GRE Exam Questions & Answers 2026 (21–30)

Question 1

Rectangle B has a length of six units and a width of four units. Rectangle A, on the other hand, has twice the length and 1/2 the width of Rectangle B.

What is the perimeter of rectangle A?

TheoryPractice · Accepted Answer

28 units. Rectangle A has two sides measuring 12 units each (twice the length of Rectangle B) and two sides measuring 2 units each (half the width of Rectangle B).

Therefore, the perimeter of Rectangle A is calculated as 12 + 12 + 2 + 2, which equals 28 units.

Question 2

Let universal set U be the set of all people. Let b represent Brittany.

Let A be the set of people who like James Blunt, B, the set of people who like John Legend, and C, the set of people who like Pharrell Williams.

True or false: Brittany likes James Blunt, John Legend, and Pharrell Williams.

Statement 1: b \in A \cap B \cap C

Statement 2: b \in A \cup B \cup C

TheoryPractice · Accepted Answer

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.. Assume Statement 1 alone. A \cap B \cap C is the intersection of the sets A, B, and C. Since Brittany falls in this intersection, she falls in all three sets, and it follows that she likes all three of James Blunt, John Legend, and Pharrell Williams.

Assume Statement 2 alone. A \cup B \cup C is the union of the three sets. Since Brittany falls in this union, she falls in one, two, or all three sets, which means that she likes at least one of James Blunt, John Legend, and Pharrell Williams. However, more information is needed before it can be established whether or not she likes all three.

Question 3

Simplify: \sqrt{-243}

TheoryPractice · Accepted Answer

9i\sqrt{3}. Step 1: Split the \sqrt{-243} into \sqrt{243}\cdot\sqrt{-1}.

Step 2: Recall that \sqrt{-1}=i, so let's replace it.

We now have: \sqrt{243}\cdot i.

Step 3: Simplify \sqrt{243}. To do this, we look at the number on the inside.

243=3\cdot3\cdot3\cdot3\cdot3.

Step 4: Take the factorization of 243 and take out any pairs of numbers. For any pair of numbers that we find, we only take 1 of the numbers out.

We have a pair of 3's, so a 3 is outside the radical.
We have another pair of 3's, so one more three is put outside the radical.

We need to multiply everything that we bring outside: 3*3=9

Step 5: The i goes with the 9...9i

Step 6: The last 3 after taking out pairs gets put back into a square root and is written right after the i

It will look something like this: 9i\sqrt{3}

Question 4

Select the correct option.

TheoryPractice · Accepted Answer

Quantity A is greater. We know that circumference of a circle equals 2πr.

Here, r= 3 so the circumference equals 18 approximately.

Now, Perimeter of a square is equal to sum of all its sides, which in this case is 16.

Therefore, quantity A is greater.

Question 5

x – y = 9 11 – y = 2 Quantity A: x Quantity B: 0

TheoryPractice · Accepted Answer

Quantity A is greater.. First solve the second equation:

11 − y = 2 y = 9

Next, substitute this value of y in the first equation:

x − 9 = 9 x = 18

Of the two quantities, x is greater.

Question 6

Master Chef Alan has written 3 recipe books, and each book contains 15 different recipes. As a result, he prepares a dish every day using a recipe from one of his books.

Today, what is the probability that Master Chef Alan will cook the 4th dish from the 3rd book?

TheoryPractice · Accepted Answer

1/45. There are a total of 45 recipes, with each book containing 15 recipes.

Therefore, the probability of Master Chef Alan cooking the 4th dish from the 3rd book today is 1/45.

It is important to focus on the relevant information and not get distracted.

Additionally, all the recipes are unique and different from each other.

Question 7

The figure shown consists of four straight lines and 24 individual angles.

What is the smallest number of angles that need to be explicitly measured in order for the measurements of all the angles in the figure to be determined?

TheoryPractice · Accepted Answer

3.

The way to proceed is to label one angle at a time as 'measured,' and then fill in the angles that you can calculate based on the angles already 'known.' In the diagram above, the first angle measured is marked with an arc, and the other angles that you can infer are marked with dots. The three inferred angles can be known because of the fact that two angles that make up a straight line add up to 180°, and also because vertical angles are equal to each other.

The second diagram shows a second angle measured, and the angles that can be inferred. The 3 other angles at the intersection of the lines making up the measured angle are known for the same reason as before. The other four angles on the lower left are known because the three angles that make up a triangle add up to 180°.

In the third diagram, a third angle is measured, and the angles surrounding it (in the cluster labeled 1 on the diagram) are determined, as before. Then, the angles near the label 2 can be calculated, because three out of four of the angles in the quadrilateral at bottom right are already known. Finally, the angles marked in the cluster marked 3 can be deduced because now two of the three angles of the smallest triangle are known. Therefore, the minimum number of angle measurements required is 3.

Question 8

For which values of p is \sum_{n=1}^\infty \frac{1}{n^p} convergent?

TheoryPractice · Accepted Answer

only p \gt 1. We can solve this problem quite simply with the integral test. We know that if

\int_1^{\infty} \frac{1}{x^p} dx

converges, then our series converges.

We can rewrite the integral as

\int x^{-p} dx

and then use our formula for the antiderivative of power functions to get that the integral equals

\frac{x^{1-p}}{1-p} \Big|_{x=1}^\infty.

We know that this only goes to zero if 1-p \lt 0. Subtracting p from both sides, we get

1 \lt p.

Question 9

What happens to the difference in the areas of the parallelogram and the rectangle with sides a and b, as α and β in the parallelogram approach 90º?

TheoryPractice · Accepted Answer

the difference decreases and approaches zero. As α and β in the parallelogram approach 90º, the values of sin α and sin β approach 1. The formula for the area of the parallelogram, therefore, approximately equals that of the rectangle, as shown here: Area of rectangle: A = a \cdot b Area of parallelogram: A = a \cdot b \cdot (approx.\;1)= approx.\;a \cdot b The difference continues to decrease as the angles become closer to 90º, until they become equal at 90º; at which point, parallelogram STUV would have actually become a rectangle (or parallelogram with right angles). (Remember that the formulas for the area of a rectangle and parallelogram are not the same.).

Question 10

A brick wall contains 52 bricks in its bottom row and 49 bricks in the next row up from the bottom row. Each subsequent row contains 3 fewer bricks than the row immediately below it. If the wall contains 16 rows, how many bricks total make up the wall?

TheoryPractice · Accepted Answer

472. We know the number of terms (16) and the first term (52) in this series, but to find the sum using the formula n/2(a_1+a_n), we also need to know the final term (a_n). Using the formula for a the last term of an arithmetic sequence (a_n = a_0+nd) with a common difference (d) of 3 and a zeroth term of 55 (3 greater than the first term) gives a_n = 55-3(16), or a_n = 7. We can now use the sum formula: replacing n with 16, a_1 with 52, and a_n with 7, the sum equals 16/2 (52 + 7), which simplifies to 472.

GRE Exam Questions & Answers 2026 (21–30)

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