### All GMAT Math Resources

## Example Questions

### Example Question #1 : Dsq: Graphing Complex Numbers

Let and be real numbers.

What is the product of and its complex conjugate?

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

The complex conjugate of an imaginary number is , and

.

Therefore, Statement 1 alone, which gives that , provides sufficient information to answer the question, whereas Statement 2 provides unhelpful information.

### Example Question #1 : Dsq: Graphing Complex Numbers

Let and be real numbers.

From the number , subtract its complex conjugate. What is the result?

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

**Correct answer:**

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

The complex conjugate of an imaginary number is , and

.

Therefore, it is necessary and sufficient to know in order to answer the question. Statement 1 does not give this value, and is unhelpful here; Statement 2 does give this value.

### Example Question #1 : Dsq: Graphing Complex Numbers

Let and be real numbers.

What is the sum of and its complex conjugate?

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

The complex conjugate of an imaginary number is , and the sum of the two numbers is

.

Therefore, it is necessary and sufficient to know in order to answer the question. Statement 1 alone gives this information; Statement 2 does not, and it is unhelpful.

### Example Question #1 : Dsq: Graphing Complex Numbers

Let be a positive integer.

True or false:

Statement 1: is a prime number.

Statement 2: is a two-digit number ending in a 7.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

If is a positive integer, then if and only if is a multiple of 4.

It follows that if , cannot be a prime number. Also, every multiple of 4 is even, so as an even number, cannot end in 7. Contrapositively, if Statement 1 is true and is prime, or if Statement 2 is true and if ends in 7, it follows that is not a multiple of 4, and .

### Example Question #5 : Dsq: Graphing Complex Numbers

Let and be real numbers.

What is the product of and its complex conjugate?

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

The complex conjugate of an imaginary number is , and

.

Therefore, it is necessary and sufficient to know the values of both and in order to answer the problem. Each statement alone gives only one of these values, so each statement alone provides insufficient information; the two together give both, so the two statements together provide sufficient information.

### Example Question #6 : Dsq: Graphing Complex Numbers

Let and be real numbers.

What is the sum of and its complex conjugate?

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

The complex conjugate of an imaginary number is , and

.

We show, however, that the two statements are insufficient to determine the sum by examining two scenarios.

Case 1: .

, and since , . The conditions of both statements are satisfied.

The sum of the numbers is .

Case 2: .

, and since , . The conditions of both statements are satisfied.

The sum of the numbers is .

In both cases, the conditions of both statements are satisfied, but the sum of the number and its complex conjugate differs between the two.

### Example Question #1 : Dsq: Graphing Complex Numbers

Let and be real numbers.

What is the product of and its complex conjugate?

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

The complex conjugate of an imaginary number is , and

.

Statement 1 alone provides insufficient information, as seen in these two scenarios, both of which feature values of and that add up to 12:

Case 1:

Then , and the product of this number and its complex conjugate is .

Case 2:

Then , and the product of this number and its complex conjugate is .

The two cases result in different products.

For a similar reason, Statement 2 alone provides insufficient information.

If both statements are assumed to be true, they form a system of equations that can be solved as follows:

Backsolve:

Since we know that and , then we know that the desired product is .

### Example Question #8 : Dsq: Graphing Complex Numbers

Let and be real numbers.

From the number , subtract its complex conjugate. What is the result?

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

The complex conjugate of an imaginary number is , and

.

Therefore, it is necessary and sufficient to know in order to answer the question. Neither statement alone gives this information. However, the first statement can be rewritten by factoring out as a difference of squares:

Since , then by substitution,

A system of linear equations has now been formed; subtract both sides of the equations as follows:

We need go no further; since , the desired difference is .

### Example Question #1 : Dsq: Graphing Complex Numbers

Let be a positive integer.

Evaluate .

Statement 1: is a multiple of 16.

Statement 2: is a multiple of 20.

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

EITHER statement ALONE is sufficient to answer the question.

The value of , a positive integer, is equal to , where is the remainder of the division of by 4. Either statement alone is enough to prove that is divisible by 4, since, if a number is divisible by a given number (16 or 20 in these statements), it is divisible by any factor of that number (with 4 being a factor of both).

### Example Question #10 : Dsq: Graphing Complex Numbers

Let and be real numbers.

What is the sum of and its complex conjugate?

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

The complex conjugate of an imaginary number is , and

.

Therefore, it is necessary and sufficient to know in order to answer the question. Neither statement alone gives this information. However, the two statements together form a linear system that can be solved as follows:

We need go no further; since , this is the desired sum.

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