### All SAT II Math II Resources

## Example Questions

### Example Question #11 : 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 #21 : Number Theory

denotes the complex conjugate of .

If , then evaluate .

**Possible Answers:**

**Correct answer:**

By the difference of squares pattern,

If , then .

Consequently:

Therefore,

### Example Question #21 : Number Theory

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 #12 : 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 #11 : 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

Four

One

Three

Two

**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.

### Example Question #1 : Elementary Operations

Add the following numbers:

**Possible Answers:**

**Correct answer:**

Add the ones digits.

Add the tens digits with the two as the carryover.

Add the hundreds digits with the two as the carryover.

Combine this number with the ones digits of the previous calculations.

The answer is:

### Example Question #1 : Mathematical Relationships

Add the numbers:

**Possible Answers:**

**Correct answer:**

Add the ones digits.

Carry the one, and add the tens digits.

Carry the two, and add the hundreds digits.

Combine this number with the ones digit of the other numbers.

The answer is:

### Example Question #1 : Mathematical Relationships

Add the numbers:

**Possible Answers:**

**Correct answer:**

Add the ones digits.

Carry the two, and add the tens digits.

Carry the two, and add the hundreds places.

Combine all ones digits.

The answer is:

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