UGC NET Questions (Paper – 1)

B is the definition of a GP (constant common ratio). C is true for positive terms when r > 1. D is true for positive terms when 0 < r < 1. A is false (that describes an AP), and E is false (a GP is not generally an AP). Therefore B, C and D only are correct. Option A: Option A is incomplete because it omits D, the key decreasing-case when 0 < r < 1. Option B: Option B is incorrect since it includes E, which falsely claims every GP is an AP. Option C: Option C is correct because it includes exactly the true statements (B, C, D) and excludes A and E. Option D: Option D is wrong because it includes A, which incorrectly describes a GP as having a fixed additive difference.

In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor called the common ratio. Here, 6÷3=2, 12÷6=2 and 24÷12=2, so the ratio between consecutive terms is consistently 2. This confirms that the common ratio is 2. Hence, option B correctly identifies the multiplier generating this sequence.

Option A:
The common ratio is 2, meaning each term is double the previous term. Starting from 3, multiplying by 2 gives 6, then 12, then 24, which matches the sequence given. In mathematical reasoning, recognising this multiplicative pattern helps to predict future terms and solve related problems. Therefore, 2 is the precise value that fits the definition of common ratio in this context.

Option B:
A common ratio of 3 would require that each term be three times the previous term, producing 3, 9, 27, 81 and so on. This does not match the actual sequence where the next term is only double the previous one. Since the ratios 6÷3, 12÷6 and 24÷12 are all 2, not 3, this option is incorrect.

Option C:
A common ratio of 4 would produce 3, 12, 48, 192, which is very different from the given numbers. None of the consecutive quotients in the sequence equals 4. Therefore, 4 cannot describe how the terms are generated, making this option wrong.

Option D:
A common ratio of 1.5 would create terms like 3, 4.5, 6.75, 10.125, which again do not appear in the question. The actual sequence remains strictly integer and doubles each time, unlike the fractional increase implied by 1.5. Hence, 1.5 is not suitable for this pattern.

The given series multiplies each term by a constant factor to obtain the next one. Specifically, 9 ÷ 3 = 3, 27 ÷ 9 = 3 and 81 ÷ 27 = 3, so the common ratio is 3. To continue the pattern, we multiply 81 by 3 to obtain the next term. This gives 81 × 3 = 243, which preserves the geometric progression.

Option A:
Option A, 162, arises if someone incorrectly multiplies by 2 instead of 3, giving 81 × 2. However, this contradicts the established common ratio of 3 used in all previous steps.

Option B:
Option B, 200, does not correspond to any simple multiple of 81 consistent with the progression and seems unrelated to the pattern.

Option C:
Option C, 225, is 15² and may be chosen due to familiarity with square numbers, but it does not follow from multiplying 81 by the common ratio 3.

Option D:
Option D maintains the constant ratio of 3 between consecutive terms. By applying 81 × 3, we get 243, which is the only option that respects the geometric structure of the series.

A and B define AP and GP. D is true because the sign of the common difference determines whether an AP increases, decreases, or stays constant. E is true because multiplying by −1 flips a non-zero term’s sign each step, alternating between a and −a. C is false (Fibonacci-type recurrence is generally neither AP nor GP). Hence A, B, D and E only are correct.

Option A:
Option A is incomplete as it omits E, a standard GP special case often tested.

Option B:
Option B is correct because it combines the core definitions with typical behaviour of APs and a specific GP example, and excludes C, which is false.

Option C:
Option C is wrong because it omits A, the definition of an arithmetic sequence.

Option D:
Option D is incorrect because it omits B, the definition of a geometric sequence.