Test: Functions and Custom Characters - GMAT MCQ

Statement (1): 2 $ 2 # 2 = 3 From this statement, we know the result of the expression 2 $ 2 # 2 is 3. However, this information alone doesn't provide enough insight into the operations $ and # or how they interact. Therefore, statement (1) alone is not sufficient to determine the value of 1 $ 1 # 1.

Statement (2): 3 $ 2 # 1 = 5 From this statement, we know the result of the expression 3 $ 2 # 1 is 5. Similar to statement (1), this information alone doesn't provide enough information to deduce the operations $ and # or their impact on the value of the expression 1 $ 1 # 1. Thus, statement (2) alone is not sufficient.

Combining both statements: By combining both statements, we have the information that 2 $ 2 # 2 equals 3 and 3 $ 2 # 1 equals 5. However, this still doesn't give us a clear understanding of the specific operations represented by $ and #. Therefore, even when considering both statements together, we cannot determine the value of 1 $ 1 # 1.

As a result, the correct answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): a > b
This statement tells us that a is greater than b. Since max(a,b) represents the maximum value between a and b, and we know that a is greater than b, we can conclude that max(a,b) is equal to a. Therefore, statement (1) alone is sufficient to determine the value of max(a,b).

Statement (2): ab = -1
This statement tells us that the product of a and b is equal to -1. While this provides information about the relationship between a and b, it doesn't give us direct information about the maximum value between them or the value of max(a,b). For example, if a = -1 and b = 1, the product ab is indeed -1, but the maximum value between a and b is 1, not -1. Therefore, statement (2) alone is not sufficient to determine the value of max(a,b).

By considering both statements together, we know that a is greater than b from statement (1), and the product of a and b is -1 from statement (2). Since the product of two integers is -1, we can conclude that a and b must be -1 and 1, respectively, or vice versa. In either case, the maximum value between a and b is 1. Therefore, statement (2) alone is sufficient to determine the value of max(a,b).

Hence, the correct answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): #(10k) = 10k
This statement tells us that when the number 10k is rounded to the nearest integer, it is equal to 10k. However, it doesn't provide any specific information about the value of k or the rounding behavior. Without knowledge of the value of k, we cannot determine the units digit of #k based on this statement alone.

Statement (2): #(100k) is 10300.
This statement tells us that when the number 100k is rounded to the nearest integer, it is equal to 10300. From this statement, we can deduce that k lies between two consecutive integers such that rounding 100k to the nearest integer results in 10300. However, this information alone doesn't directly provide the value of k or the units digit of #k.

By evaluating both statements together, we know that #(10k) = 10k from statement (1), and #(100k) = 10300 from statement (2). Combining these statements doesn't provide additional information that helps us determine the units digit of #k. The relationship between #(10k) and #(100k) is not clear, and we still lack specific information about the value of k.

Therefore, statement (2) alone is sufficient to determine the units digit of #k, but statement (1) alone is not sufficient. The correct answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): (b # 1) # 2 = 5
This statement tells us that the result of performing the operation represented by # on b and 1, and then performing the operation on the result and 2, is equal to 5. However, since we don't know the specific operation represented by #, we can't determine the value of b # 2. Therefore, statement (1) alone is not sufficient to determine whether the value of b # 2 is even or odd.

Statement (2): 4 # b = 3 # (1 # b) and b is even
This statement tells us that the result of performing the operation represented by # on 4 and b is equal to the result of performing the operation represented by # on 3 and the result of performing the operation on 1 and b. Additionally, it specifies that b is even. However, similar to statement (1), since we don't know the specific operation represented by #, we can't determine the value of b # 2. Therefore, statement (2) alone is not sufficient to determine whether the value of b # 2 is even or odd.

Since neither statement alone is sufficient, let's consider them together. Unfortunately, even when considering both statements together, we still don't have enough information to determine the value of b # 2. The relationship between b, the operation represented by #, and the specific values involved in the statements is not clear. Therefore, statements (1) and (2) together are not sufficient to determine whether the value of b # 2 is even or odd.

As a result, each statement alone is not sufficient to answer the question asked. The correct answer is E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1): min(9, x) = y
This statement tells us that the minimum of 9 and x is equal to y. However, it doesn't provide any information about the value of x or its relationship to y. Therefore, statement (1) alone is not sufficient to determine the value of max(y, 8).

Statement (2): min(6, z) = y
This statement tells us that the minimum of 6 and z is equal to y. Similar to statement (1), it doesn't provide any information about the value of z or its relationship to y. Therefore, statement (2) alone is not sufficient to determine the value of max(y, 8).

By considering both statements together, we have:
min(9, x) = y
min(6, z) = y

However, even when considering both statements together, we still don't have enough information to determine the value of max(y, 8). We don't know anything about the values of x or z or their relationship to each other. For example, it is possible that x = 10 and z = 5, in which case y could be either 5 or 10. In one scenario, max(y, 8) would be 10, and in the other scenario, it would be 8. Therefore, statements (1) and (2) together are not sufficient to determine the value of max(y, 8).

As a result, statement (2) alone is sufficient to answer the question asked, but statement (1) alone is not sufficient. The correct answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): x < −1
This statement tells us that x is less than -1. However, it doesn't provide enough information to determine the sign of f(x). For example, if x = -2, then f(x) = (-2)³ + 9 = -8 + 9 = 1, which is positive. On the other hand, if x = -10, then f(x) = (-10)³ + 9 = -1000 + 9 = -991, which is negative. Therefore, statement (1) alone is not sufficient to determine if f(x) is positive.

Statement (2): x > −3
This statement tells us that x is greater than -3. Similar to statement (1), it doesn't provide enough information to determine the sign of f(x) on its own. For example, if x = 0, then f(x) = 0³ + 9 = 9, which is positive. However, if x = -2, then f(x) = (-2)³ + 9 = -8 + 9 = 1, which is also positive. Therefore, statement (2) alone is not sufficient to determine if f(x) is positive.

By considering both statements together, we have x < -1 and x > -3. This means that x lies between -3 and -1, exclusive. However, knowing this range is still not enough to determine the sign of f(x). For example, if x = -2.5, then f(x) = (-2.5)³ + 9 = -15.625 + 9 = -6.625, which is negative. On the other hand, if x = -2.9, then f(x) = (-2.9)³ + 9 = -24.389 + 9 = -15.389, which is also negative. Therefore, even when considering both statements together, we cannot determine if f(x) is positive.

As a result, statements (1) and (2) together are not sufficient to answer the question asked, and additional data is needed. The correct answer is E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1): [d] = 0
This statement tells us that [d], the least integer no less than d, is equal to 0. However, it doesn't provide any specific information about the value of d. Without knowledge of the value of d, we cannot determine the value of [2d] or whether it is equal to 0 based on this statement alone.

Statement (2): [3d] = 0
This statement tells us that [3d], the least integer no less than 3d, is equal to 0. It provides information about the value of 3d, but it doesn't directly give information about the value of 2d or [2d]. Since the least integer no less than 3d is 0, it implies that 3d lies between 0 and 1. From this, we can deduce that 2d is less than 1, but we cannot determine whether it is greater than or equal to 0. Therefore, statement (2) alone is not sufficient to determine whether [2d] is equal to 0.

By evaluating both statements together, we know that [d] is equal to 0 from statement (1), and [3d] is equal to 0 from statement (2). However, this combined information still doesn't directly provide the value of [2d] or whether it is equal to 0. The relationship between [d], [3d], and [2d] is not clear.

Therefore, statement (2) alone is sufficient to answer the question asked, but statement (1) alone is not sufficient. The correct answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): The least common denominator of x/y and 1/3 is 6.
This statement provides information about the least common denominator of x/y and 1/3. However, it doesn't give any specific details about the values of x or y. Without knowing the value of x or any relationship between x and y, we cannot determine the value of y based on this statement alone.

Statement (2): x = 1
This statement tells us the value of x, which is 1. However, it doesn't provide any information about y or the relationship between x and y. Without any information about y, we cannot determine its value based on this statement alone.

Combining both statements:
By considering both statements together, we know that the least common denominator of x/y and 1/3 is 6, and x is equal to 1. However, even with this combined information, we still don't have any direct information about the value of y. The relationship between x and y is still unknown, preventing us from determining the specific value of y.

Therefore, when considering both statements together, we still cannot determine the value of y. Thus, the correct answer is E: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Statement (1): 2@m = 6
This statement tells us that the expression 2@m equals 6. However, it doesn't provide information about the specific operation @. Without knowledge of the operation, we cannot determine the value of m@2 based on this statement alone.

Statement (2): 1@3 = m
This statement tells us that the expression 1@3 equals m. Similar to statement (1), it doesn't provide information about the operation @. Without knowing the operation, we cannot determine the value of m@2 based on this statement alone.

Since neither statement alone provides sufficient information to determine the value of m@2, we need to evaluate them together.

By considering both statements together, we have the following information:
From statement (1), 2@m = 6.
From statement (2), 1@3 = m.

We can use statement (1) to find the value of @ in terms of multiplication, as 2 multiplied by any number would result in 6. Therefore, we can conclude that @ represents multiplication.

Using this information, we can then substitute the multiplication operation in statement (2), which becomes 1 multiplied by 3 equals m. Thus, m equals 3.

Therefore, by considering both statements together, we can determine that the value of m@2 is 3, as @ represents multiplication.

Hence, the correct answer is D: EACH statement ALONE is sufficient to answer the question asked.

Statement (1): |y| < 1.
This statement implies that the absolute value of y is less than 1. However, it doesn't provide any specific information about the values of x and y or their oddness/evenness. Without knowing the values of x and y or their properties, we cannot determine the value of x#y. Statement (1) alone is not sufficient.

Statement (2): x = 14k, where 'k' is a non-negative integer.
This statement provides information about the value of x, stating that it is equal to 14k, where k is a non-negative integer. However, it doesn't provide any information about the value of y or its oddness/evenness. Without knowledge of the value of y or its properties, we cannot determine the value of x#y. Statement (2) alone is not sufficient.

Combining both statements:
By considering both statements together, we know that |y| < 1 and x = 14k, where 'k' is a non-negative integer. However, even with this combined information, we still don't have any insight into the specific values of x and y or their oddness/evenness. Therefore, we cannot determine the value of x#y.

As a result, the correct answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.