Data-Sufficiency Questions

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GMAT Quantitative › Data-Sufficiency Questions

Questions 1 - 10

Let be a positive integer.

True or false: $i^{N} = 1$

Statement 1: is a prime number.

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

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.

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.

Explanation

If is a positive integer, then $i^{N} = 1$ if and only if is a multiple of 4.

It follows that if $i^{N} = 1$ , 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 $i^{N} e 1$ .

Find the equation of the line .

The slope of line is .
Line goes through point .

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

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 to answer the question.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Explanation

Statement 1: We're given the slope line AB, because we are ask for the equation of the line we need more than just the slope of the line. Therefore, this information alone is not sufficient to write an actual equation.

Statement 2: Using the information from statement 1 and the points provided in this statement, we can answer the question.

$y-y_{1}=m(x-x_{1})$

The table in a hall has a length of feet. What is its perimeter?

I) The tabletop is exactly two and a half feet above the floor.

II) The area of the table is four times three more than the width.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Both statements are needed to answer the question.

Either statement is sufficient to answer the question.

Explanation

To find perimeter we need length and width. We are given the length.

I) Is irrelevant.

II) We are given a way of relating area and width. Since we know that area is length times width, we can use II to set up an equation where we substitute in the known length along with the given statement to solve for our width

$A=l\cdot w$

$A=4\cdot (w+3)= 14w$

Solve the second one for w and you're good to go!

$\frac{12}{10}=\frac{6}{5}=w$ .

Therefore the perimeter would be:

$P=2l+2w \rightarrow P=2(14)+2\left(\frac{6}{5}\right)$

1. If the arithmetic mean of five different numbers is 50, how many of the numbers are greater than 50?

(1) None of the five numbers is greater than 100.

(2) Three of the five numbers are 24, 25 and 26, respectively.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

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

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

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Explanation

For statement (1), there are different combinations that satisfy the condition. For example, the five numbers can be $48,49,50,51,\ or\ 52,$ or the five numbers can be . Therefore, we cannot determine how many of the numbers are greater than by knowing the first statement.

For statement (2), even though we know three of them, the two unknown numbers can both be greater than , or one smaller and one greater. Thus statement (2) is not sufficient.

Putting the two statements together, we know that the sum of the two unknown numbers is

$50\cdot 5-\left ( 24+25+26 \right )=175$

Since none of them is greater than 100, both of them have to be greater than 50. Therefore when we combine the two statements, we know that there are two numbers that are greater than 50.

What is the height of the right triangle?

The area of the right triangle is .
The base of the right triangle measures .

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

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 to answer the question.

Statements 1 and 2 are not sufficient, and additional data is needed to answer the question.

Explanation

Statement 1:

$A=\frac{1}{2}\times b\times h$

$10=\frac{1}{2} \times b \times h$

$20=b \times h$

More information is required to answer the question because our base and height can be and or and

Statement 2: We're given the base so we can narrow down the information from Statement 1 to and . If the base is , then the height must be .

Both statements taken together are sufficient to answer the question, but neither statement alone is sufficient.

Let and be real numbers.

What is the product of and its complex conjugate?

Statement 1:

Statement 2: $b = \sqrt{2}$

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.

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 insufficient to answer the question.

Explanation

The complex conjugate of an imaginary number is , and

$(a + bi)(a - bi) = a^{2}+b^{2}$ .

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.

You are given two distinct lines, and on the coordinate plane. Are they parallel lines, perpendicular lines, or neither of these?

Statement 1: The absolute value of the slope of Line is 1.

Statement 2: The absolute value of the slope of Line is 1.

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.

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

Assume both statements are true. Then three things are possible:

Case 1: Both lines will have slope 1, or

Case 2: Both lines will have slope

In either case, since the lines have the same slope, they are parallel.

Case 3: One line has slope 1 and one has slope

In this case the lines are perpendicular.

The two statements therefore provide insufficient information.

$\bigtriangleup ABC$ is a right triangle with right angle $\angle B$ . Evaluate .

Statement 1: and $m \angle A = 30^{\circ }$ .

Statement 2: and $m \angle C = 60^{\circ }$ .

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.

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.

Explanation

Either statement alone is sufficient.

From either statement alone, it can be determined that $m \angle A = 30^{\circ }$ and $m \angle C = 60^{\circ }$ ; each statement gives one angle measure, and the other can be calculated by subtracting the first from $90^{\circ }$ , since the acute angles of a right triangle are complementary.

Also, since $\angle B$ is the right angle, $\overline{AC}$ is the hypotenuse, and $\overline{BC}$ , opposite the $30 ^{\circ }$ angle, the shorter leg of a 30-60-90 triangle. From either statement alone, the 30-60-90 Theorem can be used to find the length of longer leg $\overline{AB}$ . From Statement 1 alone, $\overline{AB}$ has length $\frac{\sqrt{3}}{2}$ times that of the hypotenuse, or $\frac{\sqrt{3}}{2} \cdot 22 = 11\sqrt{3}$ . From Statement 2 alone, $\overline{AB}$ has length $\sqrt{3}$ of the shorter leg, or $11\sqrt{3}$ .

You are given two distinct lines, and on the coordinate plane. Are they parallel lines, perpendicular lines, or neither of these?

Statement 1: The absolute value of the slope of Line is 1.

Statement 2: The absolute value of the slope of Line is 1.

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.

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

Assume both statements are true. Then three things are possible:

Case 1: Both lines will have slope 1, or

Case 2: Both lines will have slope

In either case, since the lines have the same slope, they are parallel.

Case 3: One line has slope 1 and one has slope

In this case the lines are perpendicular.

The two statements therefore provide insufficient information.

A bag contains red, yellow and green marbles. There are marbles total.

I) There are green marbles.

II) The number of yellow marbles is half of one less than the number of green marbles.

What are the odds of picking a red followed by a green followed by a yellow? Assume no replacement.

Both statements are needed to answer the question.

Neither statement is sufficient to answer the question. More information is needed.

Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.

Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.

Either statement is sufficient to answer the question.

Explanation

In order to calculate the probability the question asks for, we need to know the number of each color of marble.

I) Gives us the number of greens.

II) Gives us clues which will allow us to find the number of reds and yellows.

We need both statements to answer this question.

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