### All SAT II Math I Resources

## Example Questions

### Example Question #1 : Real And Complex Numbers

Evaulate:

**Possible Answers:**

**Correct answer:**

Multiply both numerator and denominator by , then divide termwise:

### Example Question #1 : Real And Complex Numbers

Which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

By the power of a product property,

### Example Question #1 : Real And Complex Numbers

Multiply:

**Possible Answers:**

None of the other responses gives the correct answer.

**Correct answer:**

### Example Question #2 : Real And Complex Numbers

Which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

By the power of a product property,

### Example Question #1 : Real And Complex Numbers

Which of the following is equal to ?

**Possible Answers:**

The expression is undefined.

**Correct answer:**

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

Therefore,

### Example Question #1 : Real And Complex Numbers

What is the conjugate for the complex number

**Possible Answers:**

**Correct answer:**

To find the conjugate of the complex number of the form , change the sign on the complex term. The complex part of the problem is so changing the sign would make it a . The sign in the real part of the number, the 3 in this case, does not change sign.

### Example Question #2 : Real And Complex Numbers

denotes the complex conjugate of .

If , then evaluate .

**Possible Answers:**

**Correct answer:**

By the difference of squares pattern,

If , then . As a result:

Therefore,

### Example Question #3 : Real And Complex Numbers

Which answer choice has the greatest real number value?

**Possible Answers:**

**Correct answer:**

Recall the definition of and its exponents

because then

.

We can generalize this to say

Any time is a multiple of 4 then . For any other value of we get a smaller value.

For the correct answer each of the terms equal

So:

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.

### Example Question #2 : Real And Complex Numbers

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

.

Evaluate .

**Possible Answers:**

**Correct answer:**

Let , where all variables represent real quantities.

Then

Since

,

if follows that

and

Also, by definition,

It is known that and , but without further information, nothing can be determined about 0r . Therefore, cannot be evaluated.

### Example Question #1 : Real And Complex Numbers

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

and .

Evaluate .

**Possible Answers:**

None of these

**Correct answer:**

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

Therefore,

Substituting:

Also,

Substituting:

Therefore,