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SAT II Math II : Real and Complex Numbers

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

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All SAT II Math II Resources

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Example Questions

Example Question #2 : Number Theory

Multiply: $\left ( 4+ 7i\right )(4 - 7i)$

Possible Answers:

16 - 49i

16 + 49i

None of the other responses is correct.

Correct answer:

None of the other responses is correct.

Explanation:

This is the product of a complex number and its complex conjugate. They can be multiplied using the pattern

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

with A = 4, B = 7

$\left (4 + 7i \right )\left (4 - 7i \right ) = 4^{2} + 7^{2} = 16 + 49 = 65$

This is not among the given responses.

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Example Question #4 : Number Theory

Multiply:

$-3 i \cdot 8i \cdot 5i \cdot 6i$

Possible Answers:

720

None of the other responses is correct.

-720i

720 i

Correct answer:

Explanation:

$-3 i \cdot 8i \cdot 5i \cdot 6i$

$= \left (-3 \cdot 8 \cdot 5 \cdot 6 \right ) \cdot\left ( i \cdot i \cdot i \cdot i \right )$

$= \left (-720 \right ) \cdot\left ( i ^{4}\right )$

$-720 \cdot 1 = -720$

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Example Question #11 : Sat Subject Test In Math Ii

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:

61+60i

11+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 #1 : Real And Complex Numbers

Multiply:

7i (5 + 9i)

Possible Answers:

-35 + 63i

35 - 63i

63- 35i

-63+ 35i

-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:

256i

16i

256

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 #4 : Real And Complex Numbers

Multiply:

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

Possible Answers:

84 - 13i

84 +13i

-84 + 13i

-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 #5 : Real And Complex Numbers

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 #9 : Real And Complex Numbers

Multiply:

8i (5-7i)

Possible Answers:

56- 40i

-40 + 56i

40 + 56i

-56+ 40i

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:

-168i

None of the other responses is correct.

168i

168

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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All SAT II Math II Resources

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SAT II Math II : Real and Complex Numbers

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

All SAT II Math II Resources

Example Questions

Example Question #2 : Number Theory

Example Question #4 : Number Theory

Example Question #11 : Sat Subject Test In Math Ii

Example Question #4 : Real And Complex Numbers

Example Question #1 : Real And Complex Numbers

Example Question #1 : Real And Complex Numbers

Example Question #4 : Real And Complex Numbers

Example Question #5 : Real And Complex Numbers

Example Question #9 : Real And Complex Numbers

Example Question #2 : Real And Complex Numbers

All SAT II Math II Resources

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