SAT II Math I : Real and Complex Numbers

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

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Example Questions

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Example Question #1 : Real And Complex Numbers

Evaulate:

Possible Answers:

Correct answer:

Explanation:

Multiply both numerator and denominator by , then divide termwise:

Example Question #2 : Real And Complex Numbers

Which of the following is equal to  ?

Possible Answers:

Correct answer:

Explanation:

By the power of a product property, 

Example Question #3 : Real And Complex Numbers

Multiply: 

Possible Answers:

None of the other responses gives the correct answer.

Correct answer:

Explanation:

Example Question #4 : Real And Complex Numbers

Which of the following is equal to ?

Possible Answers:

Correct answer:

Explanation:

By the power of a product property, 

Example Question #5 : Real And Complex Numbers

Which of the following is equal to  ?

Possible Answers:

The expression is undefined.

Correct answer:

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 #6 : Real And Complex Numbers

What is the conjugate for the complex number 

Possible Answers:

Correct answer:

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 #7 : Real And Complex Numbers

 denotes the complex conjugate of .

If , then evaluate .

Possible Answers:

Correct answer:

Explanation:

By the difference of squares pattern, 

If , then . As a result:

Therefore, 

Example Question #8 : Real And Complex Numbers

Which answer choice has the greatest real number value?

Possible Answers:

Correct answer:

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 #9 : Real And Complex Numbers

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

.

Evaluate .

Possible Answers:

Correct answer:

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 #10 : Real And Complex Numbers

Let  be a complex number.  denotes the complex conjugate of 

 and .

Evaluate .

Possible Answers:

None of these

Correct answer:

Explanation:

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

Therefore, 

Substituting:

Also,

Substituting:

Therefore, 

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