### All SAT II Math II Resources

## Example Questions

### Example Question #11 : Real And Complex Numbers

Evaluate:

**Possible Answers:**

**Correct answer:**

Use the square of a sum pattern

where :

### Example Question #11 : Real And Complex Numbers

is a complex number; denotes the complex conjugate of .

Which of the following *could* be the value of ?

**Possible Answers:**

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

**Correct answer:**

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

The product of a complex number and its complex conjugate is

Setting and accordingly for each of the four choices, we want to find the choice for which :

For each given value of , .

### Example Question #12 : Real And Complex Numbers

denotes the complex conjugate of .

If , then evaluate .

**Possible Answers:**

**Correct answer:**

Applying the Power of a Product Rule:

The complex conjugate of an imaginary number is ; the product of the two is

, so, setting in the above pattern:

Consequently,

### Example Question #11 : Real And Complex Numbers

Let and be complex numbers. and denote their complex conjugates.

Evaluate .

**Possible Answers:**

**Correct answer:**

Knowing the actual values of and is not necessary to solve this problem. The product of the complex conjugates of two numbers is equal to the complex conjugate of the product of the numbers; that is,

We are given that . is therefore the conjugate of , or .

### Example Question #13 : Real And Complex Numbers

denotes the complex conjugate of .

If , then evaluate .

**Possible Answers:**

**Correct answer:**

By the difference of squares pattern,

If , then .

Consequently:

Therefore,

### Example Question #13 : Real And Complex Numbers

The fraction is equivalent to which of the following?

**Possible Answers:**

Undefined

**Correct answer:**

Start by multiplying the fraction by .

Since , we can then simplify the fraction:

Thus, the fraction is equivalent to .

### Example Question #14 : Real And Complex Numbers

The fraction is equivalent to which of the following?

**Possible Answers:**

**Correct answer:**

Start by multiplying both the denominator and the numerator by the conjugate of , which is .

Next, recall , and combine like terms.

Finally, simplify the fraction.

### Example Question #11 : Real And Complex Numbers

Evaluate

**Possible Answers:**

**Correct answer:**

To evaluate a power of , divide the exponent by 4 and note the remainder.

The remainder is 3, so

Consequently, using the Product of Powers Rule:

### Example Question #12 : Real And Complex Numbers

Let be a complex number. denotes the complex conjugate of .

and .

How many of the following expressions could be equal to ?

(a)

(b)

(c)

(d)

**Possible Answers:**

None

Three

Two

One

Four

**Correct answer:**

Two

is a complex number, so for some real ; also, .

Therefore,

Substituting:

Therefore, we can eliminate choices (c) and (d).

Also, the product

Setting and substituting 10 for , we get

Therefore, either or - making two the correct response.

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