Let the population of the planet be x.
Statement (1) implies that 0.6x inhabitants own personal computing devices. Of these, 0.5x inhabitants also own heaters. So, 0.1x inhabitants (or 10% of the inhabitants) own only personal computing devices. Thus, statement (1) ALONE is sufficient.
Statement (2) does not provide sufficient data to answer the question. Using statement (2), the percentage of the inhabitants who own only heaters is found to be 20%. Thus, statement (2) ALONE is not sufficient.

Statement (1) gives x = 2 or −2 because 2² = 4 and (−2)² = 4. Thus, statement (1) ALONE is not sufficient.
Statement (2) specifies that x = 2 because 2³ = 8 and (−2)³ = −8. Thus, statement (2) ALONE is sufficient.

Note that Speed = Distance traveled / Time taken.
Statement (1) provides some information on the time taken, but nothing about the distance traveled. So, statement (1) ALONE is not sufficient.
Statement (2) provides some information on the distance traveled, but nothing about the time taken. So, statement (2) ALONE is not sufficient.
Consider both statements together. Statement (2) specifies that Town R lies midway between Town P and Town Q; so, the distance traveled by both trains is the same. Both trains start at the same time, but Train A crosses Town R before Train B does according to statement (1); so, time taken by Train A is less than that taken by Train B. Since speed = distance / time, Train A travels at a higher speed. Thus, BOTH statements TOGETHER are sufficient.

Statement (1) specifies that x − 3y = 27 (on dividing the given equation by 3). Thus, statement (1) ALONE is sufficient.
Statement (2) specifies that x − 3y = 27 (on dividing the given equation by 2). Thus, statement (2) ALONE is sufficient.

Statement (1) specifies the distance between points A and B, but states nothing about point C. Thus, statement (1) ALONE is not sufficient.
Statement (2) specifies the distance between points B and C, but states nothing about point A. Thus, statement (2) ALONE is not sufficient.
When both statements are considered together, the data they provide give rise to two distinct possibilities, which are not sufficient to answer the question.
(a) If point B lies between points A and C, then the required distance is 140 miles.
(b) If point B does not lie between A and C but lies beyond C, then the required distance is 60 miles.
Thus, statements (1) and (2) TOGETHER are NOT sufficient.

Statement (1) specifies that q = 3 or q = −3, because 3⁴ = 81 and (−3)⁴ = 81. Thus, statement (1) ALONE is not sufficient.
Statement (2) specifies that |q| = 3 giving q = 3 or −3. Thus, statement (2) ALONE is not sufficient.
When both statements are taken together, one still obtains q = 3 or −3. Thus, statements (1) and (2) TOGETHER are NOT sufficient.

For two triangles to be similar, they must have the same angles and their corresponding sides must bear a constant ratio to each other.
Statement (1) does not guarantee that the two triangles are similar because their angles (other than the right-angle) could differ. Thus, statement (1) ALONE is not sufficient.
Statement (2) also does not guarantee similarity because the triangles having the same area could have different angles. Thus, statement (2) ALONE is not sufficient.
Even when the two statements are considered together, they do not guarantee similarity of the triangles because their angles (other than the right-angle) may differ and/or their corresponding sides may not be in the right proportion. Thus, statements (1) and (2) TOGETHER are NOT sufficient.
Note that area of triangle = ½ base × altitude. So, triangles having the same area have the product of the base and altitude equal; however, the base and altitude individually could be different.

Average (arithmetic mean) score on the test = Total marks of all the students / Number of students.
Since the number of students is given in the question, only the total marks of all the students is required.
Statement (1) does not imply that the average score on the test is 50. For example, the average could be 48 if 20 students scored 80 marks each and the other 20 students scored 16 marks each. On the other hand, the average could be 54 if 20 students scored 60 marks each and the other 20 students 48 marks each. In other words, the total marks of all the students and the average cannot be determined. Thus, statement (1) ALONE is not sufficient.
Statement (2) specifies the total marks of all the students. Using the formula, the average score on the test is 2000 / 40 = 50. Thus, statement (2) ALONE is sufficient.

Statement (1) does not provide sufficient information to determine whether 7 is a factor of the integer S because 7S has to be divisible by 7. Thus, statement (1) ALONE is not sufficient.
Statement (2) specifies that 2S is divisible by 7. Since 2 is not divisible by 7, S must be divisible by 7. Thus, statement (2) ALONE is sufficient.

Let p and q be the slopes of lines P and Q, respectively. Since the two lines are perpendicular to each other, their slopes are related by pq = −1.
Statement (1) specifies that p + q = 0 or p = −q. Eliminating p gives q² = 1. So, q = 1 or q = −1. Thus, statement (1) ALONE is not sufficient.
Statement (2) specifies that |p| = |q|, which implies p = −q. As before, q = 1 or q = −1. Thus, statement (2) ALONE is not sufficient.
Since statements (1) and (2) contain the same information, combining them does not help in answering the question. Thus, statements (1) and (2)TOGETHER are NOT sufficient.

Now, 3n(n² − 1) = 3n(n − 1)(n + 1).
For 3n(n² − 1) to be divisible by 12, it is necessary that n(n − 1)(n + 1) be divisible by 4.
Statement (1) specifies that n is odd, which implies that (n − 1) and (n + 1) are even numbers and are each divisible by 2. So, (n − 1)(n + 1) is divisible by 4. Thus, statement (1) ALONE is sufficient.
Statement (2) specifies that n is a multiple of 3. Consider an even multiple of 3 [because an odd multiple has been already considered in statement (1)]. If n is even, then (n − 1) and (n + 1) are odd integers. Since odd integers are not divisible by 2, n must be divisible by 4 for n(n − 1)(n + 1) to be divisible by 4. However, not all even multiples of 3 are divisible by 4. Thus, statement (2) ALONE is not sufficient.
Alternative Approach :
Since statement (1) specifies that n is odd, let n = 2m + 1, where m is a positive integer. Then,
3n(n² − 1) = 3(2m + 1)(4m² + 4m + 1 − 1) = 12(2m + 1)(m² + m), which is divisible by 12.
Thus, statement (1) ALONE is sufficient.
Since statement (2) specifies that n is a multiple of 3, let n = 3m, where m is a positive integer. Then,
3n(n² − 1) = 9m(9m² − 1), which is divisible by 9 but not necessarily 12.
Thus, statement (2) ALONE is not sufficient.

Statement (1) states that one of the angles measures twice another angle.
So, there are two possibilities as indicated below.
(a) The angles are 90^o, 45^o and 45^o (with 90 = 2 x 45).
(b) The angles are 90^o, 60^o and 30^o (with 60 = 2 x 30).
Thus, statement (1) ALONE is not sufficient.
Statement (2) states the angles of the triangle are in arithmetic progression. So, if the largest angle measures 90^o, then the other two angles measure (90 − d)^o and (90 − 2d)^o. Imposing the condition that their sum must be 180^o gives 270 − 3d = 180 or d = 90/3 = 30. The smallest angle measures (90 − 2d)^o or 30^o. Thus, statement (2) ALONE is sufficient.

Let the pencil distribution be represented by (a, b, c), where Box A has a pencils, Box B has b pencils and Box C has c pencils.
Each box has at least 1 pencil. So, out of a total of 8 pencils, 5 pencils now need to be distributed among 3 boxes.
Given that 1, 2, 3, 5 and 7 are the only prime numbers less than 8, statement (1) gives rise to four possibilities: (6, 1, 1), (4, 2, 2), (4, 3, 1) and (4, 1, 3).Thus, statement (1) ALONE is not sufficient.
Statement (2) gives rise to three possibilities: (6, 1, 1), (4, 2, 2) and (2, 3, 3). Thus, statement (2) ALONE is not sufficient.
When both statements are considered together, two possibilities still exist: (6, 1, 1) and (4, 2, 2). Thus, statements (1) and (2) TOGETHER are NOT sufficient.

Let P, A and L be the amounts that Pablo, Argente and Lars received, respectively. The question states that P + A + L = 6000.
Statement (1) gives P = 2A + L. Since the amounts cannot be negative, P must be larger than A and L according to the equation. Thus, statement (1) ALONE is sufficient.
Statement (2) gives L = P − 2A, which may be rearranged to obtain P = 2A + L. As before, the amounts cannot be negative and so P must be larger than A and L. Thus, statement (2) ALONE is sufficient.
Therefore, EACH statement ALONE is sufficient.

Statement (1) implies that the distance between the centres of the two circles is equal to the radius of the larger circle (r₁) minus the radius of the smaller circle (r₂). So, r₁ − r₂ = 4 − 2 = 2 inches. Thus, statement (1) ALONE is sufficient.
Statement (2) implies that the distance between the centres of the two circles is equal to the radius of the smaller circle (r₂ = 2 inches). Thus,
statement (2) ALONE is sufficient.
Therefore, EACH statement ALONE is sufficient.