# GMAT Math : Graphing

## Example Questions

### Example Question #11 : Coordinate Geometry

Let  and  be real numbers.

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

Statement 1:

Statement 2:

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.

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.

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

Explanation:

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 #12 : Coordinate Geometry

Let  and  be real numbers.

What is the sum of  and its complex conjugate?

Statement 1:

Statement 2:

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.

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

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.

Explanation:

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 #13 : Coordinate Geometry

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.

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.

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

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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 #641 : Data Sufficiency Questions

Let  and  be real numbers.

What is the product of  and its complex conjugate?

Statement 1:

Statement 2:

BOTH statements TOGETHER are insufficient to answer the question.

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

EITHER 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.

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

Explanation:

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 #15 : Coordinate Geometry

Let  and  be real numbers.

What is the sum of  and its complex conjugate?

Statement 1:

Statement 2:

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

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient 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.

Explanation:

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 #16 : Coordinate Geometry

Let  and  be real numbers.

What is the product of  and its complex conjugate?

Statement 1:

Statement 2:

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

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT 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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

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 #17 : Coordinate Geometry

Let  and  be real numbers.

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

Statement 1:

Statement 2:

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

EITHER statement ALONE is 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.

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

Explanation:

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.

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.

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.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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 #19 : Coordinate Geometry

Let  and  be real numbers.

What is the sum of  and its complex conjugate?

Statement 1:

Statement 2:

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

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient 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 sufficient to answer the question, but NEITHER 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.

Explanation:

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.

### Example Question #20 : Coordinate Geometry

Let  and  be real numbers.

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

Statement 1:

Statement 2:

BOTH statements TOGETHER are insufficient to answer the question.

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 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.

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

Explanation:

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 , the desired sum is .

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