GMAT Math : Problem-Solving Questions

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Example Questions

Example Question #3 : Functions/Series

A sequence begins as follows:

8, 12, ...

It is formed the same way that the Fibonacci sequence is formed. What are the next two numbers in the sequence?

Possible Answers:

18, 27

20, 35

20, 32

8, 4

16, 20

Correct answer:

20, 32

Explanation:

Each term of the Fibonacci sequence is formed by adding the previous two terms. Therefore, do the same to form this sequence:

8 + 12= 20

12+20=32

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Example Question #1 : Understanding Functions

Give the inverse of f (x) = 5x - 7

Possible Answers:

$f^{-1}(x)= \frac{1}{5}x - \frac{7}{5}$

$f^{-1}(x)=- \frac{1}{5}x + \frac{7}{5}$

$f^{-1}(x)= -\frac{1}{5}x +7$

$f^{-1}(x)= \frac{1}{5}x + \frac{7}{5}$

$f^{-1}(x)= \frac{1}{5}x +7$

Correct answer:

$f^{-1}(x)= \frac{1}{5}x + \frac{7}{5}$

Explanation:

The easiest way to find the inverse of f (x) is to replace f (x) in the definition with , switch with , and solve for in the new equation.

y = 5x - 7

x = 5y - 7

x + 7 = 5y - 7 + 7

x + 7 = 5y

$\frac{1}{5} \cdot\left ( x + 7 \right ) = \frac{1}{5} \cdot 5y$

$\frac{1}{5} x + \frac{7}{5} = y$

$f^{-1}(x)= \frac{1}{5}x + \frac{7}{5}$

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Example Question #2 : Understanding Functions

Define $f (x) = x^{3} - 8$ . Give $f^{-1} (x)$

Possible Answers:

$f^{-1}(x) = 2- \sqrt[3]{x }$

$f^{-1}(x) = \sqrt[3]{x +2}$

$f^{-1}(x) = 8+ \sqrt[3]{x }$

$f^{-1}(x) = \sqrt[3]{x +8}$

$f^{-1}(x) = 2+ \sqrt[3]{x }$

Correct answer:

$f^{-1}(x) = \sqrt[3]{x +8}$

Explanation:

The easiest way to find the inverse of f (x) is to replace f (x) in the definition with , switch with , and solve for in the new equation.

$y = x^{3} - 8$

$x= y^{3} - 8$

$x +8= y^{3} - 8 +8$

$x +8= y^{3}$

$\sqrt[3]{x +8}= \sqrt[3]{y^{3}}$

$y = \sqrt[3]{x +8}$

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Example Question #5 : Functions/Series

Define $f(x) = x^{2}- 4$ and $g(x) = x^{2} + 4$ .

Give the definition of $(f \circ g)(x)$ .

Possible Answers:

$(f \circ g)(x) = x^{4}+8x^{2}+12$

$(f \circ g)(x) = x^{4} -16$

$(f \circ g)(x) = x^{4} - 8x^{2} +20$

$(f \circ g)(x) = x^{4}+8x^{2}-12$

$(f \circ g)(x) = x^{4}$

Correct answer:

$(f \circ g)(x) = x^{4}+8x^{2}+12$

Explanation:

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

$= f(x^{2}+4)$

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

$=x^{4} + 8x^{2}+16-4$

$=x^{4} + 8x^{2}+12$

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Example Question #2 : Understanding Functions

Define g (x) = 5x - 2 .

If g (A) = 11 , evaluate .

Possible Answers:

A=2.6

A=57

A=53

A=1.8

A=4.2

Correct answer:

A=2.6

Explanation:

Solve for in this equation:

g (A) = 5A - 2 = 11

5A - 2 + 2 = 11+ 2

5A = 13

$5A \div 5 = 13\div 5$

A = 2.6

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Example Question #11 : Functions/Series

Define the operation $\Theta$ as follows:

$a \; \Theta \; b = ab + 100$

Solve for : $4\; \Theta \; x = 72$

Possible Answers:

x = -9

x = -18

x = 18

x = -7

x = 9

Correct answer:

x = -7

Explanation:

$4\; \Theta \; x = 72$

4x + 100 = 72

4x + 100 -100 = 72-100

4x = -28

$4x \div 4 = -28 \div 4$

x = -7

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Example Question #11 : Functions/Series

Define $f(x) = A (x-B)^{3}$ , where $A \neq 0, B \neq 0$ .

Evaluate $f^{-1} (1)$ in terms of and .

Possible Answers:

$f^{-1} (1)= B + \frac{1}{\sqrt[3]{A}}$

$f^{-1} (1)= -B + \frac{1}{\sqrt[3]{A}}$

$f^{-1} (1)= B + \sqrt[3]{A}$

$f^{-1} (1)= B - \frac{1}{\sqrt[3]{A}}$

$f^{-1} (1)= B - \sqrt[3]{A}$

Correct answer:

$f^{-1} (1)= B + \frac{1}{\sqrt[3]{A}}$

Explanation:

This is equivalent to asking for the value of for which f(x) = 1 , so we solve for in the following equation:

f(x) = 1

$A (x-B)^{3} =1$

$\frac{A (x-B)^{3}}{A} =\frac{1}{A}$

$(x-B)^{3} =\frac{1}{A}$

$\sqrt[3]{(x-B)^{3} }=\sqrt[3]{\frac{1}{A}}$

$x-B =\sqrt[3]{\frac{1}{A}}$

$x-B = \frac{1}{\sqrt[3]{A}}$

$x-B + B = B + \frac{1}{\sqrt[3]{A}}$

$x = B + \frac{1}{\sqrt[3]{A}}$

Therefore, $f^{-1} (1)= B + \frac{1}{\sqrt[3]{A}}$ .

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Example Question #13 : Functions/Series

Define an operation $\odot$ as follows:

For any real numbers a,b ,

$a \odot b = (a-1) (b-1)$

Evaluate $\left (5 \odot 5 \right )\odot 5$ .

Possible Answers:

100

Correct answer:

Explanation:

$\left (5 \odot 5 \right )\odot 5$

$= \left [(5 -1) (5 -1) \right ] \odot 5$

$= (4 \cdot 4 ) \odot 5$

$= 16 \odot 5$

$= \left (16-1 \right ) \left (5-1 \right )$

$= 15 \cdot4 = 60$

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Example Question #14 : Functions/Series

An infinite sequence begins as follows:

1, 2, -3, 4, 5, -6, 7,8,-9 ...

Assuming this pattern continues infinitely, what is the sum of the 1000th, 1001st and 1002nd terms?

Possible Answers:

999

1,001

1,000

Correct answer:

999

Explanation:

This can be seen as a sequence in which the Nth term is equal to if is not divisible by 3, and otherwise. Since 1,000 and 1,001 are not multiples of 3, but 1,002 is, the 1000th, 1001st, and 1002nd terms are, respectively,

1000,1001,-1002

and their sum is

$1000+ 1001+ \left (-1002 \right ) = 999$

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Example Question #15 : Functions/Series

Define $f(x) = \sqrt{x}$ . What is $\left (f \circ f \right )(x)$ ?

Possible Answers:

$\left (f \circ f \right )(x) = \sqrt[3]{x}$

$\left (f \circ f \right )(x) = 2\sqrt{x}$

$\left (f \circ f \right )(x) = x$

$\left (f \circ f \right )(x) = \sqrt[4]{x}$

$\left (f \circ f \right )(x) = x^{2}$

Correct answer:

$\left (f \circ f \right )(x) = \sqrt[4]{x}$

Explanation:

This can best be solved by rewriting $\sqrt{ x}$ as $x^{\frac{1}{2}}$ and using the power of a power property.

$\left (f \circ f \right )(x) = f (f(x)) = \left [f(x) \right]^{\frac{1}{2}} =\left ( x^{\frac{1}{2}} \right )^{\frac{1}{2}}=x^{\frac{1}{2}\cdot \frac{1}{2} } =x^{\frac{1}{4} } =\sqrt[4]{x}$

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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 #3 : Functions/Series

Example Question #1 : Understanding Functions

Example Question #2 : Understanding Functions

Example Question #5 : Functions/Series

Example Question #2 : Understanding Functions

Example Question #11 : Functions/Series

Example Question #11 : Functions/Series

Example Question #13 : Functions/Series

Example Question #14 : Functions/Series

Example Question #15 : Functions/Series

All GMAT Math Resources

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