GMAT Quantitative

GMAT Quantitative Practice Test: Practice Test 11

Practice Test 11 for GMAT Quantitative: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

What is the area of a trapezoid with a height of 7, a base of 5, and another base of 13?

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Question 1

What is the area of a trapezoid with a height of 7, a base of 5, and another base of 13?

$\dpi{100} \small 63$ (correct answer)
$\dpi{100} \small 39$
$\dpi{100} \small 29$
$\dpi{100} \small 43$
$\dpi{100} \small 51$

Explanation: $area = \frac{(b_{1}+ b_{2}\cdot h)}{2} = \frac{(5 + 13)\cdot 7}{2} = \frac{18\cdot 7}{2} = \frac{126}{2} = 63$

Question 2

Data Sufficiency Question

Solve for , , and :

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient (correct answer)
Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question
Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question
Each statement alone is sufficient
Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question

Explanation: In order to solve an equation set, one requires a number of equations equal to the number of variables. Therefore, three equations are needed and both statements are required to solve the problem.

Question 3

Consider this system of equations:

Does this system has exactly one solution?

Statement 1: The lines representing the equations are parallel.

Statement 2:

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.
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 insufficient to answer the question.

Explanation: The solution of a system of linear equations is the point at which their lines intersect; if they are parallel, then by definition, there is no such point, and the system has no solution. If , then we can rewrite the second equation as In slope-intercept form: This line has a slope of . The other equation has a line with slope of also, as can be easily seen since it is already in slope-intercept form. Since both equations have lines with the same slope, they are either the same line or parallel lines; either way, the system does not have exactly one solution.

Question 4

A line segement on the coordinate plane has endpoints and . Which of the following expressions is equal to the length of the segment?

(correct answer)

Explanation: Apply the distance formula, setting :

Question 5

Consider the function .

State whether this function is even, odd, or neither, and give the reason for your answer.

is not odd, because there exists at least one value of for which ; is not even, because there exists at least one value of for which . (correct answer)
is odd because for each value of in the domain.
is even because for each value of in the domain.
is odd because it is a polynomial of degree 3.
is even because it is a polynomial of degree 3.

Explanation: A function is odd if and only if for each value of in the domain; it is even if and only if for each value of in the domain. To disprove a function is odd or even, we need only find one value of for which the appropriate statement fails to hold. Consider : , so is not an odd function; , so is not an even function.

Question 6

Data Sufficiency Question- do not actually solve the problem

Solve for .

$\small 8vxy+9xz+10vyz=14$

1. $\small x=7$

2. $\small y=2$

Statements 1 and 2 together are not sufficient, and additional data is needed to answer the question (correct answer)
Statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question
Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question
Both statements taken together are suffienct to answer the question, but neither statement alone is sufficient
Each statement alone is sufficient

Explanation: In order to solve an equation with 4 variables, you need to know either 3 of the variables or have a system of 4 equations to solve.

Question 7

A monster is holding a brown paper bag filled with cookies. Inside the bag there are two types of cookies, chocolate chip and oatmeal raisin. If the monster pulls one cookie out at random, what is the probability that he pulls out a chocolate chip cookie.

1. There are a total of 24 cookies in the bag.

2. There are 8 more chocolate chip cookies in the bag than oatmeal raisin cookies.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (correct answer)
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.

Explanation: We need both statements to get the answer. Statement 1 alone is not enough. Having 24 total cookies does not tell us the proportions of each type of cookie. We could have 1 chocolate chip cookie and 23 oatmeal raisin, or vice versa. Statement 2 alone is not sufficient either. Knowing that there are 8 more chocolate chip cookies than oatmeal raisin without knowing the total amount in the bag does not help. It could be 1 oatmeal raisin cookie and 9 chocolate chip, or it could be 100 oatmeal raisin cookies and 108 chocolate chip cookies (although that is a pretty big bag!) Only by having both statements together can we find the right proportion. If we let be the number of chocolate chip cookies, and be the number of oatmeal raisin, we can set two equations and find the right ratio. We know that from statement 1, and from statement 2. Substituting into the first equation we get Therefore, we get 8 oatmeal raisin cookies, and 16 chocolate chip cookies. The answer is or 66.67% chance that the monster pulls out a chocolate chip cookie.

Question 8

For which of the following values of would the median and the mode of the data set be equal?

(correct answer)
None of the other answers are correct.

Explanation: If the known values are ordered from least to greatest, the set looks like this: Below are each of the choices, followed by the set that results if it is added to the above set, followed by the median - the middle element - and the mode - the most frequently occurring element. Only the addition of 11 yields a set with median and mode equal to each other.

Question 9

A baseball travels at a rate of . How long does it take to reach a fence that is away?

(correct answer)

Explanation: The first step is to convert the distance into feet: Then, divide the distance by the baseball's rate of travel:

Question 10

If , what does equal?

(correct answer)

Explanation: We can use the fact that to see that Since , we have .

Question 11

Lines

Note: Figure NOT drawn to scale.

Evaluate .

Statement 1:

Statement 2:

BOTH statements TOGETHER are insufficient 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.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient 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.

Explanation: Even with both statements, cannot be determined because the length of is missing. For example, we can have and , making ; or, we can have and , making . Neither scenario violates the conditions given.

Question 12

If an equilateral triangle has a perimeter of , what is the length of each side?

(correct answer)
Cannot be determined

Explanation: An equilateral triangle has three equal sides; therefore, to find the length of each side, divide the perimeter by :

Question 13

Simplify this expression as much as possible:

(correct answer)
The expression cannot be simplified further

Explanation:

Question 14

If an equilateral triangle has a perimeter of , what is the length of each side?

(correct answer)
Cannot be determined

Explanation: An equilateral triangle has three equal sides; therefore, to find the length of each side, divide the perimeter by :

Question 15

What is the mean of , , , , , and ?

Statement 1:

Statement 2:

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

Explanation: The mean of a data set requires you to know the sum of the elements and the number of elements; you know the latter, but neither statement alone provides any clues to the former. However, if you know both, you can add both sides of the equations as follows: Rewrite as: and divide by 9: Now you know the sum, so divide it by 6 to get the mean: .

Question 16

Find the solution set of the inequality

(correct answer)

Explanation: To solve a quadratic inequality, move all expressions to the left first: The square of a real number cannot be less than 0, so , the only solution.

Question 17

What is 15 percent of 80?

(correct answer)

Explanation: This word problem can be written as Simplifying this, we get

Question 18

A vertical parabola has two -intercepts, one at and one at .

Which of the following must be true about this parabola?

The parabola must have the line of the equation as its line of symmetry. (correct answer)
The parabola must have as its vertex.
The parabola must have its -intercept at .
The parabola must be concave downward.
None of the statements in the other choices must be true.

Explanation: A parabola with its -intercepts at and at has as its equation for some nonzero . If this is multiplied out, the equation can be rewritten as or, simplified, The sign of quadratic coefficient determines whether it is concave upward or concave downward. We do not have the sign or any way of determining it. The -coordinate of the -intercept is the contant, , but without knowing , we have no way of knowing . The -coordinate of the vertex of is the value . since , this expression becomes The -coordinate is , but without knowing , this coordinate, and the vertex itself, cannot be determined. The line of symmetry is the line ; this value was computed to be equal to 6, so the line can be determined to be .

Question 19

Determine the equation of the tangent line to the following curve at the point :

(correct answer)

Explanation: First find the slope of the tangent line by taking the derivative of the function and plugging in the x value of the given point to find the slope of the curve at that location: So the slope of the tangent line to the curve at the given point is . The next step is to plug this slope into the formula for a line, along with the coordinates of the given point, to solve for the value of the y intercept of the tangent line: We now know the slope and y intercept of the tangent line, so we can write its equation as follows:

Question 20

Some balls are placed in a large box; the balls include one ball marked "10", two balls marked "9", and so forth up to ten balls marked "1". A ball is drawn at random.

is an integer between 1 and 10 inclusive. True or false: the probability that the ball will have the number marked on it is greater than .

Statement 1: is a prime integer.

Statement 2:

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question. (correct answer)
STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.
EITHER STATEMENT ALONE provides sufficient information to answer the question.
BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.
BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

Explanation: The total number of balls in the box will be . Since , it follows that the number of balls is . The frequencies out of 55 of each outcome from 1 to 10, in order, are as follows: Their respective probabilities are their frequencies divided by 55: . The probability that the ball will be marked "5" is ; therefore, the probability that the ball will be marked with any given integer less than or equal to 5 will be greater than . The probability that the ball will be marked "6" is ; therefore, the probability that the ball will be marked with any given integer greater than or equal to 6 will be less than . Therefore, it suffices to know whether the number on the ball is less than or equal to 5. Statement 2 states that the number on the ball is less than or equal to 5, so it is sufficient to answer the question in the affirmative. Statement 1 is insufficient, since there are primes less than or equal to 5 - 2, 3, and 5 - and one prime greater than 5, which is 7.

Question 21

What is the vertical asymptote of the graph of ?

(correct answer)
The graph has no vertical asymptote.

Explanation: The graph of a logarithmic function has a vertical asymptote which can be found by finding the value at which the power is equal to 0: If , then is an undefined expression, so the vertical asymptote is .

Question 22

Define a function as follows:

Give the vertical aysmptote of the graph of .

The graph of does not have a vertical asymptote. (correct answer)

Explanation: Since any number, positive or negative, can appear as an exponent, the domain of the function is the set of all real numbers; in other words, is defined for all real values of . It is therefore impossible for the graph to have a vertical asymptote.

Question 23

Define two sets as follows:

where and are distinct positive odd integers and and are distinct positive even integers.

How many elements are contained in the set ?

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.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.

Explanation: Suppose we know , but we do not assume the second statement. If and , then , a four-element set. If If and , then , a three-element set. Therefore, we cannot make a conclusion about the size of . A similar argument can be used to show that assuming only the second statement also does not allow a conclusion. If we know both statements, however, , and we can prove that has four elements. The answer is that both statements together are sufficient to answer this question, but neither statement alone is sufficient.

Question 24

What is the volume of a cylinder that is 12 inches high and has a radius of 6 inches?

(correct answer)

Explanation:

Question 25

The perimeter of a regular octagon is two kilometers. Give its sidelength in meters.

(correct answer)

Explanation: One kilometer is equal to 1,000 meters, so two kilometers comprise 2,000 meters. A regular octagon has eight sides of equal length, so divide by 8 to get the sidelength: meters.