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SAT II Math I : Real and Complex Numbers

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

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Example Questions

Example Question #1 : Real And Complex Numbers

Evaulate:

$\frac{45-35i}{5i}$

Possible Answers:

9 - 7i

-9 - 7i

-7+9i

7 - 9i

-7 - 9i

Correct answer:

-7 - 9i

Explanation:

Multiply both numerator and denominator by , then divide termwise:

$\frac{45-35i}{5i}$

$= \frac{\left (45-35i \right )\cdot i}{5i \cdot i}$

$= \frac{ 45\cdot i-35i\cdot i }{5i ^{2}}$

$= \frac{ 45 i-35i ^{2}}{5i ^{2}}$

$= \frac{ 45 i-35(-1) }{5 (-1)}$

$= \frac{ 45 i- (-35) }{-5 }$

$= \frac{35+ 45 i }{-5 }$

$= \frac{35 }{-5 } + \frac{ 45 i }{-5 }$

= -7 - 9i

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Example Question #1 : Real And Complex Numbers

Which of the following is equal to $\left (5i \right ) ^{3}$ ?

Possible Answers:

-15i

-125i

15i

125i

Correct answer:

-125i

Explanation:

By the power of a product property,

$\left (5i \right ) ^{3} = 5^{3} \cdot i^{3} = 125(-i)= -125i$

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Example Question #2 : Real And Complex Numbers

Multiply:

$3 i \cdot 5i \cdot 7i \cdot 2i$

Possible Answers:

None of the other responses gives the correct answer.

-210i

210i

210

Correct answer:

210

Explanation:

$3 i \cdot 5i \cdot 7i \cdot 2i$

$= 3 \cdot 5 \cdot 7 \cdot 2 \cdot i \cdot i \cdot i \cdot i$

$= 210 \cdot i ^{4} = 210 \cdot 1 = 210$

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Example Question #3 : Real And Complex Numbers

Which of the following is equal to $\left (2i \right )^{6}$ ?

Possible Answers:

64i

12i

Correct answer:

Explanation:

By the power of a product property,

$\left (2i \right )^{6} = 2^{6} \cdot i^{6} = 2^{6} \cdot i^{4} \cdot i^{2} = 64 \cdot 1 \cdot (-1) = -64$

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Example Question #4 : Real And Complex Numbers

Which of the following is equal to $i^{-33}$ ?

Possible Answers:

The expression is undefined.

Correct answer:

Explanation:

$i^{-33} = \frac{1}{i^{33}}$

To raise to a power, divide the exponent by 4, note its remainder, and raise to the power of that remainder:

$33 \div 4 = 8 \textrm{ R }1$

$i^{33} = i ^{1} = i$

Therefore, $i^{-33} = \frac{1}{i^{33}} = \frac{1}{i} = \frac{1 \cdot i}{i\cdot i} = \frac{i}{-1} = -i$

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Example Question #6 : Real And Complex Numbers

What is the conjugate for the complex number 3-2i

Possible Answers:

2i-3

-2i

3+2i

Correct answer:

3+2i

Explanation:

To find the conjugate of the complex number of the form $\small a\pm bi$ , change the sign on the complex term. The complex part of the problem is $\small -2i$ so changing the sign would make it a $\small +2i$ . The sign in the real part of the number, the 3 in this case, does not change sign.

3+2i

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Example Question #7 : Real And Complex Numbers

$\overline{x}$ denotes the complex conjugate of .

If x = 5 + 2i , then evaluate $x^{2} - \overline{x}^{2}$ .

Possible Answers:

-40 i

40 i

Correct answer:

40 i

Explanation:

By the difference of squares pattern,

$x^{2} - \overline{x}^{2} = (x+\overline{x}) (x-\overline{x})$

If x = 5 + 2i , then $\overline{x} = 5- 2i$ . As a result:

$x+\overline{ x} = (5+2i) + (5- 2i) = 5+ 5 + 2i - 2i = 10$

$x-\overline{ x} = (5+2i) - (5- 2i) = 5- 5 + 2i +2i = 4 i$

Therefore,

$x^{2} - \overline{x}^{2}$

$= (x+\overline{x}) (x-\overline{x})$

$= 10 \cdot 4i$

= 40i

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Example Question #8 : Real And Complex Numbers

Which answer choice has the greatest real number value?

Possible Answers:

$i^2 + i^8 + i^{10} +i^ {16}$

$i^4 + i^6 + i^{14} +i^ {24}$

$i^{40} + i^2 + i^{13} +i^ {11}$

$i + i^3 + i^{7} +i^ {9}$

$i^4 + i^8 + i^{12} +i^ {16}$

Correct answer:

$i^4 + i^8 + i^{12} +i^ {16}$

Explanation:

Recall the definition of and its exponents

$i = \sqrt[]{-1}$

i^2 = -1

i^3 = -i

i^4 = 1

because i^4 = 1 then

$i^5 = i^4\cdot i =1\cdot i =i$ .

We can generalize this to say

$i^n =i^{n-4}$

Any time is a multiple of 4 then i^n =1 . For any other value of we get a smaller value.

For the correct answer each of the terms equal

$i^4 =i^8 =i^{12} = i^{16} =1$

So:

$i^4 + i^8 + i^{12} +i^ {16}$

1+1+1+1

Because all the alternative answer choices have 4 terms, and each answer choice has at least one term that is not equal to they must all be less than the correct answer.

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Example Question #9 : Real And Complex Numbers

Let and be complex numbers. $\overline{x}$ and $\overline{y}$ denote their complex conjugates.

x+y = 30 + 50i .

Evaluate $\overline{x} - \overline{y}$ .

Possible Answers:

60-100i

$\textup{The question cannot be answered from the information given}$

60+100i

100i

Correct answer:

$\textup{The question cannot be answered from the information given}$

Explanation:

Let x = a+ bi, y = c+ di , where all variables represent real quantities.

Then

x +y = a+ c+ bi + di

x +y =( a+ c)+ (b + d)i

Since

x+y = 30 + 50i ,

if follows that

a+ c = 30

and

b+ d = 50

Also, by definition,

$\overline{x} = a- bi, \overline{y} = c-di$

$\overline{x} - \overline{y} = (a-bi) - (c- di)$

$\overline{x} - \overline{y} = a - c-bi + di$

$\overline{x} - \overline{y} =( a - c)+ (-b + d)i$

It is known that a+ c = 30 and b+ d = 50 , but without further information, nothing can be determined about a - c 0r -b+d . Therefore, $\overline{x} - \overline{y}$ cannot be evaluated.

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Example Question #9 : Real And Complex Numbers

Let be a complex number. $\overline{x}$ denotes the complex conjugate of .

$x+\overline{ x} = 16$ and $x-\overline{ x} = 40i$ .

Evaluate .

Possible Answers:

8 + 20 i

None of these

20 - 8i

20 + 8i

8 - 20 i

Correct answer:

8 + 20 i

Explanation:

is a complex number, so x = a+ bi for some real a ,b ; also, $\overline{x} = a-bi$ .

Therefore,

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

Substituting:

$x+\overline{ x} = 16$

2a = 16

a = 8

Also,

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

Substituting:

$x-\overline{ x} = 40i$

2bi = 40 i

2b= 40

b = 20

Therefore,

x = a+ bi = 8 + 20 i

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SAT II Math I : Real and Complex Numbers

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

All SAT II Math I Resources

Example Questions

Example Question #1 : Real And Complex Numbers

Example Question #1 : Real And Complex Numbers

Example Question #2 : Real And Complex Numbers

Example Question #3 : Real And Complex Numbers

Example Question #4 : Real And Complex Numbers

Example Question #6 : Real And Complex Numbers

Example Question #7 : Real And Complex Numbers

Example Question #8 : Real And Complex Numbers

Example Question #9 : Real And Complex Numbers

Example Question #9 : Real And Complex Numbers

All SAT II Math I Resources

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