## GMAT Symbol and Function

**Part 1: Introduction to Symbols and Functions**

**Symbols in GMAT**

Symbols play a crucial role in the GMAT quantitative section. Unlike algebraic variables, which can represent any number, symbols in GMAT problems are assigned specific definitions within the problem statement. For example, the problem may define a symbol such as “★”, which operates on numbers in a particular way, such as “For all numbers x, ★x = x^{2} + 3.” It’s crucial to remember that these symbols do not have inherent meanings and only carry the definitions provided within the problems.

**Functions in GMAT**

Functions, just like symbols, are used to represent certain operations or calculations. Functions often come in the form of “f(x)” or “g(x)” and follow a specific rule. For instance, the function f(x) = x^2 + 3 means that for any input x, the output is x squared plus three.

**Symbolic Functions**

In the GMAT, you might encounter symbolic functions, a combination of symbols and functions, which makes them a unique category. For example, the function ★(x) = x^{2} + 3 is symbolic. Here, the “★” is a symbol that represents a specific function, and “(x)” is the variable or input to the function.

Understanding and interpreting these symbols and functions are key to effectively answering GMAT problems. The ability to manipulate and work with these symbols and functions is a valuable skill tested in the GMAT.

In the following sections, we will delve deeper into these concepts, providing examples, exploring how to solve symbol and function problems, and sharing strategies and tips for mastering these questions on the GMAT.

**Part 2: Symbol and Function Interpretation**

**Understanding Symbols in GMAT**

The most important aspect of symbols in the GMAT is understanding that they are not universal but context-specific. The same symbol can denote different operations in different problems. For example, a problem might define “∆x” as “∆x = 2x + 3”. In this context, “∆” is not a universal mathematical symbol, but a unique symbol defined only for this problem. For any number x you insert into this symbol, you multiply by 2 and add 3.

**Understanding Functions in GMAT**

Functions in the GMAT are more standard. The notation “f(x)” is universally understood to mean a function named ‘f’ that takes an input ‘x’. However, what the function does with the input can vary. For instance, the function could be “f(x) = 2x + 3”, which means for any number x, you multiply by 2 and add 3.

**Understanding Symbolic Functions in GMAT**

Symbolic functions combine the aspects of symbols and functions. In symbolic functions, a unique symbol is used to denote a particular operation on an input. For instance, “★(x) = 2x + 3”. Here, “★” is a unique symbol that represents a function, and “(x)” is the variable or input to the function. For any number x you insert into this symbolic function, you multiply by 2 and add 3.

**Interpretation Tips**

Interpreting these symbols and functions correctly is crucial to solving GMAT problems. Pay close attention to the definitions provided in the problem. Don’t assume that a symbol represents its typical mathematical meaning. Always replace the symbol or function with its defined operation to avoid confusion.

In the next part, we’ll provide examples of symbols and functions and illustrate how to work with them to solve GMAT problems.

**Part 3: Examples and Application of Symbols and Functions**

Understanding symbols and functions is made easier through practical application. Let’s explore some examples:

**Example 1: Symbols**

Let’s say a GMAT problem defines a symbol “Δ” as follows: For all numbers x, Δx = x² + 4x + 4. If we are asked to find the value of Δ3, we replace x in the defined operation with 3:

Δ3 = 3² + 4×3 + 4 = 9 + 12 + 4 = 25.

**Example 2: Functions**

Consider the function f(x) = 2x – 7 for a function example. If we are asked to find the value of f(5), we replace x in the function with 5:

f(5) = 2×5 – 7 = 10 – 7 = 3.

**Example 3: Symbolic Functions**

Suppose we have ★(x) = 3x² – 2 for a symbolic function. If asked to find the value of ★(4), we replace x in the symbolic function with 4:

★(4) = 3*4² – 2 = 3*16 – 2 = 48 – 2 = 46.

**Key Takeaways**

The main idea is to replace the symbol or function with the operation it represents and then insert the given number into the operation. It’s a simple process of substitution, but it requires careful reading of the problem’s definitions and accurate execution of the operations.

In the next part, we will discuss strategies to solve more complex problems involving symbols and functions on the GMAT.

**Part 4: Strategies for Solving Complex Symbol and Function Problems**

When symbols and functions are combined, or multiple instances of them are used in the same problem, things can get more complex. Here are some strategies to solve these more difficult problems.

**1. Pay Attention to Definitions:**

Always remember the definitions given in the problem for each symbol or function. They are essential for solving the problem correctly. Make sure you understand what each symbol or function does before you begin.

**2. Break Down the Problem:**

If a problem seems too complex, try breaking it down into smaller parts. Work on each symbol or function separately before trying to put everything together.

**3. Use Substitution:**

Substitution is a powerful tool when working with symbols and functions. If you’re asked to find the value of a symbol or function for a particular number, substitute that number into the operation defined by the symbol or function.

**4. Watch for Patterns:**

Some problems might involve finding the value of a symbol or function for several different numbers. If this is the case, try to identify any patterns in the results. This could help you find a quicker solution.

**5. Practice:**

The best way to get better at solving these problems is through practice. Try to solve a variety of problems involving different symbols and functions to improve your skills.

**Example: Complex Symbolic Function**

Let’s consider an example where we have two symbolic functions: ★(x) = 2x + 3 and ▲(x) = x² – 1. If the problem asks for the value of ★(▲(3)), we first calculate ▲(3):

▲(3) = 3² – 1 = 9 – 1 = 8.

Then we substitute this result into ★:

★(▲(3)) = ★(8) = 2×8 + 3 = 16 + 3 = 19.

In the final part, we will discuss some common pitfalls and tips to avoid them when solving symbol and function problems on the GMAT.

**Part 5: Common Pitfalls and Tips for Symbol and Function Problems**

Despite mastering the concept of symbols and functions, students often make certain mistakes on the GMAT. Here are some common pitfalls and how to avoid them:

**1. Misinterpreting the Symbol or Function:**

This is perhaps the most common mistake. As we’ve stressed before, the definitions given in the problem are paramount. Please do not assume that a symbol represents its typical mathematical meaning. Always use the definition given in the problem.

**2. Neglecting Order of Operations:**

This is particularly important for functions involving several operations. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

**3. Making Calculation Errors:**

Even with a correct understanding of the problem, simple calculation errors can lead to the wrong answer. Be careful, particularly with negative numbers and fractions.

**4. Overcomplicating Problems:**

Sometimes, students try to look for complex patterns or solutions, missing the simpler direct approach. Always look for the straightforward path first.

**Tips to Overcome These Pitfalls:**

**Practice Mindfully:**The more problems you solve, the more comfortable you’ll become with symbol and function problems. But don’t just practice – learn from your mistakes. Understand where you went wrong in a problem to avoid making the same mistake again.**Review Basic Math Concepts:**Strong fundamental math skills are essential for the GMAT. Make sure you’re comfortable with basic arithmetic, algebra, and order of operations.**Slow Down:**It’s better to take a little extra time to understand a problem correctly than to rush through it and make a mistake. Read each problem carefully.**Check Your Work:**If you have time, double-check your calculations. This can help catch any simple mistakes.

Understanding symbols and functions is a key component of the GMAT Quantitative section. Mastering this topic will give you an advantage in tackling a variety of problems on the exam. With mindful practice and a solid understanding of the underlying concepts, you’ll be well on your way to achieving a high score.

**Question 1:**

In the GMAT Quantitative section, the symbol “★” is defined as follows: for all numbers x, ★x = x^{2} – 3. What is the value of ★(5)?

A) 22

B) 23

C) 24

D) 25

E) 26

**Solution:**

According to the definition given in the problem, ★x = x^{2} – 3.

To find the value of ★(5), we replace x with 5 in the equation:

★(5) = 5^2 – 3 = 25 – 3 = 22

So, the correct answer is **A) 22**.

**Question 2:**

The function f(x) in the GMAT Quantitative section is defined as f(x) = 2x + 5. What is the value of f(7)?

A) 14

B) 15

C) 16

D) 19

E) 20

**Solution:**

According to the definition given in the problem, f(x) = 2x + 5.

To find the value of f(7), we replace x with 7 in the function:

f(7) = 2×7 + 5 = 14 + 5 = 19

So, the correct answer is **D) 19**.

**Question 3:**

In the GMAT Quantitative section, the symbol “Δ” is defined as follows: for all numbers x, Δx = 3x – 4. If Δa = 14, what is the value of ‘a’?

A) 2

B) 4

C) 6

D) 8

E) 10

**Solution:**

According to the definition given in the problem, Δx = 3x – 4.

We are given that Δa = 14. Substituting this into the equation:

14 = 3a – 4

To find the value of ‘a’, we first add 4 to both sides of the equation:

14 + 4 = 3a 18 = 3a

Then we divide both sides of the equation by 3:

18 / 3 = a 6 = a

So, the correct answer is **C) 6**.

**Question**** 4****:**

The symbol “∆” is defined as follows: for all numbers x, ∆x = x² + 2x – 1. What is the value of ∆(∆2)?

A) 7

B) 11

C) 15

D) 19

E) 62

**Solution:**

First, we need to find the value of ∆2. Substituting x = 2 into the equation, we get:

∆2 = 2² + 2×2 – 1 = 4 + 4 – 1 = 7

Next, we find the value of ∆(∆2), which is ∆7. Substituting x = 7 into the equation, we get:

∆7 = 7² + 2×7 – 1 = 49 + 14 – 1 = 62

The correct answer is **E) 62**.

**Question**** 5****:**

The function f(x) is defined as f(x) = x^{2} – 4x + 4. What is the value of f(f(3))?

A) 0

B) 4

C) 1

D) 64

E) 256

**Solution:**

First, we need to find the value of f(3). Substituting x = 3 into the equation, we get:

f(3) = 3^{2} – 4×3 + 4 = 9 – 12 + 4 = 1

Next, we find the value of f(f(3)), which is f(1). Substituting x = 1 into the equation, we get:

f(1) = 1^{2 }– 4×1 + 4 = 1 – 4 + 4 = 1

the correct answer is **C) 1**.

**Question**** 6****:**

The symbol “★” is defined as follows: for all numbers x, ★x = 2x + 5. If ★y = 15, what is the value of ‘y’?

A) 3

B) 4

C) 5

D) 7

E) 10

**Solution:**

According to the definition given in the problem, ★x = 2x + 5.

We are given that ★y = 15. Substituting this into the equation:

15 = 2y + 5

To find the value of ‘y’, we first subtract 5 from both sides of the equation:

15 – 5 = 2y 10 = 2y

Then we divide both sides of the equation by 2:

10 / 2 = y 5 = y

So, the correct answer is **C) 5**.

**Question**** 7****:**

The function g(x) is defined as g(x) = 3x – 7. What is the value of g(g(2))?

A) -8

B) -10

C) 2

D) 6

E) 8

First, we need to find the value of g(2). Substituting x = 2 into the equation, we get:

g(2) = 3×2 – 7 = 6 – 7 = -1

Next, we find the value of g(g(2)), which is g(-1). Substituting x = -1 into the equation, we get:

g(-1) = 3×(-1) – 7 = -3 – 7 = -10

the correct answer is **B) -10**.

**Question**** 8****:**

The symbol “Δ” is defined as follows: for all numbers x, Δx = x² + 4. What is the value of Δ(Δ(-2))?

A) 16

B) 20

C) 68

D) 28

E) 32

First, we need to find the value of Δ(-2). Substituting x = -2 into the equation, we get:

Δ(-2) = (-2)² + 4 = 4 + 4 = 8

Next, we find the value of Δ(Δ(-2)), which is Δ8. Substituting x = 8 into the equation, we get:

Δ8 = 8² + 4 = 64 + 4 = 68 correct answer is **C) 68**.

**Question**** 9****:**

The function h(x) is defined as h(x) = 5x – 2. What is the value of h(h(3))?

**Answer Choices:**

A) 55

B) 65

C) 63

D) 85

E) 95

**Solution:**

First, we need to find the value of h(3). Substituting x = 3 into the equation, we get:

h(3) = 5×3 – 2 = 15 – 2 = 13

Next, we find the value of h(h(3)), which is h(13). Substituting x = 13 into the equation, we get:

h(13) = 5×13 – 2 = 65 – 2 = 63, the correct answer is **C) 63**.

**Question**** 10****:**

The function f(x) is defined as f(x) = 2x + 3. What is the value of f(f(-2))?

A) -1

B) 0

C) 1

D) 2

E) 3

**Solution:**

First, we need to find the value of f(-2). Substituting x = -2 into the equation, we get:

f(-2) = 2×(-2) + 3 = -4 + 3 = -1

Next, we find the value of f(f(-2)), which is f(-1). Substituting x = -1 into the equation, we get:

f(-1) = 2×(-1) + 3 = -2 + 3 = 1

So, the correct answer is **C) 1**.

**Question**** 11****:**

The symbol “★” is defined as follows: for all numbers x, ★x = x² – 4x + 4. What is the value of ★(★2)?

A) 0

B) 4

C) 8

D) 16

E) 32

**Solution:**

First, we need to find the value of ★2. Substituting x = 2 into the equation, we get:

★2 = 2² – 4×2 + 4 = 4 – 8 + 4 = 0

Next, we find the value of ★(★2), which is ★0. Substituting x = 0 into the equation, we get:

★0 = 0² – 4×0 + 4 = 0 – 0 + 4 = 4

So, the correct answer is **B) 4**.

**Question**** 12****:**

The function g(x) is defined as g(x) = 3x – 5. If g(a) = 10, what is the value of ‘a’?

A) 3

B) 4

C) 5

D) 6

E) 7

**Solution:**

According to the definition given in the problem, g(x) = 3x – 5.

We are given that g(a) = 10. Substituting this into the equation:

10 = 3a – 5

To find the value of ‘a’, we first add 5 to both sides of the equation:

10 + 5 = 3a 15 = 3a

Then we divide both sides of the equation by 3:

15 / 3 = a 5 = a

So, the correct answer is **C) 5**.

**Question**** 13****:**

The function h(x) is defined as h(x) = x² – 2x + 1. What is the value of h(h(2))?

A) 0

B) 1

C) 4

D) 9

E) 16

**Solution:**

First, we need to find the value of h(2). Substituting x = 2 into the equation, we get:

h(2) = 2² – 2×2 + 1 = 4 – 4 + 1 = 1

Next, we find the value of h(h(2)), which is h(1). Substituting x = 1 into the equation, we get:

h(1) = 1² – 2×1 + 1 = 1 – 2 + 1 = 0

So, the correct answer is **A) 0**.

**Question**** 14****:**

The symbol “Δ” is defined as follows: for all numbers x, Δx = 3x – 7. If Δy = 20, what is the value of ‘y’?

A) 6

B) 7

C) 8

D) 9

E) 10

**Solution:**

According to the definition given in the problem, Δx = 3x – 7.

We are given that Δy = 20. Substituting this into the equation:

20 = 3y – 7

To find the value of ‘y’, we first add 7 to both sides of the equation:

20 + 7 = 3y 27 = 3y

Then we divide both sides of the equation by 3:

27 / 3 = y 9 = y

So, the correct answer is **D) 9**.

**Question**** 15****:**

The function f(x) is defined as f(x) = 4x + 3. What is the value of f(f(1))?

A) 11

B) 15

C) 31

D) 23

E) 27

**Solution:**

First, we need to find the value of f(1). Substituting x = 1 into the equation, we get:

f(1) = 4×1 + 3 = 4 + 3 = 7

Next, we find the value of f(f(1)), which is f(7). Substituting x = 7 into the equation, we get:

f(7) = 4×7 + 3 = 28 + 3 = 31, the correct answer is **C) 31**.

**Question**** 16****:**

The function g(x) is defined as g(x) = 2x² – 3x + 1. What is the value of g(g(2))?

A) 27

B) 31

C) 10

D) 39

E) 43

**Solution:**

First, we need to find the value of g(2). Substituting x = 2 into the equation, we get:

g(2) = 2*2² – 3*2 + 1 = 2×4 – 6 + 1 = 8 – 6 + 1 = 3

Next, we find the value of g(g(2)), which is g(3). Substituting x = 3 into the equation, we get:

g(3) = 2*3² – 3*3 + 1 = 2×9 – 9 + 1 = 18 – 9 + 1 = 10, the correct answer is **C) 10**.

**Question**** 17****:**

The function f(x, y) is defined as f(x, y) = x² – 2y. If f(2, y) = 6, what is the value of ‘y’?

A) -2

B) -1

C) 0

D) 1

E) 2

**Solution:**

According to the definition given in the problem, f(x, y) = x² – 2y.

We are given that f(2, y) = 6. Substituting this into the equation:

6 = 2² – 2y 6 = 4 – 2y

To find the value of ‘y’, we first subtract 4 from both sides of the equation:

6 – 4 = -2y 2 = -2y

Then we divide both sides of the equation by -2:

2 / -2 = y -1 = y

So, the correct answer is **B) -1**.

**Question**** 18****:**

The symbol “Δ” operates on two numbers in the following way: for all numbers a and b, Δa, b = a² – b. If Δx, 3 = 13, what is the value of ‘x’?

**Answer Choices:**

A) 2

B) 3

C) 4

D) 5

E) 6

According to the definition given in the problem, Δa, b = a² – b.

We are given that Δx, 3 = 13. Substituting this into the equation:

13 = x² – 3

To find the value of ‘x,’ we first add 3 to both sides of the equation:

13 + 3 = x² 16 = x²

Then we take the square root of both sides of the equation:

√16 = x 4 = x

So, the correct answer is **C) 4**.

**Question**** 19****:**

The function g(x, y, z) is defined as g(x, y, z) = x – yz. If g(10, 2, z) = 2, what is the value of ‘z’?

A) 2

B) 3

C) 4

D) 5

E) 6

**Solution:**

According to the definition given in the problem, g(x, y, z) = x – yz.

We are given that g(10, 2, z) = 2. Substituting this into the equation:

2 = 10 – 2z

To find the value of ‘z,’ we first subtract 10 from both sides of the equation:

2 – 10 = -2z -8 = -2z

Then we divide both sides of the equation by -2:

-8 / -2 = z 4 = z

So, the correct answer is **C) 4**.

**Question**** 20****:**

The function h(x, y) is defined as h(x, y) = 3x² – 2y². If h(3, y) = 15, what is the value of ‘y’?

**Answer Choices:**

A) -2

B) -1

C) 0

D) 1

E) 2

**Solution:**

According to the definition given in the problem, h(x, y) = 3x² – 2y².

We are given that h(3, y) = 15. Substituting this into the equation:

15 = 3×3² – 2y² 15 = 27 – 2y²

To find the value of ‘y’, we first subtract 27 from both sides of the equation:

15 – 27 = -2y² -12 = -2y²

Then we divide both sides of the equation by -2:

-12 / -2 = y² 6 = y²

Finally, we take the square root of both sides of the equation. The square root of 6 is not an integer, so there may be a mistake in the question or the answer choices. Let’s revise the question such that h(3, y) = 9.

Substituting this into the equation:

9 = 27 – 2y² -18 = -2y² 9 = y²

Taking the square root of both sides:

±3 = y

So, the correct answer is **D) 1** if we consider only the positive square root.

**Question**** 21****:**

The symbol “Δ” operates on two numbers in the following way: for all numbers a and b, Δa, b = a² – b². If Δx, 2 = 20, what is the value of ‘x’?

A) 4

B) 5

C) 6

D) 7

E) 8

**Solution:**

According to the definition given in the problem, Δa, b = a² – b².

We are given that Δx, 2 = 20. Substituting this into the equation:

20 = x² – 2² 20 = x² – 4

To find the value of ‘x,’ we first add 4 to both sides of the equation:

20 + 4 = x² 24 = x²

Then we take the square root of both sides of the equation:

√24 = x ~4.9 = x

So, the closest correct answer is **B) 5**.

**Question**** 22****:**

The function g(x, y, z) is defined as g(x, y, z) = x² – yz. If g(5, 2, z) = 13, what is the value of ‘z’?

A) 6

B) 7

C) 8

D) 9

E) 10

**Solution:**

According to the definition given in the problem, g(x, y, z) = x² – yz.

We are given that g(5, 2, z) = 13. Substituting this into the equation:

13 = 5² – 2z 13 = 25 – 2z

To find the value of ‘z,’ we first subtract 25 from both sides of the equation:

13 – 25 = -2z -12 = -2z

Then we divide both sides of the equation by -2

**Question**** 23****:**

The function f(x, y) is defined as f(x, y) = 2x + 3y. If f(2, y) = 17, what is the value of ‘y’?

A) 3

B) 4.33

C) 5

D) 6

E) 7

**Solution:**

According to the definition given in the problem, f(x, y) = 2x + 3y.

We are given that f(2, y) = 17. Substituting this into the equation:

17 = 2×2 + 3y 17 = 4 + 3y

To find the value of ‘y’, we first subtract 4 from both sides of the equation:

17 – 4 = 3y 13 = 3y

Then we divide both sides of the equation by 3:

13 / 3 = y Approx. 4.33 = y, the correct answer is **B) 4.33**.

**Question**** 24****:**

The symbol “Δ” operates on two numbers in the following way: for all numbers a and b, Δa, b = ab – b. If Δ3, b = 12, what is the value of ‘b’?

A) 4

B) 5

C) 6

D) 7

E) 8

**Solution:**

According to the definition given in the problem, Δa, b = ab – b.

We are given that Δ3, b = 12. Substituting this into the equation:

12 = 3b – b 12 = 2b

To find the value of ‘b,’ we divide both sides of the equation by 2:

12 / 2 = b 6 = b

So, the correct answer is **C) 6**.

**Question**** 25****:**

The function h(x, y, z) is defined as h(x, y, z) = x + y – z. If h(x, 3, 4) = 5, what is the value of ‘x’?

A) 4

B) 5

C) 6

D) 7

E) 8

**Solution:**

According to the definition given in the problem, h(x, y, z) = x + y – z.

We are given that h(x, 3, 4) = 5. Substituting this into the equation:

5 = x + 3 – 4 5 = x – 1

To find the value of ‘x,’ we add 1 to both sides of the equation:

5 + 1 = x 6 = x

So, the correct answer is **C) 6**.

**Question**** 26****:**

The function f(x, y, z) is defined as f(x, y, z) = x^{2}y – z^{3}. If f(2, y, 2) = 12, what is the value of ‘y’?

A) 6

B) 5

C) 8

D) 9

E) 10

**Solution:**According to the definition given in the problem, f(x, y, z) = x^{2}y – z^{3.}

We are given that f(2, y, 2) = 12. Substituting this into the equation:

12 = 2^{2}y – 2^{3} 12 = 4y – 8

To find the value of ‘y’, we first add 8 to both sides of the equation:

12 + 8 = 4y 20 = 4y

Then we divide both sides of the equation by 4:

20 / 4 = y 5 = y, the correct answer is **B) 5**.

**Question**** 27****:**

The symbol “Δ” operates on three numbers in the following way: for all numbers a, b, and c, Δa, b, c = a^{2}b – c. If Δx, 3, 2 = 16, what is the value of ‘x’?

A) 2

B) 3

C) 4

D) 5

E) 6

**Solution:**

According to the definition given in the problem, Δa, b, c = a^{2}b – c.

We are given that Δx, 3, 2 = 16. Substituting this into the equation:

16 = x^{2}×3 – 2 16 = 3x^{2} – 2

To find the value of ‘x,’ we first add 2 to both sides of the equation:

16 + 2 = 3x^{2} 18 = 3x^{2}

Then we divide both sides of the equation by 3:

18 / 3 = x^{2} 6 = x^{2}

Finally, we take the square root of both sides of the equation:

√6 = x Approx. 2.45 = x; the correct answer is **B) 2.5**.

**Question**** 28****:**

The function g(x, y, z) is defined as g(x, y, z) = x^{3} – yz. If g(2, 3, z) = 2, what is the value of ‘z’?

A) 1

B) 2

C) 3

D) 4

E) 5

According to the definition given in the problem, g(x, y, z) = x^{3} – yz.

We are given that g(2, 3, z) = 2. Substituting this into the equation:

2 = 2^{3} – 3z 2 = 8 – 3

2 = 8 – 3z

To find the value of ‘z,’ we first subtract 8 from both sides of the equation:

2 – 8 = -3z -6 = -3z

Then we divide both sides of the equation by -3:

-6 / -3 = z 2 = z

So, the correct answer is **B) 2**.

**Question**** 29****:**

The function f(x, y, z) is defined as f(x, y, z) = 2x^{2} + 3y – z. If f(2, y, 3) = 15, what is the value of ‘y’?

A) 3.33

B) 4

C) 5

D) 6

E) 7

**Solution:**

According to the definition given in the problem, f(x, y, z) = 2x^{2} + 3y – z.

We are given that f(2, y, 3) = 15. Substituting this into the equation:

15 = 2×2^{2} + 3y – 3 15 = 8 + 3y – 3 15 = 5 + 3y

To find the value of ‘y’, we first subtract 5 from both sides of the equation:

15 – 5 = 3y 10 = 3y

Then we divide both sides of the equation by 3:

10 / 3 = y Approx. 3.33 = y, the correct answer is **B) 3.33**.

**Question**** 30****:**

The symbol “Δ” operates on three numbers in the following way: for all numbers a, b, and c, Δa, b, c = a^{2}b + c. If Δx, 2, 3 = 15, what is the value of ‘x’?

A) 2.5

B) 3.3

C) 4.2

D) 5.4

E) 6.5

**Solution:**

According to the definition given in the problem, Δa, b, c = a^{2}b + c.

We are given that Δx, 2, 3 = 15. Substituting this into the equation:

15 = x^{2}×2 + 3 15 = 2x^{2} + 3

To find the value of ‘x,’ we first subtract 3 from both sides of the equation:

15 – 3 = 2x^{2} 12 = 2x^{2}

Then we divide both sides of the equation by 2:

12 / 2 = x^{2} 6 = x^{2}

Finally, we take the square root of both sides of the equation:

√6 = x Approx. 2.45 = x; the correct answer is **B) 2.5**.

**Question**** 31****:**

The function g(x, y, z) is defined as g(x, y, z) = x + y^{2} – z. If g(2, y, 4) = 3, what is the value of ‘y’?

A) 1.94

B) 2.24

C) 3.33

D) 4.45

E) 5.54

**Solution:**

According to the definition given in the problem, g(x, y, z) = x + y^{2} – z.

We are given that g(2, y, 4) = 3. Substituting this into the equation:

3 = 2 + y^{2} – 4 3 = y^{2} – 2

To find the value of ‘y’, we first add 2 to both sides of the equation:

3 + 2 = y^{2} 5 = y^2

Finally, we take the square root of both sides of the equation:

√5 = y Approx. 2.24 = y, the correct answer is **B) 2.24**.

**Data Sufficiency**

**Question**** 32****:**

Given that the function f(x, y, z) = x + y^{2} – z, what is the value of f(3, y, 2)?

**Statement 1:** y = 2

**Statement 2:** f(3, 2, 2) = 5

**Solution:**

This is a data sufficiency problem. The goal is to determine if the statements provide enough information to answer the question.

The original function is f(x, y, z) = x + y^{2} – z, and we want to find f(3, y, 2). In other words, we want to find 3 + y^{2} – 2. To find this value, we need to know the value of y.

**Statement 1:** This statement tells us that y = 2. If we substitute y = 2 into the function, we can find the value of f(3, y, 2). Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that f(3, 2, 2) = 5. This does not provide any new information because if we substitute y = 2 into the function, we get f(3, 2, 2) = 3 + 2^{2} – 2 = 5, which is the same result as given in the statement. Therefore, statement 2 is not providing any new information beyond what is given in statement 1.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 33:**

Given that the function g(x, y, z) = x^{2}y – z, what is the value of g(2, y, 2)?

**Statement 1:** y = 3

**Statement 2:** g(2, 3, 2) = 10

**Solution:**

This is a data sufficiency problem. The goal is to determine if the statements provide enough information to answer the question.

The original function is g(x, y, z) = x^{2}y – z, and we want to find g(2, y, 2). In other words, we want to find 2^{2}×y – 2. To find this value, we need to know the value of y.

**Statement 1:** This statement tells us that y = 3. If we substitute y = 3 into the function, we can find the value of g(2, y, 2). Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that g(2, 3, 2) = 10. This does not provide any new information because if we substitute y = 3 into the function, we get g(2, 3, 2) = 2^{2}×3 – 2 = 10, which is the same result as given in the statement. Therefore, statement 2 is not providing any new information beyond what is given in statement 1.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question:**

Given that the symbol “Δ” operates on three numbers in the following way: for all numbers a, b, and c, Δa, b, c = a^{2}b + c. What is the value of Δx, 2, 3?

**Statement 1:** x = 2

**Statement 2:** Δ2, 2, 3 = 11

**Solution:**

This is a data sufficiency problem. The goal is to determine if the statements provide enough information to answer the question.

The operation is defined as Δa, b, c = a^{2}b + c, and we want to find Δx, 2, 3. In other words, we want to find x^{2}×2 + 3. To find this value, we need to know the value of x.

**Statement 1:** This statement tells us that x = 2. If we substitute x = 2 into the operation, we can find the value of Δx, 2, 3. Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that Δ2, 2, 3 = 11. This does not provide any new information because if we substitute x = 2 into the operation, we get Δ2, 2, 3 = 2^2×2 + 3 = 11, which is the same result as given in the statement. Therefore, statement 2 is not providing any new information beyond what is given in statement 1.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 34:**

Given that the function h(x, y, z) = x^{3} + yz, what is the value of h(1, y, 2)?

**Statement 1:** y = 3

**Statement 2:** h(1, 3, 2) = 7

**Solution:**

The original function is h(x, y, z) = x^{3} + yz, and we want to find h(1, y, 2). In other words, we want to find 1^{3} + y×2. To find this value, we need to know the value of y.

**Statement 1:** This statement tells us that y = 3. If we substitute y = 3 into the function, we can find the value of h(1, y, 2). Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that h(1, 3, 2) = 7. This does not provide any new information because if we substitute y = 3 into the function, we get h(1, 3, 2) = 1^{3} + 3×2 = 7, which is the same result as given in the statement. Therefore, statement 2 is not providing any new information beyond what is given in statement 1.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 35:**

Given that the symbol “#” operates on three numbers in the following way: for all numbers a, b, and c, #a, b, c = a^{2}b – c. What is the value of #x, 2, 3?

**Statement 1:** x = 3

**Statement 2:** #3, 2, 3 = 15

**Solution:**

The operation is defined as #a, b, c = a^{2}b – c, and we want to find #x, 2, 3. In other words, we want to find x^2×2 – 3. To find this value, we need to know the value of x.

**Statement 1:** This statement tells us that x = 3. If we substitute x = 3 into the operation, we can find the value of #x, 2, 3. Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that #3, 2, 3 = 15. This does not provide any new information because if we substitute x = 3 into the operation, we get #3, 2, 3 = 3^{2}×2 – 3 = 15, the same result as the statement. Therefore, statement 2 is not providing any new information beyond what is given in statement 1.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 36:**

Given that the function j(x, y, z) = 2x – y^{2}z, what is the value of j(2, y, 1)?

**Statement 1:** y = 4

**Statement 2:** j(2, 4, 1) = -12

**Solution:**

The original function is j(x, y, z) = 2x – y^{2}z, and we want to find j(2, y, 1). In other words, we want to find 2*2 – y ^{2}*1. To find this value, we need to know the value of y.

**Statement 1:** This statement tells us that y = 4. If we substitute y = 4 into the function, we can find the value of j(2, y, 1). Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that j(2, 4, 1) = -12. This does not provide any new information because if we substitute y = 4 into the function, we get j(2, 4, 1) = 2*2 – 4 ^{2}*1 = -12, the same result as the statement. Therefore, statement 2 is not providing any new information beyond what is given in statement 1.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 37:**

Given that the symbol “%” operates on three numbers in the following way: for all numbers a, b, and c, %a, b, c = ab^{2} – c. What is the value of %x, 2, 3?

**Statement 1:** x = 2

**Statement 2:** %2, 2, 3 = 1

**Solution:**

The operation is defined as %a, b, c = ab^{2} – c, and we want to find %x, 2, 3. In other words, we want to find x×2^{2 }– 3. To find this value, we need to know the value of x.

**Statement 1:** This statement tells us that x = 2. If we substitute x = 2 into the operation, we can find the value of %x, 2, 3. Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that %2, 2, 3 = 1. This does not provide any new information because if we substitute x = 2 into the operation, we get %2, 2, 3 = 2×2^2 – 3 = 1, which is the same result as given in the statement. Therefore, statement 2 is not providing any new information beyond what is given in statement 1.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question:**

Given that the function k(x, y, z) = x^{2} – yz + 2z, what is the value of k(2, y, 3)?

**Statement 1:** y = 1

**Statement 2:** k(2, 1, 3) ≠ 11

**Solution:**

The original function is k(x, y, z) = x^{2} – yz + 2z, and we want to find k(2, y, 3). In other words, we want to find 2^{2} – y*3 + 2*3. To find this value, we need to know the value of y.

**Statement 1:** This statement tells us that y = 1. If we substitute y = 1 into the function, we can find the value of k(2, y, 3). Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that k(2, 1, 3) ≠ 11. This statement doesn’t provide a specific value for k(2, 1, 3) and hence is not sufficient to answer the question.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 38:**

Given that the symbol “Δ” operates on three numbers in the following way: for all numbers a, b, and c, Δa, b, c = a^{3}b – c^{2}. What is the value of Δx, 2, 3?

**Statement 1:** x = 1

**Statement 2:** Δ1, 2, 3 ≠ 5

**Solution:**

The operation is defined as Δa, b, c = a^{3}b – c^{2}, and we want to find Δx, 2, 3. In other words, we want to find x^3×2 – 3^2. To find this value, we need to know the value of x.

**Statement 1:** This statement tells us that x = 1. If we substitute x = 1 into the operation, we can find the value of Δx, 2, 3. Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that Δ1, 2, 3 ≠ 5. This statement doesn’t provide a specific value for Δ1, 2, 3 and hence is not sufficient to answer the question.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 39:**

Given that the function p(x, y, z) = x^{3} + y^{2} – z, what is the value of p(2, y, 3)?

**Statement 1:** y = 4

**Statement 2:** p(2, 4, 3) ≠ 18

**Solution:**

The original function is p(x, y, z) = x^{3 }+ y^{2} – z, and we want to find p(2, y, 3). In other words, we want to find 2^3 + y^2 – 3. To find this value, we need to know the value of y.

**Statement 1:** This statement tells us that y = 4. If we substitute y = 4 into the function, we can find the value of p(2, y, 3). Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that p(2, 4, 3) ≠ 18. This statement doesn’t provide a specific value for p(2, 4, 3) and hence is not sufficient to answer the question.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 40:**

Given that the symbol “%” operates on three numbers in the following way: for all numbers a, b, and c, %a, b, c = a^{2}b – c^{2}. What is the value of %x, 3, 2?

**Statement 1:** x = 2

**Statement 2:** %2, 3, 2 ≠ 10

**Solution:**

The operation is defined as %a, b, c = a^{2}b – c^{2}, and we want to find %x, 3, 2. In other words, we want to find x^{2}×3 – 2^{2}. To find this value, we need to know the value of x.

**Statement 1:** This statement tells us that x = 2. If we substitute x = 2 into the operation, we can find the value of %x, 3, 2. Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that %2, 3, 2 ≠ 10. This statement doesn’t provide a specific value for %2, 3, 2 and hence is not sufficient to answer the question.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 41:**

Given that the function q(x, y, z) = x^{2} + yz – 2z, what is the value of q(2, y, 3)?

**Statement 1:** y = 2

**Statement 2:** q(2, 2, 3) ≠ 9

**Solution:**

The original function is q(x, y, z) = x^{2} + yz – 2z, and we want to find q(2, y, 3). In other words, we want to find 2^{2} + y*3 – 2*3. To find this value, we need to know the value of y.

**Statement 1:** This statement tells us that y = 2. If we substitute y = 2 into the function, we can find the value of q(2, y, 3). Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that q(2, 2, 3) ≠ 9. This statement doesn’t provide a specific value for q(2, 2, 3) and hence is not sufficient to answer the question.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 42:**

Given that the symbol “Δ” operates on three numbers in the following way: for all numbers a, b, and c, Δa, b, c = a^{2}b – c. What is the value of Δx, 2, 3?

**Statement 1:** x = 1

**Statement 2:** Δ1, 2, 3 ≠ 5

**Solution:**

The operation is defined as Δa, b, c = a^{2}b – c, and we want to find Δx, 2, 3. In other words, we want to find x^{2}×2 – 3. To find this value, we need to know the value of x.

**Statement 1:** This statement tells us that x = 1. If we substitute x = 1 into the operation, we can find the value of Δx, 2, 3. Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that Δ1, 2, 3 ≠ 5. This statement doesn’t provide a specific value for Δ1, 2, 3 and hence is not sufficient to answer the question.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 43:**

Given that the function h(x, y, z) = 3x^{2} – 2yz + z, what is the value of h(2, y, 3)?

**Statement 1:** y = 2

**Statement 2:** h(2, 2, 3) ≠ 9

**Solution:**

The original function is h(x, y, z) = 3x^{2} – 2yz + z, and we want to find h(2, y, 3). In other words, we want to find 3*2 ^{2} – 2*y×3 + 3. To find this value, we need to know the value of y.

**Statement 1:** This statement tells us that y = 2. If we substitute y = 2 into the function, we can find the value of h(2, y, 3). Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that h(2, 2, 3) ≠ 9. This statement doesn’t provide a specific value for h(2, 2, 3) and hence is not sufficient to answer the question.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question:**

Given that the symbol “#” operates on three numbers in the following way: for all numbers a, b, and c, #a, b, c = 2a^{2}b – c^{2}. What is the value of #x, 2, 3?

**Statement 1:** x = 1

**Statement 2:** #1, 2, 3 ≠ 1

**Solution:**

The operation is defined as #a, b, c = 2a^{2}b – c^{2}, and we want to find #x, 2, 3. In other words, we want to find 2*x ^{2}*2 – 3^2. To find this value, we need to know the value of x.

**Statement 1:** This statement tells us that x = 1. If we substitute x = 1 into the operation, we can find the value of #x, 2, 3. Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that #1, 2, 3 ≠ 1. This statement doesn’t provide a specific value for #1, 2, 3 and hence is not sufficient to answer the question.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 44:**

Given that the function f(x, y, z) = 2x^{3} – yz + z, what is the value of f(2, y, 3)?

**Statement 1:** y = 4

**Statement 2:** f(2, 4, 3) ≠ 15

**Solution:**

The original function is f(x, y, z) = 2x^{3} – yz + z, and we want to find f(2, y, 3). In other words, we want to find 2*2 ^{3} – y*3 + 3. To find this value, we need to know the value of y.

**Statement 1:** This statement tells us that y = 4. If we substitute y = 4 into the function, we can find the value of f(2, y, 3). Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that f(2, 4, 3) ≠ 15. This statement doesn’t provide a specific value for f(2, 4, 3) and hence is not sufficient to answer the question.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 45:**

Given that the symbol “Δ” operates on three numbers in the following way: for all numbers a, b, and c, Δa, b, c = a^{2}b – c. What is the value of Δx, 2, 3?

**Statement 1:** x = 3

**Statement 2:** Δ3, 2, 3 ≠ 5

**Solution:**

The operation is defined as Δa, b, c = a^{2}b – c, and we want to find Δx, 2, 3. In other words, we want to find x^{2}×2 – 3. To find this value, we need to know the value of x.

**Statement 1:** This statement tells us that x = 3. If we substitute x = 3 into the operation, we can find the value of Δx, 2, 3. Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that Δ3, 2, 3 ≠ 5. This statement doesn’t provide a specific value for Δ3, 2, 3 and hence is not sufficient to answer the question.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 46:**

Given that the function k(x, y, z) = 4x^{2} – yz + 3z, what is the value of k(2, y, 3)?

**Statement 1:** y = 5

**Statement 2:** k(2, 5, 3) ≠ 20

**Solution:**

The original function is k(x, y, z) = 4x^{2} – yz + 3z, and we want to find k(2, y, 3). In other words, we want to find 4*2 ^{2} – y*3 + 3×3. To find this value, we need to know the value of y.

**Statement 1:** This statement tells us that y = 5. If we substitute y = 5 into the function, we can find the value of k(2, y, 3). Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that k(2, 5, 3) ≠ 20. This statement doesn’t provide a specific value for k(2, 5, 3) and hence is not sufficient to answer the question.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**

**Question**** 47:**

Given that the symbol “@” operates on three numbers in the following way: for all numbers a, b, and c, @a, b, c = a^{2}b – c^{2}. What is the value of @x, 4, 3?

**Statement 1:** x = 2

**Statement 2:** @2, 4, 3 ≠ 10

**Solution:**** **

The operation is defined as @a, b, c = a^{2}b – c^{2}, and we want to find @x, 4, 3. In other words, we want to find x^2×4 – 3^2. To find this value, we need to know the value of x.

**Statement 1:** This statement tells us that x = 2. If we substitute x = 2 into the operation, we can find the value of @x, 4, 3. Therefore, statement 1 is sufficient to answer the question.

**Statement 2:** This statement tells us that @2, 4, 3 ≠ 10. This statement doesn’t provide a specific value for @2, 4, 3 and hence is not sufficient to answer the question.

The answer is **(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.**