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GMAT Quantitative › Algebra

Questions 1 - 10

What is the value of $\dpi{100} \small 2x+2y$ ?

Statement 1: $\dpi{100} \small x-3y=4$

Statement 2: $\dpi{100} \small x+y=4$

Statement 2 ALONE is sufficient, but statement 1 is not sufficient.

Statement 1 ALONE is sufficient, but statement 2 is not sufficient.

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

EACH statement ALONE is sufficient.

Statements 1 and 2 TOGETHER are NOT sufficient.

Explanation

We know that we need 2 equations to solve for 2 variables, so it is tempting to say that both statements are needed. This is actually wrong! We aren't being asked for the individual values of x and y, instead we are being asked for the value of an expression.

$\dpi{100} \small 2x+2y$ is just $\dpi{100} \small 2\left ( x+y \right )$ , and statement 2 gives us the value of $\dpi{100} \small x+y$ . For data sufficiency questions, we don't actually have to solve the question, but if we wanted to, we would simply multiply statement 2 by 2.

$\dpi{100} \small 2x+2y=2\left ( x+y \right )=2$ * Statement 2 = $\dpi{100} \small 2\times 4=8$

Consider the equation $x^{2} +Bx = C$

How many real solutions does this equation have?

Statement 1: There exists two different real numbers such that and

Statement 2: is a positive integer.

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.

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

$x^{2} +Bx = C$ can be rewritten as $x^{2} +Bx - C = 0$

If Statement 1 holds, then the equation can be rewritten as . This equation has solution set $\left { -m,-n\right }$ , which comprises two real numbers.

If Statement 2 holds, the discriminant $B^{2} -4 \cdot1\cdot(-C)= B^{2} +4C$ is positive, being the sum of a nonnegative number and a positive number; this makes the solution set one with two real numbers.

Define .

Evaluate $(f \circ g) (9)$ .

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.

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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation

If you know that , then you can calculate:

$(f \circ g) (9) = f (; g(9); ) = f(8) = 4\cdot8 -9 = 23$

Knowing the -values of that are paired with -value 9, however, is neither necessary nor useful here.

Let be two positive integers. How many real solutions does the equation $x^{2} + Bx + C = 0$ have?

Statement 1: is a perfect square of an integer.

Statement 2: $C = \left ( \frac{B}{2} \right ) ^{2}$

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.

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 number of real solutions of the equation $Ax^{2} + Bx + C = 0$ depends on whether discriminant $B^{2} - 4A C$ is positive, zero, or negative; since , this becomes $B^{2} - 4C$ .

If we only know that is a perfect square, then we still need to know to find the number of real solutions. For example, let , a perfect square. Then the discriminant is $B^{2} - 4$ , which can be positive, zero, or negative depending on .

But if we know $C = \left ( \frac{B}{2} \right ) ^{2}$ , then the discriminant is

$B^{2} - 4C = B^{2} - 4 \cdot \left ( \frac{B}{2} \right ) ^{2} = B^{2} - 4 \cdot \frac{B^{2}}{4} = B^{2} - B^{2} = 0$

Therefore, $x^{2} + Bx + C = 0$ has one real solution.

This relation has five different ordered pairs: is it a function?

$\begin{matrix} \underline{x}&\underline{y}\ 1& A\ 2& A\ 3& A\ B&A + 1 \ 5& A + 2 \end{matrix}$

Statement 1:

Statement 2:

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.

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.

Explanation

To prove that a relation is a function, you must prove that no -coordinate is matched with more than one -coordinate. Statement 2 proves that this is false, since 5 is now matched with both and , which are different numbers regardless of . Statement 1 is irrelevant, since it does not prove or disprove this condition.

Solve for .

Statement 1:

Statement 2:

Statements 1 and 2 TOGETHER are NOT sufficient.

Statement 1 ALONE is sufficient, but statement 2 is not sufficient.

Statement 2 ALONE is sufficient, but statement 1 is not sufficient.

EACH statement ALONE is sufficient.

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

Explanation

To solve for three unknowns, we need three equations. Therefore no combination of statements 1 and 2 will provide enough information to solve for $x,y,\ and\ z$ .

Data sufficiency question- do not actually solve the question

Is $\small xy< 12$ ?

1. $\small x^2=64; 1\leq y\leq 1.5$

2. $\small x+y=6$

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

Statement 1 alone is sufficient, but statement 2 along is not sufficient to answer the question

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

Each statement alone is sufficient

Statements 1 and 2 together are not sufficient, and additional information is needed to answer the question

Explanation

From statement 1, we can conclude that $\small xy\leq 12$ but not $\small xy< 12$ . From the second statement, we can conclude that the greatest product will result from $\small 3+3=6$ or 9, which is less than 12.

is an integer. Is there a real number such that $x^{A}=B$ ?

Statement 1: is negative

Statement 2: is even

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.

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 insufficient to answer the question.

Explanation

The equivalent question is "does have a real $A \textrm{th }$ root?"

If you know only that is negative, you need to know whether is even or odd; negative numbers have real odd-numbered roots, but not real even-numbered roots.

If you know only that is even, you need to know whether is negative or nonnegative; negative numbers do not have real even-numbered roots, but nonnegative numbers do.

If you know both, however, then you know that the answer is no, since as stated before, negative numbers do not have real even-numbered roots.

Therefore, the answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.

What is the first term of the geometric sequence?

Statement 1: The sum of the second and third terms is 90.

Statement 2: The sum of the third and fourth terms is 450.

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 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

Let be the first term and be the common ratio. Then the first four terms are:

$A,AR,AR^{2},AR^{3}$ .

The two statements below are equivalent to $AR+ AR^{2} = 90$ and $AR^{2}+AR^{3} = 450$ , respectively. Neither, alone, will help you figure out or . If you know both, you can use algebra to deduce their values: