GMAT Math : Algebra

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Example Questions

Example Question #26 : Understanding Functions

Define the operation $\Cap$ as follows:

For real a,b ,

$a \Cap b = 0$ if both and are negative, and $a \Cap b = ab + 1$ otherwise.

Evaluate $\left [8 \Cap 8 \right ]+ \left [ (-8) \Cap 8 \right ]$

Possible Answers:

128

Correct answer:

Explanation:

Evaluate each of $8 \Cap 8$ and $-8 \Cap 8$ separately. Since in both cases, at least one number is positive, replace the numbers for and in the second expression.

$a \Cap b = ab + 1$

$8 \Cap 8 = 8 \cdot 8 + 1 = 64 + 1 = 65$

$a \Cap b = ab + 1$

$(-8) \Cap 8 = -8 \cdot8 + 1 = -64 + 1 = -63$

$\left [ 8 \Cap 8 \right ]+ \left [ (-8) \Cap 8 \right ] = 65 + (-63) = 2$

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Example Question #171 : Algebra

Which of the following would be a valid alternative definition of the function

$f(x) = \left | x + 1 \right | - \left | x - 1\right |$ ?

Possible Answers:

$f(x) = \left\{\begin{matrix} 2& \textrm{for } x \geq 1\\ 2x & \textrm{ for } x<1 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} 2& \textrm{for } x \geq 1\\ 2x & \textrm{ for } -1<x<1\\ -2 & \textrm{for } x \leq -1 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} -2& \textrm{for } x \geq 1\\ -2x & \textrm{ for } -1<x<1\\ 2 & \textrm{for } x \leq -1 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} 2& \textrm{for } x > 0\\ 0 & \textrm{for } x = 0\\ -2& \textrm{ for } x<0 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} -2& \textrm{for } x \geq -1\\ 2x & \textrm{ for } x<-1 \end{matrix}\right.$

Correct answer:

$f(x) = \left\{\begin{matrix} 2& \textrm{for } x \geq 1\\ 2x & \textrm{ for } -1<x<1\\ -2 & \textrm{for } x \leq -1 \end{matrix}\right.$

Explanation:

If $x \geq 1$ , then x+ 1 and x - 1 are both positive, so

$f(x) = \left | x + 1 \right | - \left | x - 1\right |$

= (x + 1) - (x-1)

= x + 1 - x+ 1

= 2

If -1 < x < 1 , then , then x+ 1 is positive and x - 1 is negative, so

$f(x) = \left | x + 1 \right | - \left | x - 1\right |$

$=x+1 - \left [-\left ( x-1\right ) \right ]$

$=x+1 - \left (-x+1\right )$

=x+1 +x - 1

= 2x

If $x \leq -1$ , then x+ 1 and x - 1 are both negative, so

$f(x) = \left | x + 1 \right | - \left | x - 1\right |$

$=-\left ( x + 1 \right )- \left [-\left ( x - 1\right ) \right ]$

$=- x - 1- \left (- x + 1\right )$

=- x - 1 + x - 1

= -2

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Example Question #172 : Algebra

The first two terms of an arithmetic sequence are

8, 24...

What is the tenth term of this sequence?

Possible Answers:

160

152

168

136

144

Correct answer:

152

Explanation:

The common difference of this arithmetic sequence is 24 - 8 = 16 , so the tenth term is

$8 + 16 (10-1) = 8 + 16 \times 9 = 152$

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Example Question #32 : Understanding Functions

The first two terms of a geometric sequence are

8, 20 ,...

What is the sum of the fourth and fifth terms?

Possible Answers:

175

124

100

$1,531 \frac{1}{4}$

$437 \frac{1}{2}$

Correct answer:

$437 \frac{1}{2}$

Explanation:

The common ratio of the sequence is $20 \div 8 = 2 \frac{1}{2 }$

The fourth and fifth terms can be found by repeated multiplication:

$20 \times 2 \frac{1}{2} = 50$

$50 \times 2 \frac{1}{2} = 125$ , the fourth term

$125 \times 2 \frac{1}{2} = 312 \frac{1}{2}$ , the fifth term

Add the fourth and fifth terms:

$125+ 312 \frac{1}{2} = 437 \frac{1}{2}$

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Example Question #1254 : Problem Solving Questions

The first six terms of a sequence are:

15, 16, 20, 29, 45, 70,...

What is the next term?

Possible Answers:

106

110

119

Correct answer:

106

Explanation:

The numbers are generated by addition as follows:

$\begin{matrix} 15 + 1 = 16\\ 16+4=20\\ 20+9 = 29\\ 29+16 = 45\\ 45+25 =70\\ \end{matrix}$

Each entry is generated by adding the next perfect square to the previous entry. Therefore, the next entry will be generated as follows:

70 + 36 = 106

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Example Question #173 : Algebra

The first six terms of a sequence are:

25, 27, 31, 37, 45, 55,...

What is the eight term?

Possible Answers:

Correct answer:

Explanation:

The terms are generated by adding an increment that increases by 2 with every term:

$\begin{matrix} 25+ 2 = 27\\ 27 + 4 = 31\\ 31+6=37\\ 37+8=45\\ 45+10=55\\ 55+12= 67\\ 67+14 = 81 \end{matrix}$

81 is the eighth term.

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Example Question #1256 : Problem Solving Questions

Define an operation $\Omega$ on the set of real numbers as follows:

For all real numbers a,b , $a \Omega b = 0$ if and only if a < 0 and b < 0 , and $a \Omega b = a b$ otherwise.

Evaluate:

$6 \Omega 6 + \left ( -6 \right )\Omega \left ( -6 \right )$

Possible Answers:

Correct answer:

Explanation:

$6 \Omega 6$ can be evaluated by using the second part of the definition, since both numbers are positive:

$6 \Omega 6 = 6 \times 6 = 36$

$\left ( -6 \right )\Omega \left ( -6 \right )$ can be evaluated by using the first part of the definition, since both numbers are negative:

$\left ( -6 \right )\Omega \left ( -6 \right ) = 0$

$6 \Omega 6 + \left ( -6 \right )\Omega \left ( -6 \right ) = 36 + 0 = 36$

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Example Question #34 : Functions/Series

Define an operation $\triangleright$ on two real numbers as follows:

For all real numbers a,b ,

$a \triangleright b = \frac{a}{b}$ if and only if $b \neq 0$ , and $a \triangleright b = a ^{2}$ otherwise.

Evaluate:

$\left (5 \triangleright 2 \right )-\left ( 2 \triangleright 5 \right )$

Possible Answers:

2.1

- 2.1

Correct answer:

2.1

Explanation:

We can evaluate both $5 \triangleright 2 \right$ and $2 \triangleright 5$ using the first part of the definition, since the second number is nonzero in both cases.

$a \triangleright b = \frac{a}{b}$

$5\triangleright 2 = \frac{5}{2} = 2.5$

$2\triangleright 5 = \frac{2}{5} = 0.4$

$\left (5 \triangleright 2 \right )-\left ( 2 \triangleright 5 \right ) = 2.5 - 0.4 = 2.1$

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Example Question #35 : Functions/Series

Define an operation $\bullet$ on the real numbers as follows:

$a \bullet b = a ^{2} (b- 10)$ for all real a,b .

Evaluate:

$10 \bullet 5$

Possible Answers:

500

Correct answer:

Explanation:

$a \bullet b = a ^{2} (b- 10)$

$10 \bullet 5 = 10 ^{2} (5- 10) = 100 \cdot (-5) = -500$

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Example Question #174 : Algebra

Define f (x) = 5x + 9 .

What is $f ^{-1}$ ?

Possible Answers:

$f ^{-1} (x)= \frac{1}{5}x - \frac{9}{5}$

$f ^{-1} (x)= \frac{1}{5}x - 9$

$f ^{-1} (x)=-5x -9$

$f ^{-1} (x)=- \frac{1}{5}x - \frac{9}{5}$

$f ^{-1} (x)=- \frac{1}{5}x + \frac{9}{5}$

Correct answer:

$f ^{-1} (x)= \frac{1}{5}x - \frac{9}{5}$

Explanation:

Replace f(x) with .

y = 5x + 9

Now exchange and and solve for in the new expression:

x = 5y + 9

x - 9 = 5y + 9 - 9

x - 9 = 5y

$\left (x - 9 \right ) \div 5 = 5y \div 5$

$\frac{1}{5}x - \frac{9}{5} = y$

$f ^{-1} (x)= \frac{1}{5}x - \frac{9}{5}$

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #26 : Understanding Functions

Example Question #171 : Algebra

Example Question #172 : Algebra

Example Question #32 : Understanding Functions

Example Question #1254 : Problem Solving Questions

Example Question #173 : Algebra

Example Question #1256 : Problem Solving Questions

Example Question #34 : Functions/Series

Example Question #35 : Functions/Series

Example Question #174 : Algebra

All GMAT Math Resources

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