UGC NET Questions (Paper – 1)

This option is correct because the sum of the interior angles of a triangle in Euclidean geometry is always 180 degrees. This can be shown using parallel lines and alternate interior angles. Every triangle can be thought of as half of a straight line angle sum. Therefore, the correct total is 180 degrees.

Option A:
90 degrees is the measure of a right angle, not the sum of all angles in a triangle. Only one angle in a right triangle is 90 degrees. Thus, this value is too small for the total.

Option B:
270 degrees exceeds the known sum of angles in a triangle and does not correspond to any standard triangle property. It is closer to the sum for different configurations, not a simple triangle. Hence, this option is not valid.

Option C:
360 degrees is the angle measure around a point, not inside a triangle. It equals the sum of interior angles of a quadrilateral, not a triangle. Therefore, it cannot be the correct answer.

Option D:
180 degrees is the established sum for any triangle in a plane. No matter the shape of the triangle, the three interior angles always add up to 180 degrees. This property is widely used in geometry problems, confirming this option as correct.

The first-level differences are 15, 25, 35 and 45. These differences themselves form an arithmetic progression with common difference 10. Thus the next difference should be 55. When 55 is added to the last term 128, we obtain 183, which continues the pattern at both the term level and the difference level.

Option A:
Option A gives 178, which would create a difference of 50 from 128. This would make the sequence of differences 15, 25, 35, 45, 50, where the last increase is 5 rather than 10. Hence 178 breaks the structure of the second-level arithmetic progression.

Option B:
Option B gives 188, corresponding to a difference of 60 from 128. That would yield differences 15, 25, 35, 45, 60, causing the final increment to jump by 15 instead of 10. Therefore 188 does not preserve the consistent behaviour among the gaps.

Option C:
Option C gives 183, leading to a difference of 55 from 128. The differences become 15, 25, 35, 45, 55, a clear arithmetic progression with constant step 10. This perfect alignment with the observed pattern makes 183 the correct next term.

Option D:
Option D gives 193, implying a difference of 65 from 128. This creates an irregular spike in the differences and abandons the neat step of 10 between them. Consequently, 193 is not a valid continuation of the series.

This option is correct because 3³ means 3 multiplied by itself three times. Calculating 3 × 3 × 3 gives 27. This follows directly from the definition of exponentiation. Therefore, 3³ equals 27.

Option A:
Squaring 3 gives 3² = 9, which is too small. The exponent 3 requires one more multiplication by 3. Hence, 9 is not the correct value for 3³.

Option B:
The result 27 comes from multiplying 3 by itself three times in succession. The sequence 3, 9 and then 27 confirms that 3³ = 27. This directly matches the required calculation, so this option is correct.

Option C:
The number 18 does not arise from any standard power of 3. It would require 3 × 6, not 3 × 3 × 3. Therefore, 18 is not correct here.

Option D:
The number 21 also does not match the cube of 3. It lies between 16 and 25 but has no connection to 3³. Thus, this option is incorrect.

This option is correct because the standard formula for the nth term of an arithmetic progression is aₙ = a + (n−1)d. Here, a is the first term and d is the constant difference between successive terms. This formula comes from adding the common difference repeatedly. Therefore, the nth term is a + (n−1)d.

Option A:
The expression a + nd adds the common difference n times, which overshoots by one step. It would correctly represent the (n+1)th term, not the nth term. Hence, this option is not correct.

Option B:
The expression a − (n−1)d would produce a decreasing sequence even if d is positive. It does not match the usual pattern for an arithmetic progression defined by adding d each time. Therefore, this option is not suitable.

Option C:
The expression nd ignores the first term a entirely and cannot generate the correct sequence unless a is zero. It fails to describe the general nth term formula. Thus, this option is incorrect.

Option D:
The formula a + (n−1)d starts from a and adds the common difference d exactly n−1 times. This accurately generates each term of the progression in order, making it the correct general term.

This option is correct because the sum of interior angles of a simple quadrilateral is known to be 360 degrees. A quadrilateral can be divided into two triangles, each having an angle sum of 180 degrees. Adding these gives 180 + 180 = 360. Therefore, the total interior angle sum is 360 degrees.

Option A:
A quadrilateral can be split into two triangles by drawing one diagonal. Each triangle has 180 degrees as the sum of its interior angles. Adding both triangle sums gives 360 degrees, so 360 is the correct total for the quadrilateral.

Option B:
180 degrees is the sum of interior angles of a single triangle, not a quadrilateral. Since a quadrilateral has more sides, its angle sum must be greater than that of a triangle. Thus, this option is not correct.

Option C:
270 degrees does not match any standard polygon angle-sum formula for quadrilaterals. It is less than the correct total of 360 degrees and therefore cannot be the right answer.

Option D:
540 degrees is the interior angle sum of a pentagon, not a quadrilateral. Using this value would assume five sides instead of four. Hence, this option is also incorrect.

This option is correct because 2 raised to the power 5 means multiplying 2 by itself five times. Calculating 2 × 2 × 2 × 2 × 2 gives 32. This follows the general definition of exponents as repeated multiplication. Therefore, 2⁵ is equal to 32.

Option A:
16 is the value of 2 raised to the power 4, not 5. It comes from 2 × 2 × 2 × 2, which has only four factors. Since the exponent here is 5, this option underestimates the value.

Option B:
32 is obtained by multiplying five factors of 2 together. The sequence is 2, 4, 8, 16 and finally 32, which corresponds to 2⁵. This exactly matches the requirement of the question, making this option correct.

Option C:
64 is the value of 2 raised to the power 6, which involves six factors of 2. Using 64 would correspond to 2⁶ rather than 2⁵. Hence, this option is too large for the given exponent.

Option D:
25 is the square of 5 and is unrelated to powers of 2 in this context. It does not come from repeated multiplication of 2. Therefore, this option does not answer the question correctly.

The first-level differences are 3, 7, 11, 15 and 19. These differences form an arithmetic progression with common difference 4. Following this pattern, the next difference should be 23. Adding 23 to the last term 57 gives 80, so 80 is the next term that keeps both levels of the pattern consistent.

Option A:
Option A gives 76, which corresponds to a difference of 19 from 57. This simply repeats the last difference and leads to difference values 3, 7, 11, 15, 19, 19, breaking the steady increase by 4. Therefore 76 cannot be the correct continuation.

Option B:
Option B gives 80, which implies a difference of 23 from 57. The differences become 3, 7, 11, 15, 19, 23, forming a clean arithmetic progression with step 4. Hence 80 is fully consistent with the structure of the sequence and is the correct answer.

Option C:
Option C gives 82, leading to a difference of 25 from 57. That would make the final increment between differences equal to 6 instead of 4, distorting the neat second-level pattern. Thus 82 is not suitable.

Option D:
Option D gives 84, implying a difference of 27 from 57, which makes the last step among differences equal to 8. This change is too abrupt and inconsistent with the established progression, so 84 cannot be accepted.

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.

An even number is defined as an integer that can be written in the form 2k, where k is an integer. This means it is exactly divisible by 2 without leaving any remainder. Numbers like −4, 0, 2 and 10 are all even because they satisfy this property. Therefore, the description in the question perfectly matches the standard definition of an even number.

Option A:
Prime numbers are integers greater than 1 that have exactly two distinct positive divisors, 1 and the number itself. While some primes like 2 are even, most primes are not characterised simply by divisibility by 2. The defining property in the question is divisibility by 2 for all such numbers, which corresponds to evenness, not primality. Hence, this option is not correct.

Option B:
Odd numbers are integers that leave a remainder of 1 when divided by 2. They can be expressed as 2k+1 for some integer k. This is the opposite of being exactly divisible by 2, so odd numbers do not fit the description given in the question. Therefore, this option must be rejected.

Option C:
Even numbers are correct because they are precisely those integers that divide by 2 with no remainder. In many aptitude questions, recognising even and odd numbers quickly helps in solving problems on divisibility and parity. Knowing this definition is foundational for reasoning about number patterns and algebraic expressions. Thus, this option accurately answers the question.

Option D:
Composite numbers have more than two positive divisors, such as 4, 6 or 12, but the question is not about the count of divisors. Some composite numbers are even and some are odd, so "composite" is not equivalent to "divisible by 2." As the stem focuses solely on divisibility by 2, composite is not the correct term.

The given sequence increases from 5 to 9, from 9 to 13, and from 13 to 17. In each step, the increase is 4, which means the common difference is 4. Such a sequence with constant difference is an example of an arithmetic progression. Therefore, adding 4 each time correctly describes the pattern generating the sequence.

Option A:
The option 4 is correct because 9−5, 13−9 and 17−13 all equal 4. In arithmetic progression questions, identifying the constant difference is the key step to predicting the next term. If we add 4 to 17, we get 21 as the next term, which fits the discovered pattern. Hence, 4 is the only value that maintains the same step throughout the sequence.

Option B:
Adding 5 each time would give 5, 10, 15, 20, which is not the given sequence. The actual terms 9, 13 and 17 would not be obtained by repeated addition of 5 starting from 5. This shows that 5 does not match the observed pattern. So, this option is incorrect.

Option C:
Adding 3 at each step would produce 5, 8, 11, 14 and so on, which again does not coincide with the given terms 9, 13 and 17. The difference between each pair of consecutive terms is clearly 4, not 3. Therefore, this option does not fit the sequence.

Option D:
Adding 2 each time gives the series 5, 7, 9, 11, which is different from the one given in the question. Although 2 is a possible constant difference in other series, it cannot describe the pattern here because the actual increments are larger. Hence, this option must be rejected.

The given numbers are 1³, 2³, 3³ and 4³, showing a sequence of consecutive cubes. Following this rule, the next term should be 5³. Calculating 5³ gives 125. Hence, 125 is the correct continuation of the cube sequence.

Option A:
Option A matches the rule because 125 is 5³ and follows naturally after 4³ = 64. The extended series 1, 8, 27, 64, 125 represents cubes of 1, 2, 3, 4 and 5 in order. This neat structure is typical in exam-based number series questions.

Option B:
Option B is 100, which is not the cube of any integer. It lies between 4³ and 5³ but does not satisfy the pure cube property. Thus, it does not fit the pattern.

Option C:
Option C is 150, which has no direct connection with consecutive cubes in this context. Choosing 150 would break the exact power relationship between the terms. Therefore, it cannot be correct.

Option D:
Option D is 216, which equals 6³ and skips the cube of 5. While 216 is a cube, the series is moving stepwise through consecutive integers, so skipping 5³ is not logical. Hence, 216 is not the correct next term here.

This option is correct because a prime number is greater than 1 and has exactly two distinct positive divisors. The number 2 is divisible only by 1 and 2, satisfying this definition. It is also the smallest number with this property. Therefore, 2 is the smallest prime number.

Option A:
The number 1 has only one positive divisor, namely itself. Because primes must have exactly two distinct positive divisors, 1 is not considered a prime number. Hence, this option is incorrect.

Option B:
The number 2 is the first integer greater than 1 that has exactly two positive divisors. It is divisible by 1 and itself and no other positive integer. This makes 2 the smallest prime number as required in the question.

Option C:
The number 3 is prime, but it is not the smallest prime. Since 2 is smaller than 3 and already satisfies the definition of a prime, 3 cannot be the correct answer. Therefore, this option is not appropriate.

Option D:
The number 0 is divisible by every non-zero integer and does not fit the standard definition of prime numbers. It is not even positive in the usual sense of positive integers. Thus, 0 cannot be the smallest prime.

This option is correct because an odd number is defined as an integer that is not divisible by 2. When an odd number is divided by 2, it always leaves a remainder of 1. This matches exactly the condition given in the question. Therefore, an integer not divisible by 2 is called an odd number.

Option A:
An odd number has the form 2k+1 for some integer k and is not divisible by 2. Dividing any odd number by 2 leaves a remainder of 1, which fits the description in the stem. Hence, this option correctly names the type of number described.

Option B:
An even number is divisible by 2 and leaves no remainder on division by 2. The question specifies numbers that are not divisible by 2, so this property does not match. Therefore, this option is not correct here.

Option C:
A prime number is defined by having exactly two distinct positive divisors, 1 and the number itself. Some primes are odd but the concept of primality does not directly refer to divisibility by 2 alone. Hence, prime is not the specific term required by the question.

Option D:
A composite number has more than two positive divisors. It can be even or odd, and its definition is unrelated to divisibility by 2 alone. Therefore, composite is not the best term for numbers that are simply not divisible by 2.

The ratio 3 : 2 indicates that out of every 5 equal parts, 3 parts represent boys and 2 parts represent girls. The total number of parts is therefore 3 + 2 = 5. If 5 parts correspond to 90 students, then 1 part equals 90 ÷ 5 = 18 students. The number of boys is 3 parts, which is 3 × 18 = 54.

Option A:
Option A, 36, corresponds to 2 parts of 18 instead of 3, which would actually represent the number of girls if the same logic is applied correctly to the ratio. Hence, it mismatches the role of boys in the ratio.

Option B:
Option B, 45, is obtained if someone wrongly assumes 90 × (1/2) but that would correspond to a hypothetical 1 : 1 ratio. It does not respect the actual 3 : 2 proportion given in the question.

Option C:
Option C follows the correct ratio method: convert the total into parts, find the value of one part and then multiply by the number of parts corresponding to boys. This systematic approach yields 54, aligning with the structure of the ratio.

Option D:
Option D, 60, would imply that boys form two thirds of the class, which does not fit a 3 : 2 ratio where boys are 3/5 of the total. It clearly exceeds the value compatible with the defined proportion.

This option is correct because the square root of a number is a value which, when squared, gives the original number. Here, 9 × 9 = 81. The positive value that satisfies this is 9. Therefore, the positive square root of 81 is 9.

Option A:
If we square 7, we get 49, which is less than 81. Thus, 7 is not the square root of 81. It does not satisfy the required condition.

Option B:
Squaring 8 gives 64, which is still less than 81. Hence, 8 is not the number whose square equals 81 and cannot be the correct answer.

Option C:
The number 9 squared gives exactly 81, so 9 is the positive square root of 81. The question asks specifically for the positive square root, so we choose 9 and not −9. Therefore, this option is correct.

Option D:
81 is the original number itself and would be the square of 9, not its square root. Squaring 81 gives a much larger number, 6561. Thus, 81 cannot be the square root of 81.