SAT MATH QUESTIONS ON COMPLEX NUMBERS

Question 1 :

3i(i + 2) - i(i - 1)

For i = √-1, which of the following is equivalent to the expression above?

A) -4 + 7i

B) -2 + 7i

C) -4 + 5i

D) -2 + 5i

Answer :

= 3i(i + 2) - i(i - 1)

= 3i² + 6i - i² + i

Since, i = √-1, we have i² = -1.

= 3(-1) + 6i - (-1) + i

= -3 + 6i + 1 + i

= -2 + 7i

The correct answer choice is (B).

Question 2 :

Which of the following is equal to (5 + 3i)(5 - 3i)?

(Note : i = √-1)

A) 21

B) 29

C) 21 - 20i

D) 29 + 20i

Answer :

= (5 + 3i)(5 - 3i)

= 5² - 15i + 15i - (3i)²

= 25 - 3²i²

= 25 - 9i²

Since, i = √-1, we have i² = -1.

= 25 - 9(-1)

= 25 + 9

= 34

Question 3 :

For i = √-1, which of the following is equal to i¹¹⁷?

A) -1

B) 1

C) -i

D) i

Answer :

= i¹¹⁷

= i^{116 + 1}

= i¹¹⁶i

= (i²)⁵⁸i

= (-1)⁵⁸i

= (1)i

= i

The correct answer choice is (D).

Question 4 :

2 + 3i + 4i² + 5i³ + 6i⁴ 

If the above expression is equivalent to (a + bi), where a and b are constants, what is the value of (a + b)?

(Note : i = √-1)

A) 2

B) 6

C) 10

D) 12

Answer :

a + bi = 2 + 3i + 4i² + 5i³ + 6i⁴

a + bi = 2 + 3i + 4i² + 5i²i + 6(i²)²

Since, i = √-1, we have i² = -1.

a + bi = 2 + 3i + 4(-1) + 5(-1)i + 6(-1)²

a + bi = 2 + 3i - 4 - 5i + 6(1)

a + bi = 2 + 3i - 4 - 5i + 6

a + bi = 4 - 2i

a = 4  and  b = -2

a + b :

= 4 + (-2)

= 4 - 2

= 2

Question 5 :

If the above expression is equivalent to (a + bi), where a and b are constants, what is the value of b?

A) -4/13

B) 4/13

C) -7/13

D) 7/13

Answer :

In the given expression, the denominator is 3i - 2.

To write the given expression in the form (a + ib), multiply both numerator and denominator of the expression by the conjugate of (3i - 2), that is (3i + 2). 

The above expression is in the form of (a + ib).

a + ib = (4/13) + (-7/13)i

The value of b is -7/13.

The correct answer choice is (C).

Question 6 :

Which of the following is equal to the expression above?

(Note : i = √-1)

A) -2 - i

B) 2 + i

C) 4 + i

D) 4 - i

Answer :

In the expression above, the common denominator is

(1 + i)(1 - i)

Multiply the numerator and denominator of the first fraction by (1 + i) and the second one by (1 - i) and simplify.

The correct answer choice is (B).

Question 7 :

A) -i

B) i

C) 0

D) 1

Answer :

The correct answer choice is (A).

Question 8 :

If 2xyi + y = 2(1 + i)² + 5(1 - i), what is the value of x and y?

Answer :

2xyi + y = 2(1 + i)² + 5(1 - i)

2xyi + y = 2(1 + i)(1 + i) + 5(1 - i)

2xyi + y = 2(1 + i + i + i²) + 5(1 - i)

2xyi + y = 2(1 + 2i - 1) + 5(1 - i)

2xyi + y = 2(2i) + 5(1 - i)

2xyi + y = 4i + 5 - 5i

2xyi + y = 5 - i

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