### All GMAT Math Resources

## Example Questions

### Example Question #1 : Exponents

Which is the greater quantity, or - or are they equal?

Statement 1:

Statement 2:

**Possible Answers:**

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 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.

**Correct answer:**

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

From Statement 1 alone,

Now assume Statement 2 alone. We show that this is insufficient with two cases:

Case 1:

; ; therefore,

Case 1:

; ; therefore,

### Example Question #2 : Exponents

Does exist?

Statement 1: and are both negative.

Statement 2: divided by 2 yields an integer.

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

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.

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.

**Correct answer:**

A logarithm can be taken of a number if and only if the number is positive. If Statement 1 alone is true, then , being the product of two negative numbers, must be positive, and exists.

Statement 2 is irrelevant; 4 and both yield integers when divided by 2, but and does not exist.

### Example Question #2 : Exponents

Johnny was assigned to write a number in scientific notation by filling the circle and the square in the pattern below with two numbers.

Johnny filled in both shapes with numbers. Did he succeed?

Statement 1: He filled in the circle with the number "10".

Statement 2: He filled in the square with a negative integer.

**Possible Answers:**

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 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:**

The number is a number written in scientific notation if and only of two conditions are true:

1)

2) is an integer

By Statement 1 Johnny filled in the circle incorrectly, since it makes .

By Statement 2, Johnny filled in the square correctly, but the statement says nothing about how he filled in the circle; Statement 2 leaves the question open.

### Example Question #3 : Exponents

Is ?

(1)

(2)

**Possible Answers:**

C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

E: Statements (1) and (2) TOGETHER are not sufficient

A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient

B: Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient

D: EACH statement ALONE is sufficient

**Correct answer:**

A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient

From statement 1 we get that and .

So the first term is positive and the second term is negative, which means that is negative; therefore the statement 1 alone allows us to answer the question.

Statement 2 tells us that . If , we have which is less than . Therefore in this case .

For , we have which is greater than . So in this case .

So statement 2 is insufficient.

Therefore the correct answer is A.

### Example Question #4 : Exponents

Solve the following rational expression:

(1)

(2)

**Possible Answers:**

Both statements TOGETHER are not sufficient.

Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

EACH statement ALONE is sufficient to answer the question

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient

Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient

**Correct answer:**

Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient

When replacing m=5 in the expression we get:

Therefore statement (1) ALONE is not sufficient.

When replacing m=2n in the expression we get:

Therefore **statement (2) ALONE is sufficient.**

### Example Question #5 : Exponents

Myoshi has been assigned to write one number in the circle and one number in the square in the diagram below in order to produce a number in scientifc notation.

.

Did Myoshi succeed?

Statement 1: Myoshi wrote in the circle.

Statement 2: Myoshi wrote in the square.

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

A number in scientific notation takes the form

where and is an integer of any sign.

Assuming Statement 1 alone, Myoshi did not succeed, since she entered an incorrect number into the circle - .

Statement 2 alone is inconclusive. Myoshi entered a correct number into the square, since is an integer. But the question is open, since it is not known whether she entered a correct number into the circle or not.

### Example Question #6 : Exponents

is a nonzero number. Is it negative or positive?

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:**

All negative numbers are less than their (positive) squares, as are all positive numbers greater than 1. Therefore, if Statement 1 is assumed, .

can be determined to be positive.

Statement 2 alone is inconclusive. For example, if , then , and if , . In both cases, , but has different signs in the two cases.

### Example Question #8 : Exponents

is a nonzero number. Is it negative or positive?

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:**

BOTH statements TOGETHER are insufficient to answer the question.

Both statements together are inufficient to produce an answer. For example,

If , then and .

If , then and .

In both cases, and , but the signs of differ between cases.

### Example Question #7 : Exponents

is a number not in the set .

Of the elements , which is the greatest?

Statement 1: is a negative number.

Statement 2:

**Possible Answers:**

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

**Correct answer:**

Statement 1 alone is inconclusive, as can be demonstrated by examining two negative values of other than .

Case 1: .

Then

is the greatest of these values.

Case 2:

Then

is the greatest of these values.

Now assume Statement 2 alone. Either or .

Case 1: .

Then , so ; similarly, .

is the greatest of the three.

Case 2: .

Odd power is negative, and even powers and are positive, so one of the latter two is the greatest. Since , it follows that . It then follows that , or .

Again, is the greatest of the three.

Statement 2 alone is sufficient, but not Statement 1.

### Example Question #8 : Exponents

Note: Figure NOT drawn to scale.

Examine the above diagram. True or false: .

Statement 1:

Statement 2: and have the same perimeter.

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question

**Correct answer:**

From Statement 1 alone, it follows by the similarity of the triangles that . These are congruent inscribed angles of a circle, which intercept congruent arcs, so . Since congruent arcs have congruent chords, .

Statement 2 alone only tells us the relative perimeters of the triangles. We have no way of determining the individual sidelengths or angle measures relative to each other, so Statement 2 alone is inconclusive.

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