Test: Functions and Custom Characters - GMAT MCQ

To find the value of x, we need to determine the greatest integer less than or equal to x and then multiply it by x.

Let's consider the given equation x[x] = 10.5. The greatest integer less than or equal to x can be represented as [x].

We are given that x[x] = 10.5. So, we can write the equation as [x] * x = 10.5.

Since the greatest integer less than or equal to x will be an integer value, let's consider [x] as an integer and solve for x.

Let's start by trying different integer values for [x]:

If we let [x] = 3, then the equation becomes 3 * x = 10.5. Solving for x, we find x = 10.5/3 = 3.5.

If we let [x] = 4, then the equation becomes 4 * x = 10.5. Solving for x, we find x = 10.5/4 = 2.625.

We can see that 3.5 satisfies the equation x[x] = 10.5, while 2.625 does not. Therefore, the value of x is 3.5.

Therefore, the correct answer is option D: 3.5.

To find the value of the expression, let's first evaluate each individual △ operation step by step.

Starting with (19 △ 9):

m = 19, n = 9

The remainder when m + n is divided by m - n:

(19 + 9) % (19 - 9) = 28 % 10 = 8

So, (19 △ 9) = 8.

Now let's move on to the next part, (19 △ (9 △ 2)):

m = 9, n = 2

The remainder when m + n is divided by m - n:

(9 + 2) % (9 - 2) = 11 % 7 = 4

So, (19 △ (9 △ 2)) = 4.

Now we can substitute these values back into the original expression:

((19 △ 9) △ 2) - (19 △ (9 △ 2))

= (8 △ 2) - (19 △ 4)

Now let's evaluate (8 △ 2):

m = 8, n = 2

The remainder when m + n is divided by m - n:

(8 + 2) % (8 - 2) = 10 % 6 = 4

So, (8 △ 2) = 4.

Substituting this back into the expression:

(4) - (19 △ 4)

Now let's evaluate (19 △ 4):

m = 19, n = 4

The remainder when m + n is divided by m - n:

(19 + 4) % (19 - 4) = 23 % 15 = 8

So, (19 △ 4) = 8.

Substituting this back into the expression:

4 - 8 = -4

Therefore, the value of the expression is -4, which corresponds to option C.

To determine which statement(s) must be true, let's evaluate the given inequality:

6 ∆ 3 ≤ 3

Since ∆ can represent addition, subtraction, multiplication, or division, let's consider each operation separately:

Addition: 6 + 3 ≤ 3
This simplifies to 9 ≤ 3, which is not true.

Subtraction: 6 - 3 ≤ 3
This simplifies to 3 ≤ 3, which is true.

Multiplication: 6 * 3 ≤ 3
This simplifies to 18 ≤ 3, which is not true.

Division: 6 / 3 ≤ 3
This simplifies to 2 ≤ 3, which is true.

From the above analysis, we can conclude that the inequality 6 ∆ 3 ≤ 3 is true only when ∆ represents either subtraction or division.

Now, let's evaluate each statement:

I. 2 ∆ 2 = 0
If ∆ represents subtraction, then 2 - 2 = 0, which is true.
If ∆ represents division, then 2 / 2 = 1, which is not true.

II. 2 ∆ 2 = 1
If ∆ represents subtraction, then 2 - 2 = 0, which is not true.
If ∆ represents division, then 2 / 2 = 1, which is true.

III. 4 ∆ 2 = 2
If ∆ represents subtraction, then 4 - 2 = 2, which is true.
If ∆ represents division, then 4 / 2 = 2, which is true.

Based on our evaluation, we find that statement III (4 ∆ 2 = 2) must be true regardless of the operation ∆ represents. Therefore, the correct answer is C: III only.

To find the value of #7 - #(-1), we need to substitute the given values into the definition of #x:

#x = √(x + 2)

Let's calculate each term separately:

#7 = √(7 + 2) = √9 = 3

#(-1) = √((-1) + 2) = √1 = 1

Now we can evaluate #7 - #(-1):

#7 - #(-1) = 3 - 1 = 2

Therefore, the value of #7 - #(-1) is 2. Thus, the correct answer is C.

To find the value of 5@/4@, we need to evaluate the expressions 5@ and 4@ separately.

Let's start with 5@. According to the given definition, 5@ is the product of all fractions of the form 1/S, where S is a positive integer not greater than 5:

5@ = (1/1) * (1/2) * (1/3) * (1/4) * (1/5)

Simplifying the expression:

5@ = 1/ (1 * 2 * 3 * 4 * 5) = 1/120

Now let's calculate 4@:

4@ = (1/1) * (1/2) * (1/3) * (1/4)

Simplifying the expression:

4@ = 1/ (1 * 2 * 3 * 4) = 1/24

Finally, we can find the value of 5@/4@:

(5@) / (4@) = (1/120) / (1/24) = (1/120) * (24/1) = 24/120 = 1/5

Therefore, the value of 5@/4@ is 1/5. Hence, the correct answer is A.

To find the value of f(6), we need to use the given recursive formula and initial condition.

According to the given formula, f(n) = f(n-1) - n.

We are also given that f(4) = 10. We can use this initial condition to find the subsequent values of f(n).

Let's calculate the values step by step:

f(4) = 10
f(5) = f(4) - 5 = 10 - 5 = 5
f(6) = f(5) - 6 = 5 - 6 = -1

Therefore, the value of f(6) is -1. Hence, the correct answer is A.

To find the value of the expression [-1.6] + [3.4] + [2.7], we need to evaluate each of the individual greatest integer functions.

[-1.6] is the greatest integer less than or equal to -1.6, which is -2.
[3.4] is the greatest integer less than or equal to 3.4, which is 3.
[2.7] is the greatest integer less than or equal to 2.7, which is 2.

Now we can add these values together:
[-1.6] + [3.4] + [2.7] = -2 + 3 + 2 = 3.

Therefore, the value of the expression is 3. Hence, the correct answer is option A.

To find the value of x, we need to determine the value of x[x] when x is substituted into the expression.

We are given that x[x] = 39.

Since [x] denotes the greatest integer less than or equal to x, we can rewrite the equation as:

[x] = 39/x

Now, let's analyze the options provided:

A: [6] = 39/6 = 6.5 (not equal to 39)
B: [6.25] = 39/6.25 = 6.24 (not equal to 39)
C: [6.35] = 39/6.35 = 6.14 (not equal to 39)
D: [6.5] = 39/6.5 = 6 (equal to 39)
E: [6.75] = 39/6.75 = 5.78 (not equal to 39)

From the given options, only option D satisfies the equation x[x] = 39.

Therefore, the value of x is 6.5. Hence, the correct answer is option D.

To find the value of v, we need to substitute the given equation and solve for v.

The operation v* is defined as v* = v - v/3.

We are given that (v*)* = 8.

Substituting the definition of v* into the equation, we have:

(v*)* = (v - v/3) - (v - v/3)/3

Simplifying the expression, we get:

(v*)* = (v - v/3) - v/3 + v/9

(v*)* = v - v/3 - v/3 + v/9

(v*)* = v - 2v/3 + v/9

(v*)* = (9v - 6v + v)/9

(v*)* = 4v/9

Since (v*)* is equal to 8, we can set up the equation:

4v/9 = 8

To solve for v, we multiply both sides of the equation by 9/4:

v = 8 * (9/4)

v = 18

Therefore, the value of v is 18. Hence, the correct answer is option B.