UGC NET Questions (Paper – 1)

Statements A and B are the precise symbolic forms of De Morgan’s laws for sets. E is also correct because it broadly describes the role of De Morgan’s laws in relating complements to unions and intersections. C and D are false since they swap union/intersection incorrectly: the complement of a union is an intersection of complements, and the complement of an intersection is a union of complements. Hence A, B and E only are correct.

Option A:
Option A is incomplete because it leaves out E and therefore omits the general conceptual summary of what De Morgan’s laws do.

Option B:
Option B is wrong because it adds C, which contradicts De Morgan’s law for complements of unions.

Option C:
Option C is correct as it selects exactly those statements that align with De Morgan’s laws and excludes C and D, which misuse union and intersection in the context of complements.

Option D:
Option D is incorrect since it omits A (a true De Morgan law), so it cannot be correct.

This option is correct because the standard formula for compound interest with annual compounding is A = P(1 + r/100)^n. The factor (1 + r/100) represents the growth factor for one year. Repeating it n times corresponds to compounding the interest every year. Therefore, this expression correctly gives the amount after n years.

Option A:
The expression P(1 + nr/100) resembles simple interest growth, not compound interest. It assumes interest grows linearly with time instead of exponentially. Thus, it does not represent annual compounding and is not correct here.

Option B:
The formula P(1 + r/100)^n multiplies the principal by the growth factor once for each year. This matches the compounding process, where each year's interest is calculated on the new amount. Hence, this option is the correct compound interest formula.

Option C:
The expression P(1 − r/100)^n models a decreasing quantity, such as depreciation, where a fixed percentage is lost each year. It is not appropriate when interest is added rather than subtracted. Therefore, it does not answer the question.

Option D:
The term P(1 − nr/100) assumes a linear decrease in the amount with time, which again is not how compound interest works. It would eventually make the amount zero or negative for large n, which is unrealistic for interest growth. Hence, this option is incorrect.

Each term is a block of four consecutive letters where the starting letters L, M, N and O move forward by one. The blocks heavily overlap, shifting one position to the right each time. After OPQR, the next starting letter must be P, and the block formed is PQRS. Thus PQRS continues the overlapping sliding pattern correctly.

Option A:
Option A, QRST, is a later block that would come after PQRS in the extended pattern. Its starting letter Q skips over P, which goes against the step by step increase of the starting position. Because the series does not skip any intermediate starting letters, QRST is not the immediate next term.

Option B:
Option B is correct because it starts at P, the letter immediately after O, and includes Q, R and S as the next three letters. It preserves both the length of the block and the one position slide of the starting letter. This exact match with the observed progression makes PQRS the right continuation.

Option C:
Option C, RSTU, begins at R, which is two steps beyond O and therefore jumps over P. While RSTU is a valid consecutive block, its position in the overall ordering is too far ahead. As the series moves in single letter increments, RSTU cannot be accepted here.

Option D:
Option D, MNOP, repeats an earlier group from the sequence and thus moves backward. The pattern does not indicate any repetition or looping behaviour. Therefore MNOP is not a valid next term.

The marked price is ₹1,000 and a discount of 20% means the shopkeeper pays 80% of 1,000, which is ₹800. This ₹800 is the cost price. He sells at 10% above the marked price, so the selling price becomes 1,000 × 1.10 = ₹1,100. The profit is 1,100 − 800 = ₹300. The profit percentage is (300 ÷ 800) × 100 = 37.5%. Thus, his profit on the cost price is 37.5%, corresponding to Option A.

Option A:
Option A, 37.5%, corresponds exactly to (300 ÷ 800) × 100. It reflects the correct gain relative to the cost price and matches the computed fraction 3/8.

Option B:
Option B, 40%, would imply a profit of 320 on a cost of 800, which would require a selling price of 1,120, not 1,100, and thus does not fit the given data.

Option C:
Option C, 42.5%, is larger than the true computed profit percentage and would require an even higher selling price than 1,120. It overstates the profit.

Option D:
Option D, 50%, corresponds to a profit of half the cost price, i.e., 400 on 800, which again is inconsistent with a selling price of 1,100.

This option is correct because an even number is defined as any integer that can be written as 2k for some integer k. Being a multiple of 2 means the number fits this form. Therefore, any multiple of 2 is an even number.

Option A:
Even numbers have 2 as a factor and are exactly divisible by 2. Writing the number as 2k shows it is a multiple of 2, which matches the condition in the question. Thus, this option is correct.

Option B:
Odd numbers are not multiples of 2 and leave a remainder of 1 when divided by 2. Since the question specifies multiples of 2, odd numbers cannot be the correct classification.

Option C:
Prime numbers are defined by having exactly two distinct positive divisors. Some primes like 2 are even, but not all multiples of 2 are prime. Many even numbers have more than two divisors, so prime is not the right general term here.

Option D:
Composite numbers have more than two positive divisors and can be even or odd. The phrase "multiple of 2" points specifically to evenness, not to the broader idea of compositeness. Hence, this option is not precise enough.

Let the number be x. The statement “three times the number decreased by 5 equals 16” translates to the equation 3x − 5 = 16. Adding 5 to both sides gives 3x = 21, and then dividing by 3 yields x = 7. Substituting back into the original relation, 3 × 7 − 5 equals 21 − 5 = 16, confirming that 7 satisfies the condition.

Option A:
Option A, 5, when substituted gives 3 × 5 − 5 = 10, which is not equal to 16. Thus, it fails to satisfy the equation implied by the verbal statement.

Option B:
Option B fits the equation exactly, turning 3x − 5 into 16 when x is 7. This closes the loop between the word problem and its algebraic representation.

Option C:
Option C, 9, would make 3 × 9 − 5 = 22, overshooting the stated result 16, so it cannot be correct.

Option D:
Option D, 11, yields 3 × 11 − 5 = 28, which is even further from 16 and clearly inconsistent.

Each term after the first is obtained using the rule "multiply the previous term by 3 and subtract 1". We have 1 × 3 − 1 = 2, 2 × 3 − 1 = 5, 5 × 3 − 1 = 14, 14 × 3 − 1 = 41 and 41 × 3 − 1 = 122. Applying this rule again gives 122 × 3 − 1 = 366 − 1 = 365. This keeps the recursive structure consistent and elegant.

Option A:
Option A gives 362, which would require subtracting 4 after multiplication or using a different constant, something not present earlier in the series. As a result, 362 breaks the pattern.

Option B:
Option B offers 364, corresponding to subtracting 2 after multiplying 122 by 3, and again changes the constant part of the rule. Thus, 364 does not align with the "minus one" pattern.

Option C:
Option C yields 365, exactly equal to 122 × 3 − 1, and extends the rule that has been consistently applied to all previous transitions. The sequence 1, 2, 5, 14, 41, 122, 365 is logically coherent. Hence, 365 is the correct next term.

Option D:
Option D suggests 367, which would involve adding 1 rather than subtracting 1 at the end of the operation. This is not supported by the existing steps. Therefore, 367 cannot be accepted as correct.

This option is correct because for n observations arranged in order, with n odd, the median is located at position (n+1)/2. This ensures that there are an equal number of observations on both sides of the median. The formula is standard in statistics for odd-sized data sets. Therefore, the median is at the ((n+1)/2)th position.

Option A:
Saying nth position would place the median at the last observation, which is the maximum value in an ascending list. This ignores the central nature of the median. Hence, nth position is not correct.

Option B:
The ((n+1)/2)th position ensures that the median lies exactly in the middle of the ordered list when n is odd. For example, if n = 5, the median is at position (5+1)/2 = 3, which has two values on each side. This matches the concept of a central value and so this option is correct.

Option C:
The (n/2)th position is only meaningful when n is even or when adjusted properly. For odd n, n/2 is not an integer and does not indicate a valid central index. Therefore, this option does not provide the correct position formula.

Option D:
The (n−1)th position lies near the end of the data and does not represent the middle. It would heavily bias toward higher values when the data are ordered ascending. Thus, this position is not the median location for an odd number of observations.

Statement A is correct because range is maximum minus minimum. Statement B accurately defines the IQR as Q3 − Q1. Statement D is also true since IQR is less sensitive to outliers than the range. Statement C is false because a single extreme value can change the range a lot, and E is false because range and IQR are generally different. Therefore, A, B and D only are correct.

Option A:
Option A is incomplete because it omits D (robustness of IQR).

Option B:
Option B is incorrect as it includes C (false).

Option C:
Option C is wrong because it includes E (false) and omits A.

Option D:
Option D is correct since it includes all correct statements and excludes C and E.

The differences between successive terms are 11, 15, 19 and 23. These differences form an arithmetic progression with a common difference of 4. Therefore the next difference should be 27. Adding 27 to 75 gives 102 as the next term in the series. This choice keeps both the series and the pattern of gaps internally consistent.

Option A:
Option A would create a difference of 23 from 75, so the differences would be 11, 15, 19, 23, 23. The last two gaps are equal, which breaks the rule that each difference increases by 4. Hence 98 cannot be justified by the underlying pattern.

Option B:
Option B uses a difference of 27, giving the sequence of gaps 11, 15, 19, 23, 27. Each step increases by 4, which matches the observed second-level arithmetic progression. As this option uniquely preserves the pattern, 102 is correctly identified as the next term.

Option C:
Option C would produce a difference of 29 from 75, giving gaps 11, 15, 19, 23, 29. The jump from 23 to 29 is 6 rather than 4, which distorts the arithmetic pattern among the differences. Therefore 104 is not a valid continuation.

Option D:
Option D corresponds to a difference of 21 from 75, so the last increment decreases instead of increasing to 27. This contradicts the steadily rising nature of the gaps. Thus 96 does not follow the logic of the series.

The pair 3 and 9 is related by the operation of squaring because 3 squared is 9. To maintain the same relationship for the second pair, we should square 4. The square of 4 is 16, so x must be 16 to keep the analogy consistent. Therefore, 3 : 9 :: 4 : 16 expresses a uniform pattern based on squaring the first number.

Option A:
Option A, 8, is obtained by doubling 4, but the first pair 3 : 9 is not formed by doubling 3; it is formed by 3². Using ×2 breaks the consistent “square the first number” rule, so 8 is not suitable.

Option B:
Option B, 12, can be seen as 3 × 4, but that mixes the two numbers of the analogy rather than applying the same operation to each first term. The clearer and more consistent pattern is to square the first term (3² = 9, 4² = 16), so 12 does not fit.

Option C:
Option C, 14, does not follow any simple or symmetric rule derived from 4 that parallels 3 → 9. It would require an arbitrary operation such as “add 10”, which has no support from the first pair, so this choice breaks the analogy.

Option D:
Option D, 16, results from squaring 4 (4² = 16), exactly mirroring the relation 3² = 9. This keeps the structure of the analogy identical in both pairs, making 16 the correct value of x.

In this series, the last two letters “AT” remain fixed, while only the first letter changes in alphabetical order. The first letters progress as B, C, D, E, so the next first letter must be F. Attaching the constant suffix “AT” to F gives FAT. Hence FAT is the only word that continues the simple first-letter progression.

Option A:
Option A, GAT, uses the correct suffix but jumps from E to G, skipping F. The rule shows every adjacent letter in order without omissions. Because it violates the immediate alphabetical succession, GAT cannot be the next term.

Option B:
Option B, HAT, goes even further, skipping both F and G. This disregards the clear pattern of moving to the very next letter at each step. Therefore HAT does not fit the orderly progression established in the series.

Option C:
Option C, EAT, simply repeats the last given term instead of advancing the pattern. The series requires a new word with the next first letter, not a repetition. As there is no indication of cycling, EAT cannot be the correct answer.

Option D:
Option D is correct because it keeps “AT” unchanged and moves from E to F exactly once. This agrees perfectly with the observed behaviour of changing only the first letter in strict alphabetical order. Therefore FAT is the logical continuation of the series.

When multiplying powers of the same base, we keep the base unchanged and add the exponents. The expression 3⁴ × 3² therefore becomes 3^(4 + 2). Adding the exponents gives 3⁶. This rule arises directly from expanding each power into repeated multiplication and then counting the total number of factors of 3.

Option A:
Option A, 3⁸, results from incorrectly multiplying the exponents instead of adding them. There is no exponent law that justifies multiplying 4 and 2 in this context.

Option B:
Option B correctly applies the exponent rule aᵐ × aⁿ = aᵐ⁺ⁿ. By adding 4 and 2, it expresses the product as a single power of 3 with exponent 6.

Option C:
Option C, 3², keeps only one of the original exponents and discards the contribution of the other factor. This contradicts the idea of combining powers.

Option D:
Option D, 3⁴, similarly ignores the factor 3² and so cannot be equivalent to the full product.

The starting letters move cyclically forward: X, Y, Z, A, so next is B. Four consecutive letters from B give BCDE.

Option A:
Option A is correct because it starts at B (after A) and keeps 4 consecutive letters.

Option B:
Option B starts at C, skipping the required start B.

Option C:
Option C jumps back and breaks the forward cyclic flow.

Option D:
Option D repeats an earlier term instead of advancing.

This option is correct because 50% means 50 out of 100. Writing this as a fraction gives 50/100. Dividing numerator and denominator by their greatest common divisor 50 gives 1/2. Therefore, 50% equals the fraction 1/2.

Option A:
The fraction 1/3 corresponds to approximately 33.33%, not 50%. It does not represent half of a whole. Hence, this option is not correct.

Option B:
The fraction 1/4 is equal to 25%, which is half of 50%. Therefore, 1/4 is too small to match 50%. So, this option is incorrect.

Option C:
The fraction 1/2 represents exactly half of a unit, which matches the idea of 50%. Converting back to a percentage gives 0.5 × 100 = 50%. Thus, this option correctly expresses 50% as a fraction.

Option D:
The fraction 2/3 corresponds to about 66.67%, which is greater than 50%. Therefore, it does not represent 50% and cannot be the correct answer.