Test: Functions and Custom Characters - GMAT MCQ

(1) [x] = 1

From this statement, we know that the value of [x] is 1. Since [x] represents the least integer greater than or equal to x, it means that x is between 1 and 2 (exclusive). Therefore, the value of [x+1] would be the least integer greater than or equal to x+1, which in this case is 2.

(2) [2x] = 1

From this statement, we know that the value of [2x] is 1. Similar to statement (1), this implies that 2x is between 1 and 2 (exclusive). So x is between 0.5 and 1 (exclusive). Therefore, the value of [x+1] would be the least integer greater than or equal to x+1, which is 2.

By analyzing each statement individually, we can see that each statement alone is sufficient to determine the value of [x+1]. Hence, the answer is (D) EACH statement ALONE is sufficient to answer the question asked.

(1) 81 # x = 1

From this statement, we know that the remainder when 81 is divided by x is 1. However, we don't have any information about x itself. The remainder could be 1 for various values of x. For example, if x is 80, the remainder would be 1, but if x is 2, the remainder would also be 1. Therefore, statement (1) alone is not sufficient to determine the value of x.

(2) x # 40 = 0

From this statement, we know that the remainder when x is divided by 40 is 0. This implies that x is a multiple of 40. However, we still don't know the specific value of x. It could be 40, 80, 120, or any other positive integer that is a multiple of 40. Therefore, statement (2) alone is not sufficient to determine the value of x.

By analyzing each statement individually, we can see that neither statement alone is sufficient to determine the value of x. Therefore, the answer is (E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

(1) a#1 is not equal to 1#a for some numbers a

From this statement, we know that for some numbers a, the operation # does not exhibit commutativity when combined with the number 1. However, this statement alone does not provide enough information to determine whether a # (b-c) is equal to (a#b) - (a#c) for all numbers a, b, and c. Therefore, statement (1) alone is not sufficient to answer the question.

(2) # represents subtraction

From this statement, we know that the operation # represents subtraction. Substituting this information into the expression, we get a - (b - c) = (a - b) - (a - c). By simplifying this equation, we can see that it holds true for all numbers a, b, and c. Therefore, statement (2) alone is sufficient to answer the question.

By analyzing each statement individually, we can see that statement (2) alone is sufficient to determine that a # (b-c) is equal to (a#b) - (a#c), but statement (1) alone is not sufficient. Hence, the answer is (D) EACH statement ALONE is sufficient to answer the question asked.

(1) Max(5, p) = 6

From this statement, we know that the maximum value between 5 and p is 6. However, this information alone does not provide enough information to determine the value of 2*(Max(p, q)). We don't know the relationship between q and p or the specific values of q and p. Therefore, statement (1) alone is not sufficient to answer the question.

(2) Max(3, q) = 3

From this statement, we know that the maximum value between 3 and q is 3. Similar to statement (1), this information alone does not provide enough information to determine the value of 2*(Max(p, q)). We don't know the relationship between p and q or the specific values of p and q. Therefore, statement (2) alone is not sufficient to answer the question.

By analyzing each statement individually, we can see that neither statement alone is sufficient to determine the value of 2*(Max(p, q)). However, by combining both statements, we can conclude that the maximum value between p and q is 6, and the maximum value between q and 3 is 3. Therefore, the maximum value of p and q is 6. Substituting this into 2*(Max(p, q)), we get 2*6 = 12. Therefore, both statements together are sufficient to answer the question.

Hence, the answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

(1) f(x) = (x³)3*(x⁵)

From this statement, we have the expression for f(x) as (x³)3*(x⁵). However, we cannot determine whether f(x) is greater than 0 solely based on this information. The value of x could be positive, negative, or zero, and the outcome of the expression would vary accordingly. Therefore, statement (1) alone is not sufficient to answer the question.

(2) x = -2

From this statement, we know the specific value of x, which is -2. However, this information alone is not enough to determine whether f(x) is greater than 0. We need the expression or function for f(x) to make that determination. Therefore, statement (2) alone is not sufficient to answer the question.

By analyzing each statement individually, we can see that neither statement alone is sufficient to determine whether f(x) is greater than 0. However, by combining both statements, we can evaluate f(x) for the specific value of x = -2. Substituting x = -2 into the expression in statement (1), we can determine whether f(x) is greater than 0.

Therefore, both statements together are sufficient to answer the question.

Hence, the answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

(1) x = y + z

From this statement, we know that x is equal to y + z. Substituting this into the expression f(x) - f(y + z), we get f(x) - f(y + z) = f(y + z) - f(y + z). Since y + z is the same on both sides of the equation, f(y + z) - f(y + z) equals 0. Therefore, statement (1) alone is sufficient to answer the question.

(2) f(0) = 1

From this statement, we know the value of f(0) is 1. However, this information does not provide any direct information about the value of f(x) - f(y + z). It doesn't provide any relationship between x, y, z, or the function f(n). Therefore, statement (2) alone is not sufficient to answer the question.

By analyzing each statement individually, we can see that statement (1) alone is sufficient to determine that f(x) - f(y + z) is equal to 0, but statement (2) alone is not sufficient. Hence, the answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

(1) 0*2 = 2

From this statement, we know that the result of 02 is 2. However, this statement does not provide any direct information about the value of 10. It only tells us the result of a different operation. Therefore, statement (1) alone is not sufficient to determine the value of 1*0.

(2) 2*0 = 2

From this statement, we know that the result of 20 is 2. Similar to statement (1), this statement does not directly provide information about the value of 10. It only tells us the result of a different operation. Therefore, statement (2) alone is not sufficient to determine the value of 1*0.

By analyzing each statement individually, we can see that neither statement alone is sufficient to determine the value of 10. We need additional information to determine the operation represented by the asterisk (). Therefore, the answer is (D) EACH statement ALONE is sufficient to answer the question asked.

(1) w < 12

From this statement, we know that w is less than 12. However, this information alone does not provide enough information to determine whether [w/3] is equal to 4. For example, if w is 11, then [w/3] would be [11/3] = [3.6667] = 4. But if w is 9, then [w/3] would be [9/3] = [3] = 3. Therefore, statement (1) alone is not sufficient to answer the question.

(2) w > 10

From this statement, we know that w is greater than 10. However, this information alone also does not provide enough information to determine whether [w/3] is equal to 4. For example, if w is 11, then [w/3] would be [11/3] = [3.6667] = 4. But if w is 12, then [w/3] would be [12/3] = [4] = 4. Therefore, statement (2) alone is not sufficient to answer the question.

By analyzing each statement individually, we can see that neither statement alone is sufficient to determine whether [w/3] is equal to 4. However, by combining both statements, we can conclude that w is between 10 and 12 exclusive (10 < w < 12). In this range, [w/3] would be the least integer greater than or equal to w/3, which is 4. Therefore, both statements together are sufficient to answer the question.

Hence, the answer is (C) BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.

(1) n = 8

From this statement, we know the value of n is 8. However, we don't have any information about the function f(x) itself. Without knowing the specific definition of the function, we cannot determine whether f(n) is greater than f(n-1) based solely on the value of n. Therefore, statement (1) alone is not sufficient to answer the question.

(2) f(n) = n-1

From this statement, we know the expression for f(n) is given by f(n) = n-1. By substituting n into the function, we can rewrite the inequality as (n-1) > (n-1) - 1, which simplifies to n-1 > n-2. This inequality holds true for any positive value of n. Therefore, we can conclude that f(n) is always greater than f(n-1) based on the given function f(n) = n-1.

By analyzing each statement individually, we can see that statement (2) alone is sufficient to determine that f(n) > f(n-1), but statement (1) alone is not sufficient. Hence, the answer is (B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

(1) |k| = 3

From this statement, we know that the absolute value of k is 3. Since the absolute value always gives a positive value, we can conclude that k is either 3 or -3.

For k = 3:
Using the given function, f(x) = 27x for x ≥ 0, we can substitute x = 3:
f(3) = 27 * 3 = 81

For k = -3:
Using the given function, f(x) = x⁴ for x < 0, we can substitute x = -3:
f(-3) = (-3)⁴ = 81

Therefore, regardless of whether k is 3 or -3, the value of f(k) is 81.

(2) k < 0

From this statement, we know that k is negative. However, this alone doesn't provide enough information to determine the value of f(k). We need to know the exact value of k to evaluate the function accurately.

By analyzing each statement individually, we can see that statement (1) alone is sufficient to determine the value of f(k), but statement (2) alone is not sufficient. Hence, the answer is (A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.