Questions & explanations
1. Is x an even integer?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (1) alone: x = 2k with integer k means x is even, so answer is yes. Sufficient. Statement (2) alone: x^2 divisible by 4 does not guarantee x is even. For example, x=2 gives x^2=4 divisible by 4, x even; x=√2 gives x^2=2 not divisible by 4, but x not integer. However, if x is integer, x^2 divisible by 4 implies x is even?, if x is odd, x^2 is odd, not divisible by 4. So for integer x, x^2 divisible by 4 implies x is even. But statement (2) does not specify x is integer. If x is not integer, the question 'is x an even integer?' is false because x is not integer. So we get a definite no?: if x is not integer, answer is no (since not integer). But if x is integer and even, answer yes; if integer and odd, answer no. So we need to consider all possibilities. For x=2, answer yes; for x=1, x^2=1 not divisible by 4, so not allowed; for x=√2, x^2=2 not divisible by 4; for x=0, x^2=0 divisible by 4, x is even integer, answer yes. So statement (2) allows both integer even and non-integer?, if x is not integer, x^2 divisible by 4 is possible? For example, x=2√2, x^2=8 divisible by 4, bu
2. What is the value of integer n?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1): primes between 10 and 20: 11, 13, 17, 19. Not unique. Insufficient. Statement (2): odd numbers with digit sum 5: 5, 23, 41, etc. Not unique. Insufficient. Together: from (1) possible: 11 (sum 2), 13 (4), 17 (8), 19 (10). Only 23 from (2) but 23 not in (1)., check: 23 is not between 10 and 20. So no overlap?, 23 is not in (1). So no integer satisfies both? That means no value, but we need a unique value. Since no integer satisfies both, the statements are contradictory? But statements are always true, so they must be consistent. However, if no integer satisfies both, then the conditions cannot both be true, but the problem assumes they are true. In DS, if statements are contradictory, answer is (e) because together they are not sufficient to determine a unique value. So answer is (e).
3. A company's revenue in 2023 was $500,000. If revenue increased by a constant percentage each year from 2021 to 2023, what was the revenue in 2021?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: EACH statement ALONE is sufficient.
Rephrase: Let R be 2021 revenue, r be annual growth rate. Then 2022 revenue = R(1+r), 2023 = R(1+r)^2 = 500,000. Statement (1): R(1+r) = 550,000. Together with R(1+r)^2 = 500,000, we get (1+r) = 500,000/550,000 = 10/11, so R = 550,000 * (11/10) = 605,000?, check: R(1+r)=550,000 and R(1+r)^2=500,000 => divide: (1+r)=500/550=10/11, then R=550,000/(10/11)=605,000. Unique. Sufficient. Statement (2): 2022 revenue = 1.1R. Then 2023 = 1.1R * (1+r) = 500,000. But we also know 2022 = R(1+r)=1.1R => 1+r=1.1, so R=500,000/(1.1^2)=413,223.14?, careful: from (2), 2022 = 1.1R, and 2023 = 2022*(1+r)=1.1R*(1+r)=500,000. But we also have 2022 = R(1+r)=1.1R => 1+r=1.1, so 2023 = 1.1R*1.1=1.21R=500,000 => R=500,000/1.21 ≈ 413,223. Unique. Sufficient. Each alone sufficient, answer (d).
4. What is the value of integer p?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) alone: primes between 10 and 20 are 11, 13, 17, 19. Not unique, so not sufficient. Statement (2) alone: multiples of 3 could be many (e.g., 3, 6, 9,...). Not sufficient. Together: from (1) p is 11,13,17,19; from (2) p is multiple of 3. None of 11,13,17,19 is a multiple of 3. So no value satisfies both. The question asks for value of p; together they give no possible integer, so the answer is that p cannot be determined uniquely. Thus even together not sufficient. Answer is (e).
5. What is the value of x?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: Statements (1) and (2) TOGETHER are NOT sufficient.
Statement (1) gives x^2 - 5x + 6 = 0, which factors to (x-2)(x-3)=0, so x = 2 or 3. This is not sufficient because there are two possible values. Statement (2) says x > 0, which alone is not sufficient as x could be any positive number. Combining both, x must be positive and either 2 or 3, but both are positive, so x could still be 2 or 3. Thus, even together, the statements do not give a unique value for x. Therefore, statements (1) and (2) together are NOT sufficient.
6. Is (x + y)^2 > x^2 + y^2?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Rephrase: (x+y)^2 = x^2 + 2xy + y^2, so the inequality becomes x^2+2xy+y^2 > x^2+y^2, which simplifies to 2xy > 0, i.e., xy > 0. So the question is: Is xy > 0? Statement (1) directly says xy > 0, so sufficient. Statement (2): x = 2y gives xy = 2y^2. Since y^2 ≥ 0, xy ≥ 0. But xy could be 0 (if y=0), giving answer no; if y≠0, xy>0, answer yes. Not sufficient. Thus (1) alone is sufficient, (2) alone is not. Answer is (a).
7. If x and y are numbers, is x > y?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) alone is insufficient because if x=-3 and y=2, then x^2=9 > 4=y^2 but x<y. Statement (2) alone is insufficient because it only tells us both are positive, but gives no relation between x and y. Together, since x>0 and y>0, the inequality x^2>y^2 implies x>y (as squaring is order-preserving for positive numbers). Thus, both statements together are sufficient, but neither alone is sufficient.
8. What is the value of x?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) alone: quadratic gives x = 2 or x = 3, two possible values, not sufficient. Statement (2) alone: x > 0 gives infinitely many values, not sufficient. Together, x must be positive and satisfy the equation, so x = 2 or 3, but still two values?, both 2 and 3 are positive, so still not unique., together they yield two values, so not sufficient. Thus answer is (e).
9. Is x an integer?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: EACH statement ALONE is sufficient.
Statement (1) alone gives x = 3.5, which is not an integer, so the answer is a definite NO; thus (1) is sufficient. Statement (2) alone says x is a prime number, and all prime numbers are integers, so the answer is a definite YES; thus (2) is also sufficient. Since each statement alone is sufficient, the correct answer is (d).
10. If x and y are integers, is x divisible by 6?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: EACH statement ALONE is sufficient.
Statement (1): x = 2y and y is divisible by 3, so y = 3k for some integer k. Then x = 2(3k) = 6k, which is divisible by 6. Sufficient.
Statement (2): x = 3y and y is divisible by 2, so y = 2m for some integer m. Then x = 3(2m) = 6m, which is divisible by 6. Sufficient.
Each statement alone is sufficient, so answer is D.
11. Is the integer n divisible by 12?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) alone: n divisible by 3 does not guarantee divisibility by 4 (e.g., n=6). Not sufficient. Statement (2) alone: n divisible by 4 does not guarantee divisibility by 3 (e.g., n=8). Not sufficient. Together: n divisible by both 3 and 4, and since 3 and 4 are coprime, n divisible by 12. Sufficient. Answer (c).
12. What is the value of x?
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
Answer: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) alone: x + y = 10 gives infinite possibilities for x (e.g., x=0, y=10; x=5, y=5). Not sufficient. Statement (2) alone: x - y = 2 also gives infinite possibilities. Not sufficient. Together: solving the system yields x = 6, y = 4, a unique value for x. Thus both together are sufficient. Answer is (c).