GMAT Math : Problem-Solving Questions

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Example Questions

Example Question #461 : Algebra

f(x) = |10-x|

g(x) = |x-10|

How many values of make

$(f \circ g )(x) = 0$

a true statement?

Possible Answers:

None

Two

One

Four

Three

Correct answer:

Two

Explanation:

$(f \circ g )(x) = f[g(x)]$ , so we want the number of values of for which

f[g(x)] = 0 .

f(x) = |10-x| , so

f[g(x)] = |10-g(x)|

Therefore, if f[g(x)] = 0 , then

|10-g(x)| = 0

10 - g(x) = 0

g(x) = 10

|x-10| = 10

Either

x - 10 = 10 , in which case x =20 , or

x - 10 = -10 , in which case x = 0 .

The correct choice is therefore two.

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Example Question #469 : Algebra

f(x) = |x+ 6|

g(x) = |x-6|

How many values of make

$(f \circ g )(x) = 3$

a true statement?

Possible Answers:

Three

None

Four

Two

One

Correct answer:

None

Explanation:

$(f \circ g )(x) = f[g(x)]$ , so we want the number of values of for which

f[g(x)] = 3

f(x) = |x+ 6|

f[g(x)] = |g(x)+ 6| = 3 , so either

g(x)+ 6 = -3 or g(x)+ 6 = 3

If the first equation is true, then

g(x)+ 6 = -3

g(x)= -9

and

|x-6| = -9 .

If the second equation is true, then

g(x)+ 6 = 3

g(x)= -3

and

|x-6| = -3 .

In each situation, the absolute value of an expression would be negative; since the absolute value of an expression cannot be negative, no solution is yielded.

There are no values of that make $(f \circ g )(x) = 3$ true; the correct response is zero.

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Example Question #1 : Arithmetic

Which set is NOT equal to the other sets?

Possible Answers:

$\left \{ 1,3,4 \right \}$

$\left \{ 3,4,3,1 \right \}$

$\left \{ 4,3,1,4 \right \}$

$\left \{ 4,1,4,3 \right \}$

$\left \{ 1,3,1,1 \right \}$

Correct answer:

$\left \{ 1,3,1,1 \right \}$

Explanation:

Order and repetition do NOT change a set. Therefore, the set we want to describe contains the numbers 1, 3, and 4. The only set that doesn't contain all 3 of these numbers is $\left \{ 1,3,1,1 \right \}$ , so it is the set that does not equal the rest of the sets.

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Example Question #1 : Understanding Arithmetic Sets

Given the sets A = {2, 3, 4, 5} and B = {3, 5, 7}, what is $A \bigcup B$ ?

Possible Answers:

$\left \{ 2,3,4,5 \right \}$

$\left \{ 2,3,4,5,6,7 \right \}$

$\left \{ 2,3,4,5,7 \right \}$

$\left \{ 3,5,7 \right \}$

$\left \{ 3,5 \right \}$

Correct answer:

$\left \{ 2,3,4,5,7 \right \}$

Explanation:

We are looking for the union of the sets. That means we want the elements of A OR B.

So $A \bigcup B$ = {2, 3, 4, 5, 7}.

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Example Question #2 : Understanding Arithmetic Sets

Given the set = {2, 3, 4, 5}, what is the value of 3+A ?

Possible Answers:

$\left \{ 5 \right \}$

$\left \{ 5,6,7,8 \right \}$

cannot be added together

$\left \{ 12,20,28 \right \}$

$\left \{ 6,9,12,15 \right \}$

Correct answer:

$\left \{ 5,6,7,8 \right \}$

Explanation:

We need to add 3 to every element in .

Then:

$3+A=\left \{ 3+2,3+3,3+4,3+5 \right \}=\left \{ 5,6,7,8 \right \}.$

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Example Question #1 : Sets

There exists two sets and . = {1, 4} and = {3, 4, 6}. What is A+B ?

Possible Answers:

$\left \{ 6,7,9,8,10,12 \right \}$

$\left \{ 1,3,4,4,6 \right \}$

$\left \{ 3,8 \right \}$

$\left \{ 2,5,8 \right \}$

$\left \{ 4,5,7,8,10 \right \}$

Correct answer:

$\left \{ 4,5,7,8,10 \right \}$

Explanation:

Add each element of to each element of .

A+B = {1 + 3, 1 + 4, 1 + 6, 4 + 3, 4 + 4, 4 + 6} = {4, 5, 7, 8, 10}

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Example Question #5 : Arithmetic

How many functions map from $X=\left \{ a,b \right \}$ to $Y=\left \{ 1,2,3 \right \}$ ?

Possible Answers:

Correct answer:

Explanation:

There are three choices for (1, 2, and 3), and similarly there are three choices for (also 1, 2, and 3). Together there are $3\cdot 3=9$ possible functions from to . Remember to multiply, NOT add.

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Example Question #2 : Sets

How many elements are in a set from which exactly 768 unique subsets can be formed?

Possible Answers:

It is not possible to form exactly 768 unique subsets.

Correct answer:

It is not possible to form exactly 768 unique subsets.

Explanation:

The number of subsets that can be formed from a set with elements is $2^{N}$ . However, $2^{9} = 512$ and $2^{10} = 1024$ , so there is no integer for which $2^{N} = 768$ . Therefore, a set with exactly 768 elements cannot exist.

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Example Question #1 : Understanding Arithmetic Sets

Let the univeraal set be the set of all positive integers.

Define the sets

$A = \left \{ 1,6,11,16,21,26,...\right\}$ ,

$B = \left \{1, 5, 9, 13,17,21,25,...\right \}$ ,

$C = \left\{1,4,7,10,13,16,19,...\right \}$ .

If the elements in $A \cap B \cap C$ were ordered in ascending order, what would be the fourth element?

Possible Answers:

421

481

241

181

Correct answer:

181

Explanation:

A,B,C are the sets of all positive integers that are one greater than a multiple of five, four, and three, respectively. Therefore, for a number to be in all three sets, and subsequently, $A \cap B \cap C$ , the number has to be one greater than a number that is a multiple of five, four, and three. Since LCM (5,4,3) = 60 , the number has to be one greater than a multiple of 60. The first four numbers that fit this description are 1, 61, 121, and 181, the last of which is the correct choice.

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Example Question #2 : Arithmetic

A six-sided die is rolled, and a coin is flipped. If the coin comes up heads, the roll is considered to be the number that appears face up on the die; if the coin comes up tails, the outcome is considered to be twice that number. What is the sample space of the experiment?

Possible Answers:

$\left \{ 2,4,6,8,10,12\right \}$

$\left \{ 1,2,3,4,5,6 ,8,10,12\right \}$

$\left \{ 2,3,4,5,6, 7,8,9,10,11,12\right \}$

$\left \{ 1,2,3,4,5,6, 7,8,9,10,11,12\right \}$

$\left \{ 2,3,4,5,6 ,8,10,12\right \}$

Correct answer:

$\left \{ 1,2,3,4,5,6 ,8,10,12\right \}$

Explanation:

If heads comes up on the coin, the number on the die is recorded. This can be any element of the set $\left \{ 1,2,3,4,5,6\right \}$ .

If tails comes up on the coin, twice the number on the die is recorded. This can be twice any element of the set $\left \{ 1,2,3,4,5,6\right \}$ - that is, any element of the set $\left \{ 2,4,6,8,10,12\right \}$ .

The sample space is the union of these two sets:

$\left \{ 1,2,3,4,5,6 ,8,10,12\right \}$

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GMAT Math : Problem-Solving Questions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #461 : Algebra

Example Question #469 : Algebra

Example Question #1 : Arithmetic

Example Question #1 : Understanding Arithmetic Sets

Example Question #2 : Understanding Arithmetic Sets

Example Question #1 : Sets

Example Question #5 : Arithmetic

Example Question #2 : Sets

Example Question #1 : Understanding Arithmetic Sets

Example Question #2 : Arithmetic

All GMAT Math Resources

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