B, D, and E as the three males. Since E is a male, the statement that E is a female is incorrect.

Explanation

To find F's profession, we analyze the distribution of passengers, their genders, and their professions across the three vehicles.

Since there are seven passengers and each vehicle has at least two passengers with exactly one male, there are exactly three males and four females.

B is a male engineer who travels only with G, a teacher, in Maruti. Thus, Maruti has only two passengers.

The remaining five passengers are A, C, D, E, and F. The males are B, D (doctor), and E. Since each vehicle has exactly one male, the remaining males D and E must be distributed between Santro and Opel.

A (female) travels in Santro. C is a lady doctor who does not travel with the sisters A and F. Therefore, C must travel in Opel, and F must travel in Santro with A.

This means Santro holds three passengers: A (female), F (female), and one male. Opel holds C (female doctor) and one male. If D (doctor) travels in Opel, it would violate the rule that no two persons of the same profession travel together because C is also a doctor. Thus, E must travel in Opel, and D (doctor) must travel in Santro.

In Santro, the three passengers (A, F, and D) must represent three different professions. Since D is a doctor, A and F must be an engineer and a teacher. Given that A is not an engineer, A must be the teacher, leaving F to be the engineer.

Explanation

Mahatma Gandhi withdrew the Non-Cooperation Movement in February 1922 following the Chauri-Chaura incident. During a clash between protesters and police in Chauri-Chaura, Gorakhpur district, Uttar Pradesh, on 4 February 1922, an angry crowd set fire to a police station, leading to the death of 22 policemen. Committed to absolute non-violence, Gandhiji felt that the movement was turning violent and that the masses were not yet fully trained for peaceful resistance, leading him to suspend the nationwide campaign.

Explanation

Let the weights of A, B, and C be represented by W_A, W_B, and W_C respectively.

From Statement I, we have W_A > W_B.

From Statement II, we have W_C > W_B.

Combining these relations shows that both A and C weigh more than B. Since no information is given to compare W_A and W_C, it cannot be determined whether C weighs more than A. Consequently, the statement that C weighs the most is uncertain.

Explanation

The given series follows an alternating logic where each subsequent term is calculated by multiplying the previous term by 2 and then alternatingly adding 2 and subtracting 3.

The pattern is as follows: 4 × 2 + 2 = 10 10 × 2 - 3 = 17 17 × 2 + 2 = 36 36 × 2 - 3 = 69 69 × 2 + 2 = 140

To find the missing term, we follow this alternating rule by multiplying the last term by 2 and subtracting 3: 140 × 2 - 3 = 277

Explanation

To find the value of the question mark, represent all terms in the equation using a common base of 2 by applying the laws of exponents.

The term 8 can be written as 2³, which gives: 8⁴ = (2³)⁴ = 2^(3 × 4) = 2¹²

The term 16 can be written as 2⁴, which gives: 16¹.6 = (2⁴)¹.6 = 2^(4 × 1.6) = 2⁶.4

Substitute these values back into the equation: 2¹² ÷ 2⁶.4 × 2⁵.4 = 2^?

Using the exponent rules aᵐ ÷ aⁿ = a^(m - n) and aᵐ × aⁿ = a^(m + n), simplify the left side of the equation: 2^(12 - 6.4 + 5.4) = 2^? 2^(5.6 + 5.4) = 2^? 2¹¹ = 2^?

Comparing the powers on both sides yields: ? = 11

Explanation

Let the total capacity of the box be represented by C. According to the given condition, the number of oranges added on each successive day is equal to the number of oranges already present in the box. This implies that the quantity of oranges in the box doubles every day.

Let Q(t) be the quantity of oranges in the box on day t. The relationship between the quantity of oranges on consecutive days is: Q(t) = 2 Q(t-1)

This can be written in reverse to find the quantity on the preceding day: Q(t-1) = ½ Q(t)

The box is completely filled on the 24th day, meaning: Q(24) = C

On the 23rd day, the box is half filled: Q(23) = ½ Q(24) = ½ C

On the 22nd day, the box is one-fourth filled: Q(22) = ½ Q(23) = ¼ C

Explanation

Mohini starts at the South-East corner of the rectangular plot and walks diagonally towards the opposite corner, which is the North-West corner. Therefore, her initial direction of movement is North-West.

When she turns left (which is a 90° turn counterclockwise), her direction changes from North-West to South-West.

Subsequently, when she turns right (which is a 90° turn clockwise), her direction changes from South-West back to North-West.

Thus, she is currently facing the North-West direction.

Explanation

From Statement I, we have x³{2} is even. This can be represented as x³{2} = 2k for some integer k, which gives x³ = 4k. If x is restricted to integers, then x³ being a multiple of 4 means x³ is even, which implies x must be even. However, if x is not restricted to integers, x could be a non-integer like ³√4, which makes x³{2} = 2 (even) but x is not an even integer. Hence, Statement I alone is not sufficient.

From Statement II, we have 2³√x is even. This can be represented as 2³√x = 2m for some integer m, which simplifies to ³√x = m. Cubing both sides, we get x = m³. Since m is an integer, x must also be an integer. However, m can be either even or odd. If m = 1, then x = 1 (odd). If m = 2, then x = 8 (even). Hence, Statement II alone is not sufficient.

Combining both statements, Statement II establishes that x is an integer. Statement I establishes that if x is an integer, x must be even because x³ is a multiple of 4. Therefore, both statements together are sufficient to determine that x is indeed even.

Explanation

To find the number of unchanged positions, write down the letters of the word IMPOSSIBLE along with their original positions from left to right: 1: I, 2: M, 3: P, 4: O, 5: S, 6: S, 7: I, 8: B, 9: L, 10: E.

The alphabetical order of these letters is B, E, I, I, L, M, O, P, S, S. Reversing this alphabetical sequence yields the reverse alphabetical order: S, S, P, O, M, L, I, I, E, B.

Aligning the original word and the reverse alphabetical arrangement side by side for comparison: Original: I, M, P, O, S, S, I, B, L, E Reverse: S, S, P, O, M, L, I, I, E, B

Comparing the letters at each position reveals that the third position (P), the fourth position (O), and the seventh position (I) have the exact same letters in both sequences. This means there are three positions that remain completely unchanged.

Explanation

Let P, Q, R, S, and T represent the number of cards held by each player.

From the first statement, if Q gives 3 cards to P, Q will have as many as T: T = Q - 3

If P gives 3 cards to Q, Q will have as many as S: S = Q + 3

From the second statement, P and Q together have 10 more cards than S and T together: P + Q = S + T + 10

Substituting the expressions for S and T gives: P + Q = (Q + 3) + (Q - 3) + 10 P = Q + 10

From the third statement, Q has 2 cards more than R: R = Q - 2

Given that the total number of cards is 133: P + Q + R + S + T = 133

Substituting all variables in terms of Q: (Q + 10) + Q + (Q - 2) + (Q + 3) + (Q - 3) = 133 5Q + 8 = 133 5Q = 125 Q = 25

Explanation

The sum of the cubes of the first n natural numbers is given by the formula: ᵢ₌₁ⁿ i³ = [ ⁿ⁽ⁿ⁺¹⁾⁄₂ ]² = ((n² + n)²)/(4)

The sum of the first n natural numbers is given by the formula: ᵢ₌₁ⁿ i = ⁿ⁽ⁿ⁺¹⁾⁄₂ = (n² + n)/(2)

Subtracting the second sum from the first yields: 1³ + 2³ + … + n³ - (1 + 2 + … + n) = ((n²+n)²)/(4) - (n²+n)/(2)

Taking the common factor of (n² + n) / 4 from the terms: (n²+n)/(4) [ (n²+n) - 2 ]

This simplifies to: ((n² + n)(n² + n - 2))/(4)

Explanation

By adding 984 to the 3-digit number 4a3, we get 13b7: 4a3 + 984 = 13b7

Writing this equation in place-value form: (400 + 10a + 3) + 984 = 1300 + 10b + 7 1387 + 10a = 1307 + 10b 10(b - a) = 80 b - a = 8

Since a and b are single-digit whole numbers, the possible pairs for (a, b) are: If a = 0, then b = 8, making the 4-digit number 1387. If a = 1, then b = 9, making the 4-digit number 1397.

We want to find the minimum positive integer c such that (13b7 + c) is completely divisible by 8.

For 13b7 = 1387: Dividing 1387 by 8 yields 1387 = 8 × 173 + 3, leaving a remainder of 3. To make (1387 + c) divisible by 8, the minimum positive integer c is 8 - 3 = 5.

For 13b7 = 1397: Dividing 1397 by 8 yields 1397 = 8 × 174 + 5, leaving a remainder of 5. To make (1397 + c) divisible by 8, the minimum positive integer c is 8 - 5 = 3.

Comparing the two possible cases, the absolute minimum value for c is 3.

Explanation

The given expression is (25)¹³ × (10)⁷ × (27)⁵.

To find the total number of prime factors, we express each base as a product of its prime factors:

25 = 5² 10 = 2 × 5 27 = 3³

Substituting these prime representations back into the expression:

((5)²)¹³ × (2 × 5)⁷ × ((3)³)⁵

Applying the laws of exponents, this simplifies to:

5²⁶ × 2⁷ × 5⁷ × 3¹⁵

Combining the terms with the same base:

2⁷ × 3¹⁵ × 5²⁶⁺⁷ = 2⁷ × 3¹⁵ × 5³³

The total number of prime factors, including their multiplicities, is the sum of the exponents of the prime bases:

7 + 15 + 33 = 55

Explanation

Using modular arithmetic, we write the relation:

29 ≡ 1 28

Raising both sides of the congruence to the power of 22:

(29)²² ≡ 1²² 28

Since 1²² = 1, we get:

(29)²² ≡ 1 28

Alternatively, using the binomial theorem, we can express the number as:

(29)²² = (28 + 1)²²

Expanding this binomial expression:

(28 + 1)²² = 22{0} (28)²² + 22{1} (28)²¹ + … + 22{21} (28)¹ + 1²²

Every term in this expansion except the last one has 28 as a factor and is thus perfectly divisible by 28. The remainder is determined solely by the last term, which is:

1²² = 1

Explanation

Any integer can be represented in one of the forms 6k, 6k+1, 6k+2, 6k+3, 6k+4, or 6k+5, where k is a non-negative integer. Since P is a prime number greater than 5, it cannot be even or divisible by 3. The forms 6k, 6k+2, and 6k+4 are even numbers, and the form 6k+3 is a multiple of 3, meaning P cannot take any of these forms. Consequently, P must be of the form 6k+1 or 6k+5. Dividing a number of the form 6k+1 by 6 yields a remainder of 1, and dividing a number of the form 6k+5 by 6 yields a remainder of 5. Thus, the remainder when P is divided by 6 is always 1 or 5.

Explanation

The sum of the squares of the first n natural numbers is given by the formula: 1² + 2² + … + n² = (n(n + 1)(2n + 1))/(6)

The sum of the first n natural numbers is given by the formula: 1 + 2 + … + n = (n(n + 1))/(2)

Subtracting the second sum from the first sum yields the given expression: 1² + … + n² - (1 + … + n) = (n(n + 1)(2n + 1))/(6) - (n(n + 1))/(2)

Factoring out the common term (n(n + 1))/(2) from both fractions: (n(n + 1))/(2) ( (2n + 1)/(3) - 1 ) = (n(n + 1))/(2) ( (2n + 1 - 3)/(3) )

(n(n + 1))/(2) ( (2n - 2)/(3) ) = (n(n + 1))/(2) · (2(n - 1))/(3)

Canceling the common factor of 2 in the numerator and denominator: (n(n + 1)(n - 1))/(3)

Using the algebraic identity (n + 1)(n - 1) = n² - 1: n(n² - 1){3}

Explanation

When two numbers N₁ and N₂ are divided by a divisor d leaving remainders r₁ and r₂ respectively, their sum N₁ + N₂ divided by d leaves a remainder r₃ given by the relation:

r₁ + r₂ = kd + r₃

where k is a positive integer. Substituting the given values:

12 + 38 = kd + 6

50 = kd + 6

kd = 44

Since a divisor must always be strictly greater than any remainder it produces, we have:

d > 12 and d > 38 ⟹ d > 38

The only factor of 44 that is greater than 38 is 44 itself (where k = 1). Hence, the divisor is 44.

Explanation

To find the number of integers between 500 and 600 where the digit 9 occurs exactly once, we analyze the three-digit numbers of the form 5ab, where a represents the tens digit and b represents the units digit.

Since the hundreds digit is fixed as 5, the digit 9 can only appear in either the tens place or the units place, but not both.

In the first case, the tens digit is 9 (a = 9). For the digit 9 to occur only once, the units digit b can be any digit from 0 to 9 except 9. This gives 9 possible values for b (namely 0, 1, 2, 3, 4, 5, 6, 7, and 8), resulting in 9 such numbers: 590, 591, 592, 593, 594, 595, 596, 597, and 598.

In the second case, the units digit is 9 (b = 9). Similarly, the tens digit a can be any digit from 0 to 9 except 9. This gives 9 possible values for a (namely 0, 1, 2, 3, 4, 5, 6, 7, and 8), resulting in 9 such numbers: 509, 519, 529, 539, 549, 559, 569, 579, and 589.

Adding the possibilities from both mutually exclusive cases, we get 9 + 9 = 18 numbers.

Explanation

Let the eight seats around the circular table be numbered 1 to 8 in clockwise order.

According to Statement I, let F sit at seat 1. Since D sits second to the right of F, D is placed at seat 3. D is opposite to E, which means E must be at seat 7. The occupied seats are now 1, 3, and 7. The remaining vacant seats are 2, 4, 5, 6, and 8. Since A is an immediate neighbour of both B and C, they must occupy three consecutive vacant seats with A in the middle. The only consecutive vacant seats available are 4, 5, and 6. Therefore, A must be at seat 5, with B and C at seats 4 and 6. The seat opposite to A (seat 5) is seat 1, which is occupied by F. Thus, Statement I alone is sufficient to determine that F is opposite to A.

According to Statement II, we only know the relative positions of G, E, and A, which does not provide enough information to determine who occupies the seat opposite to A. Thus, Statement II alone is not sufficient.

Explanation

The number of zeroes at the end of any product is determined by the number of factors of 10, which is equal to the number of pairs of 2 and 5 in its prime factorization. Therefore, the number of trailing zeroes is given by the minimum of the exponent of 2 and the exponent of 5 in the overall product.

Analyzing the prime factorization of each number in the expression to find the powers of 2 and 5: 8 contributes three 2s 12 contributes two 2s 14 contributes one 2 15 contributes one 5 27 contributes no 2s or 5s 32 contributes five 2s 25 contributes two 5s 22 contributes one 2 40 contributes three 2s and one 5 75 contributes two 5s 125 contributes three 5s 212 contributes two 2s

Summing the exponents of 2: Power of 2 = 3 + 2 + 1 + 5 + 1 + 3 + 2 = 17

Summing the exponents of 5: Power of 5 = 1 + 2 + 1 + 2 + 3 = 9

Since the number of zeroes is the minimum of these two powers, we compute min(17, 9) = 9.

Explanation

The set of natural numbers N consists of positive counting integers, namely {1, 2, 3, …}. The set of whole numbers W includes all natural numbers along with zero, which means N is a subset of W. The set of integers I contains all whole numbers and negative integers, which means W is a subset of I. The set of rational numbers Q consists of all numbers that can be expressed as a fraction p/q of two integers where q is not zero. Since any integer n can be represented as n/1, I is a subset of Q. Finally, the set of real numbers R comprises all rational and irrational numbers, which means Q is a subset of R. This establishes the sequential subset relation N⊂ W⊂ I⊂ Q⊂ R.

Explanation

Let the number be represented as N = 527k + 37, where k is an integer.

To find the remainder when N is divided by 17, we analyze the expression modulo 17: N ≡ 527k + 37 17

Since 527 is a multiple of 17 (17 × 31 = 527), the term 527k is completely divisible by 17 and leaves a remainder of 0. This simplifies the congruence to: N ≡ 37 17

Dividing 37 by 17 yields a quotient of 2 and a remainder of 3, as 37 = 17 × 2 + 3. Thus, the remainder when the number is divided by 17 is 3.

Explanation

Convert each radical to a fractional exponent of a (a>1, so larger exponent ⇒ larger value). I = ∛(∜(a³)) = a^(3/12) = a^(1/4) ≈ a^0.250. II = ∛(⁵√(a⁴)) = a^(4/15) ≈ a^0.267. III = √(∛a) = a^(1/6) ≈ a^0.167. IV = ⁵√(a³) = a^(3/5) = a^0.600. Ascending order of exponents: 1/6 < 1/4 < 4/15 < 3/5, i.e., III < I < II < IV.

Explanation

The unit digit of a product of numbers is determined solely by the product of the unit digits of the individual numbers.

Let the unknown digit represented by * be d. The unit digits of the given numbers 31, 49, and 34 are 1, 9, and 4 respectively.

The product of these known unit digits is: 1 × 9 × 4 = 36

Since the unit digit of 36 is 6, the unit digit of the entire product is given by the relation: d × 6 ≡ 2 (mod 10)

To solve for the single digit d, we divide the congruence by gcd(6, 10) = 2: 3d ≡ 1 (mod 5)

Multiplying both sides by the modular inverse of 3 modulo 5 (which is 2): d ≡ 2 (mod 5)

For a single digit d where 0 ≤ d ≤ 9, the possible values are: d = 2 d = 2 + 5 = 7

Therefore, the digit in place of * will always be 2 or 7.

Explanation

Let the four prime numbers in ascending order be a, b, c, and d.

The product of the first three numbers is given as: a · b · c = 385

The product of the last three numbers is given as: b · c · d = 1001

Since b and c are common to both products, their product b · c is the greatest common divisor of 385 and 1001.

Finding the prime factorization of both products: 385 = 5 · 7 · 11 1001 = 7 · 11 · 13

Comparing the prime factors for a · b · c = 5 · 7 · 11 in ascending order, we get: a = 5 b = 7 c = 11

Comparing the prime factors for b · c · d = 7 · 11 · 13 in ascending order, we get: b = 7 c = 11 d = 13

The four prime numbers are 5, 7, 11, and 13. The third of these prime numbers is 11.

Explanation

The product of odd integers is always an odd integer. If n⁵ is an odd number, then n must be an odd integer because any power of an even integer is always even. Since n is an odd integer, raising it to any positive integer power will also yield an odd integer. Thus, n³ = n × n × n is a product of three odd integers, which must be odd. Therefore, the statement that n³ is even is definitely false.

Explanation

To determine the total number of students in the class, we need to find the sum of male students, denoted as m, and female students, denoted as f, which is m + f.

Statement I gives the relation m = f + 10. This is a single linear equation with two variables, which has infinitely many possible solutions. Thus, Statement I alone is not sufficient to find the total number of students.

Statement II states that the average of the obtained marks of male and female students is equal. This relates solely to the marks of the students and does not provide any information regarding the physical count of the students. Thus, Statement II alone is not sufficient.

Combining both statements, we have the relation m = f + 10 and the equality of their average marks. Since the average marks do not provide any numerical value or constraint on the count of students, we are still left with one equation, m = f + 10, for two variables. A unique solution for the total number of students m + f cannot be determined. Therefore, even both statements together are not sufficient to answer the question.

Explanation

To determine Usha's salary, we need a specific monetary value because the initial ratio 5 : 4 only gives the relative proportion. Statement II provides the absolute value of Preeti's salary as ¥ 80,000 per month. Using the ratio, Usha's salary can be directly calculated as ⁵⁄₄ × 80,000 = ¥ 100,000. Statement I merely reiterates the ratio in percentage form (80% of Usha's salary is equivalent to the 4 : 5 ratio) and does not provide any numerical value to find the actual salary. Thus, Statement II alone is sufficient.

Explanation

By using the principles of modular arithmetic, we can express the number 19 in terms of its remainder when divided by 20:

19 ≡ -1 20

Raising both sides of this congruence to the power of 51:

19⁵¹ ≡ (-1)⁵¹ 20

Since 51 is an odd exponent, we have:

(-1)⁵¹ = -1

Thus:

19⁵¹ ≡ -1 20

To convert a negative remainder to a positive remainder within the standard divisor range, we add the divisor 20 to the negative remainder:

-1 + 20 = 19

Explanation

In the triad (256, 8, 529), the sum of the square roots of the first and third numbers is √(256) + √(529) = 16 + 23 = 39. This does not equal the square of the middle number, which is 8² = 64. The group fails to satisfy the rule where the sum of the square roots of the first and third numbers equals the square of the second number.

Explanation

A proposition is a declarative sentence that is either true or false, but not both.

The statement 3 + 5 = 10 is a declarative sentence. Although the mathematical assertion is incorrect, it has a definite truth value of false, which satisfies the definition of a proposition.

In contrast, the statement x + y = 15 is an open sentence containing variables x and y. Because no values are specified for these variables, it is impossible to determine whether the statement is true or false. Consequently, it does not have a fixed truth value and does not qualify as a proposition.

Explanation

A proposition is a declarative sentence that is either true or false, but not both.

Sentence I is an open sentence rather than a proposition. Its truth value depends on the variable x, and because it is not quantified with a universal or existential quantifier, nor is x given a specific value, it cannot be classified as definitively true or false.

Sentence II is a question, which is an interrogative sentence. Since questions do not declare facts and lack a truth value, they are not propositions.

Explanation

The logic followed in the given analogy is to subtract the sum of the digits of the number from the number itself, and then square the resulting difference.

For the first pair: The number is 21. The sum of its digits is 2 + 1 = 3. Subtracting the sum from the number gives 21 - 3 = 18. Squaring this result gives 18² = 324.

For the second pair: The number is 32. The sum of its digits is 3 + 2 = 5. Subtracting the sum from the number gives 32 - 5 = 27. Squaring this result gives 27² = 729.

Applying the same pattern to the fifth number: The number is 43. The sum of its digits is 4 + 3 = 7. Subtracting the sum from the number gives 43 - 7 = 36. Squaring this result gives 36² = 1296.

Explanation

In the alphabetical sequence, the letters of the group TRP have positions T = 20, R = 18, and P = 16. This represents a constant decremental pattern of 2 between consecutive letters. The other groups in the set follow a decremental pattern of 1 followed by 2. This makes TRP the odd group of letters.

Explanation

The statement that some lawyers are engineers implies there is a non-empty intersection between the set of lawyers and the set of engineers. Since all engineers are brokers, the entire set of engineers is a subset of the set of brokers. Therefore, the common region between lawyers and engineers is also part of the set of brokers, which guarantees that some brokers are indeed lawyers, making conclusion II correct.

The statement that all doctors are lawyers means the set of doctors is entirely contained within the set of lawyers. Assuming the standard syllogistic rule of existential import where the subject class is non-empty, it directly follows that some of the lawyers are doctors, which makes conclusion IV correct.

Explanation

Reduced eye contact is a key non-verbal signal of withdrawal, disinterest, or a desire to disengage from a discussion. In interpersonal communication, maintaining eye contact signifies active listening, participation, and attentiveness. When an individual significantly decreases or avoids eye contact, it indicates a psychological retreat or emotional distancing, as they actively minimize connection and communication with other participants in the conversation. This behavior serves as a subconscious mechanism to limit engagement and signal a transition out of the active discussion.

Explanation

When facing interpersonal or leadership challenges in a new workplace, the most constructive and professional initial step is to gather contextual information and understand team dynamics. Speaking with established team members who have experience working under the same manager helps in gaining valuable insights into the manager's behavioral patterns, expectations, and communication preferences. This approach allows a new employee to learn successful adaptation strategies and coping mechanisms from experienced colleagues, enabling them to build a better working relationship and navigate the organizational culture effectively.

Explanation

The code is determined by two distinct rules for the letter and the number.

For the letter part of the code: The letter is obtained by shifting the first letter of the word backward by 3 positions in the alphabetical order. In DUSTILY, the first letter is D (4th letter). Shifting back by 3 gives 4 - 3 = 1, which corresponds to A. In FICTION, the first letter is F (6th letter). Shifting back by 3 gives 6 - 3 = 3, which corresponds to C. Applying this to VIOLENT, the first letter is V (22nd letter). Shifting back by 3 gives 22 - 3 = 19, which corresponds to S.

For the number part of the code: The number is the sum of the alphabetical position values of all letters in the word except the first letter. In DUSTILY, the sum of position values of U, S, T, I, L, Y is 21 + 19 + 20 + 9 + 12 + 25 = 106. In FICTION, the sum of position values of I, C, T, I, O, N is 9 + 3 + 20 + 9 + 15 + 14 = 70. Applying this to VIOLENT, the sum of position values of I, O, L, E, N, T is 9 + 15 + 12 + 5 + 14 + 20 = 75.

Combining these two parts, VIOLENT is coded as S75.

Explanation

The coding pattern represents each letter by the single-digit sum of its alphabetical position value. For MATCH, the positional values are M = 13, A = 1, T = 20, C = 3, and H = 8. Summing the digits of these positions gives 1 + 3 = 4, 1, 2 + 0 = 2, 3, and 8, resulting in the code 41238. For FLOOR, the positional values are F = 6, L = 12, O = 15, O = 15, and R = 18. Summing the digits gives 6, 1 + 2 = 3, 1 + 5 = 6, 1 + 5 = 6, and 1 + 8 = 9, which matches the code 63669. Applying this rule to DEVELOP, the alphabetical positions are D = 4, E = 5, V = 22, E = 5, L = 12, O = 15, and P = 16. Summing the digits of these positions yields 4, 5, 2 + 2 = 4, 5, 1 + 2 = 3, 1 + 5 = 6, and 1 + 6 = 7, which gives the code 4545367.

Explanation

The pattern followed in the given pairs is as follows:

For the first letter, the alphabetical position of the letter increases by 4. For the second letter, the alphabetical position of the letter increases by 11. For the number, the resulting number is the cube of the starting number.

Applying this pattern to the first and second terms: T (20) + 4 = X (24) R (18) + 11 = 29, which corresponds to C (3) after wrapping around the 26-letter alphabet. 4³ = 64 Thus, TR4 becomes XC64.

Applying the pattern to the third and fourth terms: A (1) + 4 = E (5) C (3) + 11 = N (14) 3³ = 27 Thus, AC3 becomes EN27.

Applying the same pattern to the fifth term FP6: F (6) + 4 = J (10) P (16) + 11 = 27, which corresponds to A (1) after wrapping around the alphabet. 6³ = 216 Thus, FP6 becomes JA216.

Explanation

The word gullible is used to describe a person who is easily persuaded, tricked, or cheated because they are overly trusting, naive, or quick to believe what others tell them. This characteristic makes them highly susceptible to deception.

Explanation

The phrasal verb hammer out means to reach an agreement, solution, or compromise after a long, intense, and detailed discussion or negotiation. In the given sentence, the context of negotiations continuing into their fifth day indicates a prolonged and arduous process of working through differences to finalize an agreement. Therefore, hammer out is the appropriate phrasal verb to complete the sentence.

Explanation

The word indignant means feeling or showing anger, resentment, or annoyance, typically at what is perceived as unfair treatment. Its opposite is cool, which means calm, composed, unruffled, and free from excitement or passion. While wrathful and angry are synonyms of indignant, and indifferent indicates a lack of interest or concern, cool directly represents the opposite emotional temperament of being calm and collected instead of heated and upset.

Explanation

The word blundering is defined as making a clumsy, careless, or stupid mistake. In the passage, it describes the error-prone actions and arguments of less disciplined minds who misstep during controversies. Replacing it with committing mistake perfectly captures this sense of making errors or acts of oversight.

Explanation

The passage is an excerpt from John Henry Newman's famous work, The Idea of a University, in which he outlines the characteristics of a gentleman. In the original text, the final sentence reads: "If he be an unbeliever, he will be too profound and large-minded to ridicule religion or to act against it; he is too wise to be a dogmatist or fanatic in his infidelity." Therefore, the correct spelling of the garbled word in the passage is infidelity, which refers to a lack of religious belief or skepticism in matters of faith.

Explanation

In the context of the passage, the word controversy refers to an intellectual dispute, public disagreement, or clash of opposing viewpoints where parties present reasoning to defend their opinions. A debate is defined as a formal discussion or argument on a particular topic in which opposing arguments are put forward, aligning perfectly with the passage's discussion of arguments, adversaries, and opponents.

Explanation

The word appalling means causing dismay, horror, or shock, and is used to describe something that is extremely bad, unpleasant, or distressing. The word dreadful shares this exact meaning, referring to something that causes great dread, fear, or is extremely disagreeable and shocking. Therefore, dreadful is the closest synonym to appalling in the context of describing an unpleasantly sudden and harsh change in weather.

Explanation

प्रत्यक्ष शब्द का निर्माण प्रति उपसर्ग और अक्ष मूल शब्द के मेल से हुआ है। यहाँ यण स्वर संधि का नियम लागू होता है। यण संधि के नियमानुसार जब इ या ई के बाद कोई अन्य असमान स्वर आता है, तो इ या ई का य् हो जाता है। इस प्रकार प्रति + अक्ष के योग से प्रत्यक्ष शब्द बनता है, जिसमें प्रति उपसर्ग स्पष्ट रूप से निहित है।

Explanation

उड्डयन शब्द का शुद्ध सन्धि-विच्छेद उत् + डयन है। यह व्यंजन सन्धि का एक प्रमुख उदाहरण है। व्यंजन सन्धि के नियमानुसार, यदि त् या द् के बाद ड आए, तो त् या द् का परिवर्तन ड् में हो जाता है। इस नियम के तहत जब उत् उपसर्ग का मेल डयन शब्द से होता है, तब त् के स्थान पर ड् होकर उड्डयन शब्द का निर्माण होता है।

Explanation

अब्ज (यहाँ अज्ज) शब्द का अर्थ कमल होता है। तामरस शब्द भी कमल का पर्यायवाची है। अब्ज और तामरस दोनों ही शब्द कमल के अत्यंत प्रसिद्ध पर्यायवाची हैं। कमल के अन्य महत्वपूर्ण पर्यायवाची शब्दों में जलज, पंकज, नीरज, वारिज, सरोज, अरविंद, पद्म, शतदल, राजीव, उत्पल और अम्बुज शामिल हैं।

Explanation

निरपेक्ष शब्द का विलोम शब्द सापेक्ष होता है। निरपेक्ष का अर्थ किसी भी अन्य वस्तु, स्थिति या व्यक्ति की अपेक्षा न रखने वाला यानी पूर्णतः स्वतंत्र अथवा निरपेक्ष (absolute) होता है, जबकि सापेक्ष का अर्थ किसी अन्य वस्तु या परिस्थिति की अपेक्षा रखने वाला अथवा उससे संबद्ध (relative) होता है।

Explanation

'मनु' शब्द में अपत्यवाचक तद्धित प्रत्यय 'अ' लगाने से 'मानव' शब्द बनता है।

संस्कृत और हिन्दी व्याकरण के नियम के अनुसार, जब किसी शब्द के अंत में 'उ' स्वर हो और उसमें 'अ' प्रत्यय जोड़ा जाता है, तो अंत का 'उ' स्वर 'अव्' में परिवर्तित हो जाता है तथा शब्द के प्रथम स्वर में 'वृद्धि' नियम के तहत वृद्धि हो जाती है (जैसे 'अ' का 'आ' हो जाता है)।

इस नियम के अनुसार: मनु + अ = मानव

यहाँ 'म' का प्रथम स्वर 'अ' बढ़कर 'आ' (मा) हो जाता है, और 'नु' का 'उ' बदलकर 'अव्' (नव) हो जाता है। इस प्रकार 'मनु' और 'अ' के मेल से 'मानव' शब्द का निर्माण होता है।

Explanation

उन्मीलित अलंकार वहाँ होता है जहाँ दो समान रंग या रूप वाली वस्तुएँ अत्यधिक सादृश्य के कारण आपस में मिलकर एकाकार हो जाती हैं और उनका भेद छिप जाता है, किंतु बाद में किसी बाह्य कारण या रंग-परिवर्तन के कारण उन दोनों का भेद स्पष्ट रूप से प्रकट हो जाता है.

प्रस्तुत दोहे में नायिका के अत्यंत गोरे मुख पर सफेद चंदन की बेंदी लगी हुई है. गोरे रंग और चंदन के श्वेत रंग की अत्यधिक समानता के कारण वह बेंदी मुख में मिलकर छिप जाती है और दिखाई नहीं देती. परंतु जैसे-जैसे नायिका पर मदिरापान या यौवन जनित मद की लालिमा चढ़ती है, वैसे-वैसे मुख लाल होने लगता है और उस लाल रंग की पृष्ठभूमि में सफेद चंदन की बेंदी स्पष्ट रूप से उभरकर दिखने लगती है. यहाँ मद-लाली के चढ़ने रूपी विशेष कारण से छिपी हुई बेंदी और गोरे मुख का अंतर उजागर हो गया है, जिसके कारण इसमें उन्मीलित अलंकार है.

Explanation

गद्यांश की पहली पंक्ति में स्पष्ट रूप से उल्लेख किया गया है कि नई आलोचना को अंग्रेजी में New Criticism कहा जाता है। इस साहित्यिक आंदोलन ने अपना यह नाम सन् 1941 ई. में प्रकाशित जॉन क्रोंरसम की पुस्तक द न्यू क्रिटिसज्म से ग्रहण किया था।

Explanation

प्रस्तुत अपठित गद्यांश के अनुसार, 'द न्यू क्रिटिसज्म' (The New Criticism) नामक पुस्तक के लेखक जॉन क्रोंरसम हैं। गद्यांश की दूसरी पंक्ति में स्पष्ट रूप से उल्लेख किया गया है कि नई आलोचना (New Criticism) आंदोलन ने अपना यह नाम सन् 1941 ई. में प्रकाशित जॉन क्रोंरसम की पुस्तक 'द न्यू क्रिटिसज्म' से ग्रहण किया था।

Explanation

गद्यांश में स्पष्ट रूप से उल्लेखित है कि नई आलोचना का जोर इस बात पर था कि साहित्यिक कृतियाँ अपनी संरचना में पूर्णतया आत्मनिर्भर होती हैं और वे अपनी एक निरपेक्ष और स्वायत्त सौन्दर्यपरक सत्ता निर्मित करती हैं। इसके अनुसार साहित्यिक व्याख्याओं का सरोकार कृति की बाह्य परिस्थितियों से न होकर पाठ के सूक्ष्म और विस्तृत भाषिक विश्लेषण पर केंद्रित था, जो संरचना की आत्मनिर्भरता को प्रतिपादित करता है।

Explanation

गद्यांश में स्पष्ट रूप से उल्लेख किया गया है कि 'नई आलोचना' 'टेक्स्ट' या पाठ को आधार मानते हुए कृति के सूक्ष्म और विस्तृत भाषिक विश्लेषण अर्थात् 'क्लोज रीडिंग' को महत्त्व देती थी। इसके अनुसार साहित्यिक व्याख्याओं का मुख्य सरोकार कृति की बाह्य परिस्थितियों अथवा लेखक के मंतव्य से न होकर केवल पाठ की आंतरिक संरचना और उसके अंतर्निहित संबंध-सूत्रों से होता है।

Explanation

गद्यांश के अनुसार, 'नई आलोचना' का जन्म उन्नीसवीं सदी की उस पुरानी आलोचना-दृष्टि की प्रतिक्रिया में हुआ था जिसमें साहित्यिक कृति पर विचार करते समय उसके ऐतिहासिक सन्दर्भ, दार्शनिक आधार, रचनाकार के जीवनवृत्त, उसके घोषित उद्देश्यों, साहित्यिक सन्दर्भ-स्रोतों तथा राजनीतिक एवं सामाजिक अभिप्रायों को ही सर्वोपरि माना जाता था। इसके विपरीत, नई आलोचना ने साहित्यिक कृतियों को अपनी संरचना में पूर्णतया आत्मनिर्भर और स्वायत्त माना तथा पाठ के सूक्ष्म एवं विस्तृत भाषिक विश्लेषण को महत्त्व दिया।

Explanation

दिए गए गद्यांश के अनुसार अमेरिकी विश्वविद्यालयों और कॉलेजों में सन् 1960 ई. के दशक के अन्त तक 'द न्यू क्रिटिसज्म' (नई आलोचना) नामक साहित्यिक प्रविधि का बोलबाला था। गद्यांश की प्रारंभिक पंक्तियों में स्पष्ट रूप से उल्लेख किया गया है कि बीसवीं सदी के मध्यवर्ती दशकों में अमेरिकी साहित्यिक आलोचना के क्षेत्र में 'द न्यू क्रिटिसज्म' नामक रूपवादी अनुशीलन का आन्दोलन सक्रिय था और इसी प्रविधि का प्रभाव सन् 1960 ई. के दशक के अन्त तक विश्वविद्यालयों और कॉलेजों में बना रहा।

Explanation

पेट में दाढ़ी होना (प्रश्न में दाही मुद्रित) मुहावरे का सही अर्थ बचपन में ही बहुत चतुर-चालाक होना होता है। दाढ़ी परिपक्वता और उम्र का संकेत होती है, इसलिए जब किसी बच्चे के संदर्भ में इस मुहावरे का प्रयोग किया जाता है, तो इसका तात्पर्य यह होता है कि वह अपनी आयु से कहीं अधिक बुद्धिमान, चतुर और व्यावहारिक समझ रखता है।