**Chapter 1 Complex Numbers Miscellaneous Exercise 1**

## Chapter 1 Complex Numbers Miscellaneous Exercise 1

**(I) Select the correct answer from the given alternatives.**

**Question 1.If n is an odd positive integer, then the value of 1 + is:(A) -4i(B) 0(C) 4i(D) 4Answer:**

(B) 0

Hint:

= 1 â€“ 1 + 1 â€“ 1 â€¦..(n odd positive integer)

= 0

**Question 2.(A) -2(B) 1(C) 0(D) -1Answer:**

(D) -1

Hint:

**Question 3.âˆš-3 âˆš-6 is equal to(A) -3âˆš2(B) 3âˆš2(C) 3âˆš2 i(D) -3âˆš2 iAnswer:**

(A) -3âˆš2

Hint:

âˆš-3 âˆš-6

= (âˆš3 i) (âˆš6 i)

= 3âˆš2 (-1)

= -3âˆš2

**Question 4.If Ï‰ is a complex cube root of unity, then the value of is:(A) -1(B) 1(C) 0(D) 3Answer:**
(C) 0

Hint:

**Question 5.(A) cos 2Î¸(B) 2cos 2Î¸(C) 2cos Î¸(D) 2sin Î¸Answer:**

(B) 2cos 2Î¸

Hint:

**Question 6.If Ï‰(â‰ 1) is a cube root of unity and (1 + Ï‰ = A + BÏ‰, then A and B are respectively the numbers(A) 0, 1(B) 1, 1(C) 1, 0(D) -1, 1Answer:**

(B) 1, 1

Hint:

= 1 + Ï‰

A = 1, B = 1

**Question 7.**

Hint:

**Question 8.(A) -Î¸(B) Î¸(C) Ï€ â€“ Î¸(D) Ï€ + Î¸Answer:**

(A) -Î¸

Hint:

**Question 9.**

Hint:

**Question 10.If z = x + iy and |z â€“ zi| = 1, then(A) z lies on X-axis(B) z lies on Y-axis(C) z lies on a rectangle(D) z lies on a circleAnswer:**

(D) z lies on a circle

Hint:

|z â€“ zi | = |z| |1 â€“ i| = 1

**(II) Answer the following:**

**Question 1.Simplify the following and express in the form a + ib.(i) 3 + âˆš-64Solution:**

3 + âˆš-64

= 3 + âˆš64 âˆš-1

= 3 + 8i

**(iii) (2 + 3i) (1 â€“ 4i)Solution:**

(2 + 3i)(1 â€“ 4i)

= 2 â€“ 8i + 3i â€“ 12

= 2 â€“ 5i â€“ 12(-1) â€¦..[âˆµ = -1]

= 14 â€“ 5i

Solution:

= (-8 + 6i)(3 + i)

= -24 â€“ 8i + 18i + 6i^{2}

= -24 + 10i + 6(-1)

= -24 + 10i â€“ 6

= -30 + 10i

Solution:

Solution:

Solution:

Solution:

Solution:

**Question 2.Solve the following equations for x, y âˆˆ R(i) (4 â€“ 5i)x + (2 + 3i)y = 10 â€“ 7iSolution:**

(4 â€“ 5i)x + (2 + 3i)y = 10 â€“ 7i

(4x + 2y) + (3y â€“ 5x) i = 10 â€“ 7i

Equating real and imaginary parts, we get

4x + 2y= 10 i.e., 2x + y = 5 â€¦â€¦(i)

and 3y â€“ 5x = -7 â€¦â€¦(ii)

Equation (i) Ã— 3 â€“ equation (ii) gives

11x = 22

âˆ´ x = 2

Putting x = 2 in (i), we get

2(2) + y = 5

âˆ´ y = 1

âˆ´ x = 2 and y = 1

x + iy = (7 â€“ i)(2 + 3i)

x + iy = 14 + 21i â€“ 2i â€“ 3

x + iy = 14 + 19i â€“ 3(-1)

x + iy = 17 + 19i

Equating real and imaginary parts, we get

âˆ´ x = 17 and y = 19

**(iii) (x + iy) (5 + 6i) = 2 + 3iSolution:**

2x + 2yi â€“ y + xi = 10

(2x â€“ y) + (x + 2y)i = 10 + 0 . i

Equating real and imaginary parts, we get

2x â€“ y = 10 â€¦â€¦(i)

and x + 2y = 0 â€¦â€¦..(ii)

Equation (i) Ã— 2 + equation (ii) gives, we get

5x = 20

âˆ´ x = 4

Putting x = 4 in (i), we get

2(4) â€“ y = 10

y = 8 â€“ 10

âˆ´ y = -2

âˆ´ x = 4 and y = -2

**Question 3.Evaluate(i) (1 â€“ i + Solution:**

**Question 4.Find the value of(i) + 2 â€“ 3x + 21, if x = 1 + 2iSolution:**

**(ii) + 9 + 35 â€“ x + 164, if x = -5 + 4iSolution:**

**Question 5.Find the square roots of(i) -16 + 30iSolution:**

**(ii) 15 â€“ 8iSolution:**

**(iv) 18iSolution:**

**(v) 3 â€“ 4iSolution:**

**(vi) 6 + 8iSolution:**

**Question 6.Find the modulus and argument of each complex number and express it in the polar form.(i) 8 + 15iSolution:**

**(ii) 6 â€“ iSolution:**

Solution:

Solution:

**(v) 2iSolution:**

**(vi) -3iSolution:**

Solution:

**Question 7.Represent 1 + 21, 2 â€“ i, -3 â€“ 2i, -2 + 3i by points in Argandâ€™s diagram.Solution:**

The complex numbers 1 + 2i, 2 â€“ i, -3 â€“ 2i, -2 + 3i will be represented by the points A(1, 2), B(2, -1), C(-3, -2), D(-2, 3) respectively as shown below:

**Question 8.Solution:**

**Question 9.Find the real numbers x and y such that**

(3x + y) + 2(x + y)i = 5 + 6i

Equating real and imaginary parts, we get

3x + y = 5 â€¦â€¦(i)

and 2(x + y) = 6

i.e., x + y = 3 â€¦â€¦.(ii)

Subtracting (ii) from (i), we get

2x = 2

âˆ´ x = 1

Putting x = 1 in (ii), we get

1 + y = 3

âˆ´ y = 2

âˆ´ x = 1, y = 2

**Question 10.Solution:**

**Question 11.Solution:**

**Question 12.Convert the complex numbers in polar form and also in exponential form.Solution:**

**(ii) z = -6 + âˆš2 iSolution:**

z = -6 + âˆš2 i

âˆ´ a = -6, b = âˆš2

i.e. a < 0, b > 0

Solution:

**Question 13.Solution:**

**Question 14.Solution:**

**Question 15.Solution:**

**Question 16.SimplifySolution:**

Solution:

Solution:

**Question 17.Solution:**

**Question 18.If Î± and Î² are complex cube roots of unity, prove that (1 â€“ Î±) (1 â€“ Î²) (1 â€“ ) (1 â€“) = 9.Solution:**

Î± and Î² are the complex cube roots of unity.

**Question 19.**

**Question 20.If Ï‰ is the cube root of unity, then find the value of **

Solution:

If Ï‰ is the complex cube root of unity, then