SAT II Math I : Real and Complex Numbers

Example Questions

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Example Question #1 : Real And Complex Numbers

Evaulate:

Explanation:

Multiply both numerator and denominator by , then divide termwise:

Example Question #1 : Real And Complex Numbers

Which of the following is equal to  ?

Explanation:

By the power of a product property,

Example Question #3 : Real And Complex Numbers

Multiply:

None of the other responses gives the correct answer.

Explanation:

Example Question #1 : Real And Complex Numbers

Which of the following is equal to ?

Explanation:

By the power of a product property,

Example Question #2 : Real And Complex Numbers

Which of the following is equal to  ?

The expression is undefined.

Explanation:

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

Therefore,

Example Question #3 : Real And Complex Numbers

What is the conjugate for the complex number

Explanation:

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 #1 : Real And Complex Numbers

denotes the complex conjugate of .

If , then evaluate .

Explanation:

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?

Explanation:

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 #1 : Real And Complex Numbers

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

.

Evaluate .

Explanation:

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 #3 : Real And Complex Numbers

Let  be a complex number.  denotes the complex conjugate of

and .

Evaluate .

None of these

Explanation:

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

Therefore,

Substituting:

Also,

Substituting:

Therefore,

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