Test: Functions and Custom Characters - GMAT MCQ

Statement (1): [2x] = 0

In this case, [2x] will be 0 if and only if 2x is between 0 and 1 (exclusive), since [2x] represents the greatest integer less than or equal to 2x. Therefore, for [2x] to be 0, x must be between 0 and 0.5 (exclusive). Hence, statement (1) alone is sufficient to conclude that [x] = 0.

Statement (2): [3x] = 0

Similarly, [3x] will be 0 if and only if 3x is between 0 and 1 (exclusive). This means x must be between 0 and 1/3 (exclusive) for [3x] to be 0. Consequently, statement (2) alone is sufficient to conclude that [x] = 0.

Since both statements (1) and (2) independently lead to the conclusion that [x] = 0, each statement alone is sufficient to answer the question. Therefore, the correct answer is D: EACH statement ALONE is sufficient to answer the question asked.

To find the value of f(4), let's use the given recursive equation:

f(n) = f(n - 1) - n

We can start by finding f(3) using the equation.

Statement (1): f(3) = 14

Using the recursive equation, we have:

f(3) = f(2) - 3
f(2) = f(1) - 2
f(1) = f(0) - 1

Substituting f(1) and f(2) into the equation for f(3), we get:

f(3) = (f(0) - 1) - 2 - 3
f(3) = f(0) - 6

We know that f(3) = 14 from Statement (1), so we can solve for f(0):

14 = f(0) - 6
f(0) = 20

Now, we can use the recursive equation to find f(4):

f(4) = f(3) - 4
f(4) = 14 - 4
f(4) = 10

Therefore, from Statement (1) alone, we can determine that f(4) is equal to 10.

Statement (2): f(6) = -1

Using the same process as above, we can find f(4) if we know f(6), but Statement (2) does not provide any information about f(4). Therefore, Statement (2) alone is not sufficient to determine the value of f(4).

Since Statement (1) alone is sufficient and Statement (2) alone is not sufficient, the answer is D: EACH statement ALONE is sufficient to answer the question asked.

Statement (1): Min(x, 60) = x

If Min(x, 60) = x, it means x is the smaller value between x and 60. Therefore, the expression Max(x, 60) will always be equal to x. We can write the expression as follows: (Max(x, 60) + Min(40, x)) / 2 = (x + Min(40, x)) / 2.

Since Max(x, 60) is always equal to x, we can simplify the expression: (x + Min(40, x)) / 2 = (x + x) / 2 = 2x / 2 = x.

Therefore, from Statement (1) alone, we can conclude that the average of Max(x, 60) and Min(40, x) is x.

Statement (2): Max(40, x) = x

If Max(40, x) = x, it means x is the greater value between x and 40. However, we cannot determine the value of Min(40, x) based on this information alone. The value of x can be greater than or equal to 40, in which case Min(40, x) would be 40. Or, x could be less than 40, in which case Min(40, x) would be x. Therefore, we cannot determine the average of Max(x, 60) and Min(40, x) solely from Statement (2).

Since Statement (1) alone is sufficient to determine the average as x, but Statement (2) alone is not sufficient, the answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

To find the value of x in the equation f(x) = 3x - 5, we can evaluate each statement separately:

Statement (1): f(x) = 22

Substituting 22 for f(x) in the equation, we have:

22 = 3x - 5

Solving for x:

3x = 27
x = 9

Therefore, from Statement (1) alone, we can determine that the value of x is 9.

Statement (2): f(x²) = 22

Substituting x² for x in the equation, we have:

f(x²) = 3(x²) - 5

This statement does not provide a specific value for f(x²), and we cannot solve for x based on this equation alone. Therefore, Statement (2) alone is not sufficient to determine the value of x.

Since Statement (1) alone is sufficient to determine the value of x as 9, but Statement (2) alone is not sufficient, the answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): f(2) = 81

Substituting 2 for x in the function, we have:

f(2) = p * 2 = 81

This equation alone does not provide enough information to determine the value of p or f(1). Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): f(3) = -729

Substituting 3 for x in the function, we have:

f(3) = p * 3 = -729

This equation alone provides the information to determine the value of p. Dividing both sides of the equation by 3, we find:

p = -729 / 3 = -243

Now we can find the value of f(1) by substituting 1 for x in the function:

f(1) = p * 1 = -243 * 1 = -243

Therefore, from Statement (2) alone, we can determine that the value of f(1) is -243.

Since Statement (2) alone is sufficient to determine the value of f(1), but Statement (1) alone is not sufficient, the answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

To determine the value of 6 ∗ 2, we need to identify whether the symbol ∗ represents multiplication or division. Let's evaluate each statement separately:

Statement (I): (5 ∗ 5) - (12 ∗ 2) = 1

This equation indicates that the result of (5 ∗ 5) minus (12 ∗ 2) is equal to 1. However, this equation alone does not provide enough information to determine the specific operation represented by ∗. Therefore, Statement (I) alone is not sufficient to answer the question.

Statement (II): (9 ∗ 3) - (5 ∗ 5) = 2

This equation indicates that the result of (9 ∗ 3) minus (5 ∗ 5) is equal to 2. Similarly to Statement (I), this equation alone does not provide enough information to determine the specific operation represented by ∗. Therefore, Statement (II) alone is not sufficient to answer the question.

When we consider both statements together, we have:

(5 ∗ 5) - (12 ∗ 2) = 1
(9 ∗ 3) - (5 ∗ 5) = 2

Although the specific operation represented by ∗ is unknown, we can see that both equations involve the same operands (5, 12, 2, 9, and 3) and the same result (1 and 2). This suggests that the operation ∗ might be division, as multiplication would yield different results. By assuming that ∗ represents division, we can check if the equations hold true:

(5 ÷ 5) - (12 ÷ 2) = 1 → 1 - 6 = 1 → False
(9 ÷ 3) - (5 ÷ 5) = 2 → 3 - 1 = 2 → False

Since the equations do not hold true when assuming ∗ represents division, we can conclude that ∗ represents multiplication. Therefore, 6 ∗ 2 denotes 6 multiplied by 2, resulting in 12.

Since Statement (I) alone is not sufficient to determine the operation ∗, but it provides additional information about the equation, it is sufficient to answer the question. Therefore, the answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

To find the value of f(1), we need to determine the value of the constant a in the function f(n) = an. Let's evaluate each statement separately:

Statement (1): f(2) = 100

Substituting 2 for n in the function, we have:

f(2) = a * 2 = 100

This equation alone does not provide enough information to determine the value of a or f(1). Therefore, Statement (1) alone is not sufficient to answer the question.

Statement (2): f(3) = -1,000

Substituting 3 for n in the function, we have:

f(3) = a * 3 = -1,000

This equation alone provides the information to determine the value of a. Dividing both sides of the equation by 3, we find:

a = -1,000 / 3 = -333.33...

Now we can find the value of f(1) by substituting 1 for n in the function:

f(1) = a * 1 = -333.33... * 1 = -333.33...

Therefore, from Statement (2) alone, we can determine that the value of f(1) is approximately -333.33....

Since Statement (2) alone is sufficient to determine the value of f(1), but Statement (1) alone is not sufficient, the answer is B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Statement (1): If r = s, then r θ s = 0.

From this statement, we know that if r is equal to s, then the result of the operation θ between r and s is 0. However, this statement does not provide specific information about the operation θ. We cannot determine which operation (+, −, ×, or ÷) corresponds to θ based solely on this statement.

Statement (2): If r ≠ s, then r θ s ≠ s θ r.

From this statement, we know that if r is not equal to s, then the result of the operation θ between r and s is not equal to the result of the operation θ between s and r. This statement provides a rule about the commutativity of the operation θ, indicating that the operation θ is not commutative. However, we still do not know the specific operation θ.

When we consider both statements together, we know that if r = s, then r θ s = 0, and if r ≠ s, then r θ s ≠ s θ r. These statements do not provide enough information to determine the exact operation θ. We only know that the operation θ must satisfy these conditions.

Therefore, Statement (1) alone is sufficient to establish a rule about the result of the operation θ, but we cannot determine the specific operation θ. The correct answer is A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Statement (1): 3&a = 13

Substituting 13 for 3&a in the equation, we have:

13 = a² - 2(3)
13 = a² - 6
a² = 19
a = ±√19

From Statement (1) alone, we can determine the possible values of a as ±√19. However, this information is not sufficient to determine the value of a&2.

Statement (2): The value of a is either 2 or −2.

From this statement alone, we know that the value of a can be either 2 or −2. However, this information alone is not sufficient to determine the value of a&2.

When we consider both statements together, we know that a can be either ±√19 and that the value of a can be either 2 or −2. However, the value of a&2 is not affected by the possible values of a being ±√19. Since a&2 only depends on the arithmetic operation defined by the equation x&y = x² − 2y, we can determine that a&2 will have a unique value regardless of the specific values of a.

Therefore, each statement alone is sufficient to determine the value of a&2. The correct answer is D: EACH statement ALONE is sufficient to answer the question asked.

Statement (1): m > 2

From this statement, we know that m is greater than 2. However, we do not have any specific information about the decimal part of m or its relationship to the integer part. Therefore, we cannot determine the value of [m] based solely on this statement.

Statement (2): m < 3

From this statement, we know that m is less than 3. Similar to Statement (1), we do not have any specific information about the decimal part of m or its relationship to the integer part. Hence, we cannot determine the value of [m] based solely on this statement.

When we consider both statements together, we know that m is greater than 2 and less than 3. Since there are no integers between 2 and 3, we can conclude that the only possible value for [m] is 2.

Therefore, both statements together are sufficient to determine that [m] is equal to 2. The correct answer is C: BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.