PROBLEMS ON COMPLEX NUMBERS FOR SAT

Problem 1 :

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

(5 - 3i) - (-2 + 5i) ?

(a)  3 - 8i    (b)  3 + 2i     (c) 7 - 8i    (d)  7 + 2i

Solution :

= (5 - 3i) - (-2 + 5i)

Distributing the negative, we get

= 5 - 3i + 2 - 5i

= 7 - 8i

So, option c is correct.

Problem 2 :

Given that i = √-1, which of the following is equal to i(i + 1) ?

(a)  i - 2    (b)  i - 1     (c) i + 1    (d)  0

Solution :

= i(i + 1)

= i² + i

= -1 + i

So, option b is correct.

Problem 3 :

i⁴ + 3i² + 2

Which of the following is equal to the expression above.

(a)  i    (b)  - 1     (c) 0     (d)  1

Solution :

= i⁴ + 3i² + 2

= (i²)² + 3i² + 2

= (-1)² + 3(-1) + 2

= 1 - 3 + 2

= 0

Problem 4 :

(6 + 2i) (2 + 5i)

If the expression above is equivalent to a + ib, where a and b are constants. What is the value of a?

(a)  2    (b)  12     (c) 22     (d)  34

Solution :

= (6 + 2i) (2 + 5i)

= 12 + 30i + 4i + 10

= 22 + 34i

a + ib = 22 + 34i

Comparing the corresponding terms, we get

a = 22 and b = 34

So, answer is option d.

Problem 5 :

Which of the following is equal to 

3(i + 2) - 2(5 - 4i)?

(a)  16 - 5i    (b)  -4 + 7i     (c) -4 + 11i     (d)  16 + 11i

Solution :

= 3(i + 2) - 2(5 - 4i)

= 3i + 6 - 10 + 8i

= 3i + 8i + 6 - 10

= 11i - 4

So, option c is correct.

Problem 6 :

Which of the following equivalent to 3i(i + 2) - i(i - 1) ?

(a)  -4 + 7i    (b)  -2 + 7i     (c) -4 + 5i     (d)  -2 + 5i

Solution :

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

= 3i² + 6i - i² + i

By combining real and imaginary, we get

= 7i + 2i²

= 7i + 2(-1)

= 7i - 2

= -2 + 7i

So, option b is correct.

Problem 7 :

Which of the following is equivalent to i⁹³ ?

(a)  -1    (b)  1     (c) -i     (d)  i

Solution :

= i⁹³

= i⁹² (i)

= (i²)⁴⁶ (i)

= (-1)⁴⁶ (i)

= 1 (i)

= i

So, the answer is i.

Problem 8 :

Which of the following complex number is equivalent to

(3 - i)² ?

(a)  8 - 6i    (b)  8 + 6i     (c) 10 - 6i     (d)  10 + 6i

Solution :

= (3 - i)² 

Expand this using algebraic identity

(a - b)² = a² - 2ab + b²

= 3² - 2(3)i + i²

= 9 - 6i + (-1)

= 9 - 1 - 6i

= 8 - 6i

Problem 9 :

(5 - 2i)(4 - 3i)

Which of the following is equal to the expression above ?

(a)  14 - 7i    (b)  14  - 23i     (c) 26 + 7i     (d)  26 - 23i

Solution :

= (5 - 2i)(4 - 3i)

= 20 - 15i - 8i + 6i²

= 20 - 23i + 6(-1)

= 20 - 23i - 6

= 14 - 23i

So, option b is correct,

Problem 10 :

Which of the following is equal to 

(a)  -i    (b)  i     (c) 0     (d)  1

Solution :

Option a is correct.

Problem 11 :

Which of the following is equal to 

(a)  -i    (b)  i     (c) (-5/4)i     (d)  (3/4) - (5/4)i

Solution :

So, option a is correct.

Problem 12 :

Which of the following complex number is equivalent to 

(2 - i)/(2 + i) ?

Solution :