Algebra - GMAT Quantitative

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Question

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.

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Answer

$x^{2} +Bx = C$ 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.

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Question

True or false: $\left \{ a_{n} \right \}$ . is an arithmetic sequence.

Statement 1: $a_{4} - a_{3} = a_{2} -a_{1}$

Statement 2: $a_{6}- a_{5} = a_{8} - a_{7}$

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Answer

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

The two statements together demonstrate that two such differences are equal to each other, and that two other such differences are equal to each other. No information, however, is given about any other differences, of which there are infinitely many. Therefore, the question of whether the sequence is arithmetic or not is unresolved.

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Question

Define $f (x) = \left\{\begin{matrix} 1\textrm{ if } x< 0\ x \textrm{ if } x \geq 0 \end{matrix}\right.$

True or false: .

Statement 1: is a positive number

Statement 2: is a negative number

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Answer

Assume both statements. Then and . But this does not answer the question. For example,

If , then

making true.

But if , then

making false.

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Question

A relation comprises ten ordered pairs. Is it a function?

Statement 1: The domain of the relation is $\left \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \right \}$ .

Statement 2: The range of the relation is $\left \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \right \}$ .

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Answer

The relation comprises ten ordered pairs. If Statement 1 alone is known, then the domain comprises ten elements, each of which must appear in exactly one ordered pair. Therefore, no domain element is matched with more than one range element, and the relation is a function.

If Statement 2 alone is known, then the range comprises ten elements, each of which must appear in exactly one ordered pair. But nothing is known about the domain. If no domain element is repeated among the ordered pairs, the relation is a function; otherwise, the relation is not a function.

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Question

True or false: $\left \{ a_{n} \right \}$ , is an arithmetic sequence.

Statement 1: $a_{1}+ a_{4} \ne a_{2} + a_{3}$

Statement 2: $a_{6}- a_{5} = a_{8} - a_{7}$

Tap to reveal answer

Answer

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

Assume Statement 1 alone.

$a_{1}+ a_{4} \ne a_{2} + a_{3}$

$a_{1}+ a_{4} - a_{1} - a_{3} \ne a_{2} + a_{3} - a_{1} - a_{3}$

$a_{4} - a_{3} \ne a_{2} - a_{1}$ ,

meaning that at least two such differences are unequal, and proving the sequence is not arithmetic.

Statement 2 alone only proves that two such differences are equal, but says nothing about any of the other (infinitely many) such differences. Therefore, it leaves the question unresolved.

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Question

True or false: $\left \{ a_{n} \right \}$ . is an arithmetic sequence.

Statement 1: $a_{4} - a_{3} = a_{2} -a_{1}$

Statement 2: $a_{6}- a_{5} = a_{8} - a_{7}$

Tap to reveal answer

Answer

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

The two statements together demonstrate that two such differences are equal to each other, and that two other such differences are equal to each other. No information, however, is given about any other differences, of which there are infinitely many. Therefore, the question of whether the sequence is arithmetic or not is unresolved.

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Question

Define $f (x) = \left\{\begin{matrix} 1\textrm{ if } x< 0\ x \textrm{ if } x \geq 0 \end{matrix}\right.$

True or false: .

Statement 1: is a positive number

Statement 2: is a negative number

Tap to reveal answer

Answer

Assume both statements. Then and . But this does not answer the question. For example,

If , then

making true.

But if , then

making false.

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Question

Define $f (x) = \left\{\begin{matrix} 0\textrm{ if } x< 0\ x \textrm{ if } x \geq 0 \end{matrix}\right.$

Is greater than, less than, or equal to ?

Statement 1: is a positive number.

Statement 2: is a negative number.

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Answer

One must have information about both and to answer the question, so neither statement is sufficient by itself.

Now assume both statements to be true. Then, being positive and being negative, .

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Question

A relation comprises ten ordered pairs. Is it a function?

Statement 1: Its domain is $\left \{ 1,3,5, 7, 9, 11\right \}$ .

Statement 2: The line passes through its graph twice.

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Answer

If Statement 1 alone is assumed, then, since there are only six domain elements and ten points in the relation, at least one of the domain elements must match with more than one range element. This forces the relation to not be a function.

If Statement 2 alone is assumed, then, since is a vertical line that passes through the graph twice, the relation fails the vertical line test and is therefore not a function.

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Question

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x^{4}, x^{5}, x^{6}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right | > 1$

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Answer

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

Case 1: .

Then

$x^{4} = (-2)^{4} = 16$

$x^{5} = (-2)^{5} = -32$

$x^{6} = (-2)^{6} = 64$

$x^{6}$ is the greatest of these values.

Case 2: $x = -\frac{1}{2}$

Then

$x ^{4}=\left ( -\frac{1}{2} \right )^{4} = \frac{1}{16}$

$x ^{5}=\left ( -\frac{1}{2} \right )^{5} = -\frac{1}{32}$

$x ^{6}=\left ( -\frac{1}{2} \right )^{6} = \frac{1}{64}$

$x^{4}$ is the greatest of these values.

Now assume Statement 2 alone. Either or .

Case 1: .

Then $x \cdot x^{4} > 1\cdot x^{4}$ , so $x^{5} > x^{4}$ ; similarly, $x^{6} > x^{5}$ .

$x^{6}$ is the greatest of the three.

Case 2: .

Odd power $x^{5}$ is negative, and even powers $x^{4}$ and $x^{6}$ are positive, so one of the latter two is the greatest. Since , it follows that $x^{2} > 1$ . It then follows that $x^{2} \cdot x^{4} > 1 \cdot x^{4}$ , or $x^{6} > x^{4}$ .

Again, $x^{6}$ is the greatest of the three.

Statement 2 alone is sufficient, but not Statement 1.

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Question

Define $f (x) = \left\{\begin{matrix} 0\textrm{ if } x< 0\ x \textrm{ if } x \geq 0 \end{matrix}\right.$

Is greater than, less than, or equal to ?

Statement 1: is a positive number.

Statement 2: is a negative number.

Tap to reveal answer

Answer

One must have information about both and to answer the question, so neither statement is sufficient by itself.

Now assume both statements to be true. Then, being positive and being negative, .

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Question

True or false: $\left \{ a_{n} \right \}$ , is an arithmetic sequence.

Statement 1: $a_{1}+ a_{4} \ne a_{2} + a_{3}$

Statement 2: $a_{6}- a_{5} = a_{8} - a_{7}$

Tap to reveal answer

Answer

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

Assume Statement 1 alone.

$a_{1}+ a_{4} \ne a_{2} + a_{3}$

$a_{1}+ a_{4} - a_{1} - a_{3} \ne a_{2} + a_{3} - a_{1} - a_{3}$

$a_{4} - a_{3} \ne a_{2} - a_{1}$ ,

meaning that at least two such differences are unequal, and proving the sequence is not arithmetic.

Statement 2 alone only proves that two such differences are equal, but says nothing about any of the other (infinitely many) such differences. Therefore, it leaves the question unresolved.

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Question

True or false: $\left \{ a_{n} \right \}$ . is an arithmetic sequence.

Statement 1: $a_{4} - a_{3} = a_{2} -a_{1}$

Statement 2: $a_{6}- a_{5} = a_{8} - a_{7}$

Tap to reveal answer

Answer

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

The two statements together demonstrate that two such differences are equal to each other, and that two other such differences are equal to each other. No information, however, is given about any other differences, of which there are infinitely many. Therefore, the question of whether the sequence is arithmetic or not is unresolved.

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Question

Define $f (x) = \left\{\begin{matrix} 1\textrm{ if } x< 0\ x \textrm{ if } x \geq 0 \end{matrix}\right.$

True or false: .

Statement 1: is a positive number

Statement 2: is a negative number

Tap to reveal answer

Answer

Assume both statements. Then and . But this does not answer the question. For example,

If , then

making true.

But if , then

making false.

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Question

Define $f (x) = \left\{\begin{matrix} 0\textrm{ if } x< 0\ x \textrm{ if } x \geq 0 \end{matrix}\right.$

Is greater than, less than, or equal to ?

Statement 1: is a positive number.

Statement 2: is a negative number.

Tap to reveal answer

Answer

One must have information about both and to answer the question, so neither statement is sufficient by itself.

Now assume both statements to be true. Then, being positive and being negative, .

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Question

A relation comprises ten ordered pairs. Is it a function?

Statement 1: Its domain is $\left \{ 1,3,5, 7, 9, 11\right \}$ .

Statement 2: The line passes through its graph twice.

Tap to reveal answer

Answer

If Statement 1 alone is assumed, then, since there are only six domain elements and ten points in the relation, at least one of the domain elements must match with more than one range element. This forces the relation to not be a function.

If Statement 2 alone is assumed, then, since is a vertical line that passes through the graph twice, the relation fails the vertical line test and is therefore not a function.

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Question

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x^{4}, x^{5}, x^{6}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right | > 1$

Tap to reveal answer

Answer

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

Case 1: .

Then

$x^{4} = (-2)^{4} = 16$

$x^{5} = (-2)^{5} = -32$

$x^{6} = (-2)^{6} = 64$

$x^{6}$ is the greatest of these values.

Case 2: $x = -\frac{1}{2}$

Then

$x ^{4}=\left ( -\frac{1}{2} \right )^{4} = \frac{1}{16}$

$x ^{5}=\left ( -\frac{1}{2} \right )^{5} = -\frac{1}{32}$

$x ^{6}=\left ( -\frac{1}{2} \right )^{6} = \frac{1}{64}$

$x^{4}$ is the greatest of these values.

Now assume Statement 2 alone. Either or .

Case 1: .

Then $x \cdot x^{4} > 1\cdot x^{4}$ , so $x^{5} > x^{4}$ ; similarly, $x^{6} > x^{5}$ .

$x^{6}$ is the greatest of the three.

Case 2: .

Odd power $x^{5}$ is negative, and even powers $x^{4}$ and $x^{6}$ are positive, so one of the latter two is the greatest. Since , it follows that $x^{2} > 1$ . It then follows that $x^{2} \cdot x^{4} > 1 \cdot x^{4}$ , or $x^{6} > x^{4}$ .

Again, $x^{6}$ is the greatest of the three.

Statement 2 alone is sufficient, but not Statement 1.

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Question

Define $f (x) = \left\{\begin{matrix} 0\textrm{ if } x< 0\ x \textrm{ if } x \geq 0 \end{matrix}\right.$

Is greater than, less than, or equal to ?

Statement 1: is a positive number.

Statement 2: is a negative number.

Tap to reveal answer

Answer

One must have information about both and to answer the question, so neither statement is sufficient by itself.

Now assume both statements to be true. Then, being positive and being negative, .

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Question

True or false: $\left \{ a_{n} \right \}$ , is an arithmetic sequence.

Statement 1: $a_{1}+ a_{4} \ne a_{2} + a_{3}$

Statement 2: $a_{6}- a_{5} = a_{8} - a_{7}$

Tap to reveal answer

Answer

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

Assume Statement 1 alone.

$a_{1}+ a_{4} \ne a_{2} + a_{3}$

$a_{1}+ a_{4} - a_{1} - a_{3} \ne a_{2} + a_{3} - a_{1} - a_{3}$

$a_{4} - a_{3} \ne a_{2} - a_{1}$ ,

meaning that at least two such differences are unequal, and proving the sequence is not arithmetic.

Statement 2 alone only proves that two such differences are equal, but says nothing about any of the other (infinitely many) such differences. Therefore, it leaves the question unresolved.

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Question

True or false: $\left \{ a_{n} \right \}$ . is an arithmetic sequence.

Statement 1: $a_{4} - a_{3} = a_{2} -a_{1}$

Statement 2: $a_{6}- a_{5} = a_{8} - a_{7}$

Tap to reveal answer

Answer

An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.

The two statements together demonstrate that two such differences are equal to each other, and that two other such differences are equal to each other. No information, however, is given about any other differences, of which there are infinitely many. Therefore, the question of whether the sequence is arithmetic or not is unresolved.

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