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SAT II Math II : Number Theory

Study concepts, example questions & explanations for SAT II Math II

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Example Questions

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Example Question #3 : Real And Complex Numbers

Which of the following is equal to $i^{37}$ ?

Possible Answers:

37i

-37i

Correct answer:

Explanation:

To raise to a power, divide the exponent by 4 and note the remainder.

$37 \div 4 = 9 \textrm{ R }1$

Raise to the power of that remainder:

$i^{37} = i^{1} = i$

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Example Question #4 : Real And Complex Numbers

Evaluate:

$(6 +5i)^{2}$

Possible Answers:

11+60i

61-60i

61+60i

Correct answer:

11+60i

Explanation:

Use the square of a sum pattern

$\left ( A+B\right ) = A^{2}+ 2AB + B^{2}$

where A = 6, B = 5i :

$(6 +5i)^{2} = 6^{2} + 2 \cdot 6i \cdot 5 + (5i)^{2}$

$= 6^{2} + 2 \cdot 6 \cdot 5i + 5^{2}i^{2}$

= 36+ 60i + 25 (-1)

= 36+ 60i -25

= 11+60i

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Example Question #11 : Number Theory

Multiply:

7i (5 + 9i)

Possible Answers:

-63- 35i

-63+ 35i

-35 + 63i

35 - 63i

63- 35i

Correct answer:

-63+ 35i

Explanation:

Apply the distributive property:

7i (5 + 9i)

$= 7i \cdot 5 + 7i \cdot 9i$

$= 35i + 63 i^{2}$

= 35i + 63 (-1)

= -63+ 35i

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Example Question #1 : Real And Complex Numbers

Which of the following is equal to $\left ( 4i\right )^{4}$ ?

Possible Answers:

16i

256

256i

Correct answer:

256

Explanation:

By the power of a product property,

$\left ( 4i\right )^{4} = 4^{4} \cdot i^{4} = 256 \cdot 1 = 256$

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Example Question #1 : Real And Complex Numbers

Multiply:

(6 - 7i)(7+6i)

Possible Answers:

84 +13i

-13i

84 - 13i

-84 + 13i

Correct answer:

84 - 13i

Explanation:

Use the FOIL method:

(6 - 7i)(7+6i)

$= 6 \cdot 7 +6 \cdot 6i - 7 \cdot 7i - 7i \cdot 6i$

$=42 +36i - 49i - 42i ^{2}$

=42 +36i - 49i - 42 (-1)

=42 +36i - 49i + 42

=84 - 13i

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Example Question #11 : Sat Subject Test In Math Ii

Which of the following is equal to $i^{77}$ ?

Possible Answers:

77i

-77i

Correct answer:

Explanation:

To raise to a power, divide the exponent by 4 and note the remainder.

$77 \div 4 = 19\textrm{ R }1$

Raise to the power of that remainder:

$i^{77} = i^{1} = i$

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Example Question #1 : Real And Complex Numbers

Multiply:

8i (5-7i)

Possible Answers:

-56+ 40i

40 + 56i

56- 40i

-40 + 56i

56+ 40i

Correct answer:

56+ 40i

Explanation:

Apply the distributive property:

8i (5-7i)

$= 8i \cdot 5- 8i \cdot 7i$

$= 40i- 56i^{2}$

= 40i- 56 (-1)

= 40i- (-56)

= 56+ 40i

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Example Question #2 : Real And Complex Numbers

Multiply:

$\left ( -4i\right ) (-6i)(-7i)$

Possible Answers:

None of the other responses is correct.

168i

168

-168i

Correct answer:

168i

Explanation:

$\left ( -4i\right ) (-6i)(-7i)$

$= \left ( -4\right ) (-6)(-7) \cdot i \cdot i \cdot i$

$= -168 \cdot i ^{3} = -168 (-i) = 168i$

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Example Question #11 : Real And Complex Numbers

Evaluate:

$\left ( 7+3i\right )^{2}$

Possible Answers:

40 - 42i

58 + 42i

40 + 42i

Correct answer:

40 + 42i

Explanation:

Use the square of a sum pattern

$\left ( A+B\right ) = A^{2}+ 2AB + B^{2}$

where A = 7, B = 3i :

$\left ( 7+3i\right )^{2} = 7^{2}+ 2 \cdot 7 \cdot 3i + (3i)^{2}$

$= 7^{2}+ 2 \cdot 7 \cdot 3i + 3^{2}i^{2}$

= 49 + 42i + 9 (-1)

= 49 + 42i -9

= 40 + 42i

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Example Question #12 : Number Theory

is a complex number; $\overline{x}$ denotes the complex conjugate of .

$x \cdot \overline{x} = 84$

Which of the following could be the value of ?

Possible Answers:

Any of the numbers in the other four choices could be equal to .

$8 + 2 i \sqrt{5}$

$6 - 4 i \sqrt{3}$

$7 + i \sqrt{35}$

$9 - i \sqrt{3}$

Correct answer:

Any of the numbers in the other four choices could be equal to .

Explanation:

The product of a complex number x = a+ bi and its complex conjugate $\overline{x} = a- bi$ is

$x \cdot \overline{x} = (a+ bi)(a-bi) = a^{2} + b^{2}$

Setting and accordingly for each of the four choices, we want to find the choice for which $x \cdot \overline{x} = a^{2} + b^{2} = 84$ :

$6 - 4 i \sqrt{3}$

$a = 6, b = - 4 \sqrt{3}$

$x \cdot \overline{x} = a^{2} + b^{2} = 6^{2}+ (-4 \sqrt{3})^{2} = 36 + 4 ^{2} (\sqrt{3} )^{2} = 36 + 16 \cdot 3 = 36 + 48 = 84$

$7 + i \sqrt{35}$

$a = 7, b = \sqrt{35}$

$x \cdot \overline{x} = a^{2} + b^{2} = 7^{2} + (\sqrt{35})^{2} = 49 + 35 = 84$

$8 + 2 i \sqrt{5}$

$a = 8 , b = 2 \sqrt{5}$

$x \cdot \overline{x} = a^{2} + b^{2} = 8^{2} + (2 \sqrt{5})^{2} = 64 + 2^{2} \cdot (\sqrt{5})^{2} = 64 + 4 \cdot 5 = 64 + 20 = 84$

$9 - i \sqrt{3}$

$a = 9 , b = - \sqrt{3}$

$x \cdot \overline{x} = a^{2} + b^{2} = 9^{2} + (- \sqrt{3})^{2} = 81 + 3 = 84$

For each given value of , $x \cdot \overline{x} = 84$ .

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SAT II Math II : Number Theory

Study concepts, example questions & explanations for SAT II Math II

All SAT II Math II Resources

Example Questions

Example Question #3 : Real And Complex Numbers

Example Question #4 : Real And Complex Numbers

Example Question #11 : Number Theory

Example Question #1 : Real And Complex Numbers

Example Question #1 : Real And Complex Numbers

Example Question #11 : Sat Subject Test In Math Ii

Example Question #1 : Real And Complex Numbers

Example Question #2 : Real And Complex Numbers

Example Question #11 : Real And Complex Numbers

Example Question #12 : Number Theory

All SAT II Math II Resources

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