The combination of a real number and an imaginary number is termed as a complex number. All the concepts related to complex numbers are explained in this chapter with suitable examples. Students are provided with exercise-wise solutions to help understand the concepts clearly from the exam point of view. Experts have prepared the solutions where the concepts are explained in detail, which is very helpful for preparing for their board exams. Students who find difficulty in solving problems can quickly jump to RD Sharma Class 11 Maths Solutions, download the pdf from the links given below and can start practising offline for good results.

Chapter 13 – Complex Numbers contains four exercises and the RD Sharma Solutions present in this page provide solutions to the questions present in each exercise. Now, let us have a look at the concepts discussed in this chapter.

- Integral Powers of IOTA (i).
- Imaginary quantities.
- Complex numbers.
- Equality of complex numbers.
- Addition of complex numbers.
- Properties of addition of complex numbers.

- Subtraction of complex numbers.
- Multiplication of complex numbers.
- Properties of multiplication.

- Division of complex numbers.
- Conjugate of complex numbers.
- Modulus of a complex number.
- Properties of modulus.

- Reciprocal of a complex number.
- Square roots of a complex number.
- Representation of a complex number.
- Geometrical representation of a complex number.
- Argument or amplitude of a complex number for different signs of real and imaginary parts.
- Vectorial representation of a complex number.
- Polar or trigonometrical form of a complex number.
- Multiplication of a complex number by IOTA.
- The polar form of a complex number for different signs of real and imaginary parts.

## Download the pdf of RD Sharma Solutions for Class 11 Maths Chapter 13 – Complex Numbers

### Access answers to RD Sharma Solutions for Class 11 Maths Chapter 13 – Complex Numbers

EXERCISE 13.1 PAGE NO: 13.3

**1. Evaluate the following:**

**(i) i ^{457}**

**(ii) i ^{528}**

**(iii) 1/ i ^{58}**

**(iv) i ^{37} + 1/i ^{67}**

**(v) [i ^{41} + 1/i ^{257}]**

**(vi) (i ^{77} + i ^{70} + i ^{87} + i ^{414})^{3}**

**(vii) i ^{30} + i ^{40} + i ^{60}**

**(viii) i ^{49} + i ^{68} + i ^{89} + i ^{110}**

**Solution:**

**(i)** i ^{457}

Let us simplify we get,

i^{457 }= iÂ ^{(456 + 1)}

= i ^{4(114)}Â Ã— i

= (1)^{114}Â Ã— i

= i [since i^{4}Â = 1]

**(ii)** i ^{528}

Let us simplify we get,

i ^{528}Â = i ^{4(132)}

= (1)^{132}

= 1 [since i^{4}Â = 1]

**(iii) **1/ i^{58}

Let us simplify we get,

1/ i^{58} = 1/ i ^{56+2}

= 1/ i ^{56} Ã— i^{2}

= 1/ (i^{4})^{14} Ã— i^{2}

= 1/ i^{2} [since, i^{4}Â = 1]

= 1/-1 [since, i^{2} = -1]

= -1

**(iv) **i ^{37} + 1/i ^{67}

Let us simplify we get,

i ^{37} + 1/i ^{67 }= i^{36+1} + 1/ i^{64+3}

= i + 1/i^{3} [since, i^{4}Â = 1]

= i + i/i^{4}

= i + i

= 2i

**(v) **[i ^{41} + 1/i ^{257}]

Let us simplify we get,

[i^{41}+ 1/i

^{257}] = [i

^{40+1}+ 1/ i

^{256+1}]

= [i + 1/i]^{9} [since, 1/i = -1]

= [i – i]

= 0

**(vi) **(i ^{77} + i ^{70} + i ^{87} + i ^{414})^{3}

Let us simplify we get,

(i ^{77} + i ^{70} + i ^{87} + i ^{414})^{3 }= (i^{(76 + 1)}Â + i^{(68 + 2)}Â + i^{(84 + 3)}Â + i^{(412 + 2)}Â )Â ^{3}

= (i + i^{2}Â + i^{3}Â + i^{2})^{3} [since i^{3}Â = â€“ i, i^{2}Â = â€“ 1]

= (i + (â€“ 1) + (â€“ i) + (â€“ 1))^{ 3}

= (â€“ 2)^{3}

= â€“ 8

**(vii) **i ^{30} + i ^{40} + i ^{60}

Let us simplify we get,

i ^{30} + i ^{40} + i ^{60} = i^{(28 + 2) +}Â i^{40}Â + i^{60}

= (i^{4})^{7}Â i^{2}Â + (i^{4})^{10}Â + (i^{4})^{15}

= i^{2}Â + 1^{10}Â + 1^{15}

= â€“ 1 + 1 + 1

= 1

**(viii) **i ^{49} + i ^{68} + i ^{89} + i ^{110}

Let us simplify we get,

i ^{49} + i ^{68} + i ^{89} + i ^{110} = i^{(48 + 1)}Â + i^{68}Â + i^{(88 + 1)}Â + i^{(116 + 2)}

= (i^{4})^{12}Ã—i + (i^{4})^{17}Â + (i^{4})^{22}Ã—i + (i^{4})^{29}Ã—i^{2}

= i + 1 + i â€“ 1

= 2i

**2. Show that 1 + i ^{10}Â + i^{20}Â + i^{30}Â is a real number?**

**Solution:**

Given:

1 + i^{10}Â + i^{20}Â + i^{30}Â = 1 + i^{(8 + 2) +}Â i^{20}Â + i^{(28 + 2)}

= 1 + (i^{4})^{2}Â Ã— i^{2}Â + (i^{4})^{5}Â + (i^{4})^{7}Â Ã— i^{2}

= 1 â€“ 1 + 1 â€“ 1 [since, i^{4}Â = 1, i^{2}Â = â€“ 1]

= 0

Hence , 1 + i^{10}Â + i^{20}Â + i^{30}Â is a real number.

**3. Find the values of the following expressions:**

**(i) i ^{49}Â + i^{68}Â + i^{89}Â + i^{110}**

**(ii) i ^{30}Â + i^{80}Â + i^{120}**

**(iii) i + i ^{2}Â + i^{3}Â + i^{4}**

**(iv) i ^{5}Â + i^{10}Â + i^{15}**

**(v) [i ^{592} + i^{590} + i^{588} + i^{586} + i^{584}] / [i^{582} + i^{580} + i^{578} + i^{576} + i^{574}]**

**(vi) 1 + i ^{2}Â + i^{4}Â + i^{6}Â + i^{8}Â + … + i^{20}**

**(vii) (1 + i) ^{6}Â + (1 â€“ i)^{3}**

**Solution:**

**(i)** i^{49}Â + i^{68}Â + i^{89}Â + i^{110}

Let us simplify we get,

i^{49}Â + i^{68}Â + i^{89}Â + i^{110 }= i ^{(48 + 1)}Â + i^{68}Â + i^{(88 + 1)}Â + i^{(108 + 2)}

= (i^{4})^{12}Â Ã— i + (i^{4})^{17}Â + (i^{4})^{22}Â Ã— i + (i^{4})^{27}Â Ã— i^{2}

= i + 1 + i â€“ 1 [since i^{4}Â = 1, i^{2}Â = â€“ 1]

= 2i

âˆ´ i^{49}Â + i^{68}Â + i^{89}Â + i^{110}Â = 2i

**(ii)** i^{30}Â + i^{80}Â + i^{120}

Let us simplify we get,

i^{30}Â + i^{80}Â + i^{120} = i^{(28 + 2)}Â + i^{80}Â + i^{120}

= (i^{4})^{7}Â Ã— i^{2}Â + (i^{4})^{20}Â + (i^{4})^{30}

= â€“ 1 + 1 + 1 [since i^{4}Â = 1, i^{2}Â = â€“ 1]

= 1

âˆ´ i^{30}Â + i^{80}Â + i^{120}Â = 1

**(iii) **i + i^{2}Â + i^{3}Â + i^{4}

Let us simplify we get,

i + i^{2}Â + i^{3}Â + i^{4 }= i + i^{2}Â + i^{2}Ã—i + i^{4}

= i â€“ 1 + (â€“ 1) Ã— i + 1 [since i^{4}Â = 1, i^{2}Â = â€“ 1]

= i â€“ 1 â€“ i + 1

= 0

âˆ´ i + i^{2}Â + i^{3}Â + i^{4} = 0

**(iv) **i^{5}Â + i^{10}Â + i^{15}

Let us simplify we get,

i^{5}Â + i^{10}Â + i^{15} = i^{(4 + 1)}Â + i^{(8 + 2)}Â + i^{(12 + 3)}

= (i^{4})^{1}Ã—i + (i^{4})^{2}Ã—i^{2}Â + (i^{4})^{3}Ã—i^{3}

= (i^{4})^{1}Ã—i + (i^{4})^{2}Ã—i^{2}Â + (i^{4})^{3}Ã—i^{2}Ã—i

= 1Ã—i + 1 Ã— (â€“ 1) + 1 Ã— (â€“ 1)Ã—i

= i â€“ 1 â€“ i

= â€“ 1

âˆ´ i^{5}Â + i^{10}Â + i^{15} = -1

**(v) **[i^{592} + i^{590} + i^{588} + i^{586} + i^{584}] / [i^{582} + i^{580} + i^{578} + i^{576} + i^{574}]

Let us simplify we get,

[i^{592}+ i

^{590}+ i

^{588}+ i

^{586}+ i

^{584}] / [i

^{582}+ i

^{580}+ i

^{578}+ i

^{576}+ i

^{574}]

= [i^{10} (i^{582} + i^{580} + i^{578} + i^{576} + i^{574}) / (i^{582} + i^{580} + i^{578} + i^{576} + i^{574})]

= i^{10}

= i^{8} i^{2}

= (i^{4})^{2}Â i^{2} ^{
}

= (1)^{2}Â (-1) [since i^{4}Â = 1, i^{2}Â = -1]

= -1^{Â }

âˆ´ [i^{592} + i^{590} + i^{588} + i^{586} + i^{584}] / [i^{582} + i^{580} + i^{578} + i^{576} + i^{574}] = -1

**(vi) **1 + i^{2}Â + i^{4}Â + i^{6}Â + i^{8}Â + … + i^{20}

Let us simplify we get,

1 + i^{2}Â + i^{4}Â + i^{6}Â + i^{8}Â + … + i^{20} = 1 + (â€“ 1) + 1 + (â€“ 1) + 1 + … + 1

= 1

âˆ´ 1 + i^{2}Â + i^{4}Â + i^{6}Â + i^{8}Â + … + i^{20} = 1

**(vii) **(1 + i)^{6}Â + (1 â€“ i)^{3}

Let us simplify we get,

(1 + i)^{6}Â + (1 â€“ i)^{3} = {(1 + i)^{2}Â }^{3}Â + (1 â€“ i)^{2}Â (1 â€“ i)

= {1 + i^{2}Â + 2i}^{3}Â + (1 + i^{2 â€“}Â 2i)(1 â€“ i)

= {1 â€“ 1 + 2i}^{3}Â + (1 â€“ 1 â€“ 2i)(1 â€“ i)

= (2i)^{3}Â + (â€“ 2i)(1 â€“ i)

= 8i^{3}Â + (â€“ 2i) + 2i^{2}

= â€“ 8i â€“ 2i â€“ 2 [since i^{3}Â = â€“ i, i^{2}Â = â€“ 1]

= â€“ 10 i â€“ 2

= â€“ 2(1 + 5i)

= – 2 â€“ 10i

âˆ´ (1 + i)^{6}Â + (1 â€“ i)^{3} = – 2 â€“ 10i

EXERCISE 13.2 PAGE NO: 13.31

**1. Express the following complex numbers in the standard form a + ib:**

**(i) (1 + i) (1 + 2i)**

**(ii) (3 + 2i) / (-2 + i)**

**(iii) 1/(2 + i) ^{2}**

**(iv) (1 – i) / (1 + i)**

**(v) (2 + i) ^{3} / (2 + 3i)**

**(vi) [(1 + i) (1 +âˆš3i)] / (1 – i)**

**(vii) (2 + 3i) / (4 + 5i)**

**(viii) (1 – i) ^{3} / (1 â€“ i^{3})**

**(ix) (1 + 2i) ^{-3}**

**(x) (3 â€“ 4i) / [(4 â€“ 2i) (1 + i)]**

**(xi)**

**(xii) (5 +âˆš2i) / (1-âˆš2i)**

**Solution:**

**(i) **(1 + i) (1 + 2i)

Let us simplify and express in the standard form of (a + ib),

(1 + i) (1 + 2i) = (1+i)(1+2i)

= 1(1+2i)+i(1+2i)

= 1+2i+i+2i^{2}

= 1+3i+2(-1) [since, i^{2 }= -1]

= 1+3i-2

= -1+3i

âˆ´Â The values of a, b are -1, 3.

**(ii) **(3 + 2i) / (-2 + i)

Let us simplify and express in the standard form of (a + ib),

(3 + 2i) / (-2 + i) = [(3 + 2i) / (-2 + i)] Ã— (-2-i) / (-2-i) [multiply and divide with (-2-i)]

= [3(-2-i) + 2i (-2-i)] / [(-2)^{2} â€“ (i)^{2}]

= [-6 -3i â€“ 4i -2i^{2}] / (4-i^{2})

= [-6 -7i -2(-1)] / (4 â€“ (-1)) [since, i^{2 }= -1]

= [-4 -7i] / 5

âˆ´Â The values of a, b are -4/5, -7/5

**(iii) **1/(2 + i)^{2}

Let us simplify and express in the standard form of (a + ib),

1/(2 + i)^{2} = 1/(2^{2} + i^{2} + 2(2) (i))

= 1/ (4 â€“ 1 + 4i) [since, i^{2 }= -1]

= 1/(3 + 4i) [multiply and divide with (3 â€“ 4i)]

= 1/(3 + 4i) Ã— (3 â€“ 4i)/ (3 â€“ 4i)]

= (3-4i)/ (3^{2} â€“ (4i)^{2})

= (3-4i)/ (9 â€“ 16i^{2})

= (3-4i)/ (9 â€“ 16(-1)) [since, i^{2 }= -1]

= (3-4i)/25

âˆ´Â The values of a, b are 3/25, -4/25

**(iv) **(1 – i) / (1 + i)

Let us simplify and express in the standard form of (a + ib),

(1 – i) / (1 + i) = (1 – i) / (1 + i) Ã— (1-i)/(1-i) [multiply and divide with (1-i)]

= (1^{2} + i^{2} â€“ 2(1)(i)) / (1^{2} â€“ i^{2})

= (1 + (-1) -2i) / (1 â€“ (-1))

= -2i/2

= -i

âˆ´Â The values of a, b are 0, -1

**(v) **(2 + i)^{3} / (2 + 3i)

Let us simplify and express in the standard form of (a + ib),

(2 + i)^{3} / (2 + 3i) = (2^{3} + i^{3} + 3(2)^{2}(i) + 3(i)^{2}(2)) / (2 + 3i)

= (8 + (i^{2}.i) + 3(4)(i) + 6i^{2}) / (2 + 3i)

= (8 + (-1)i + 12i + 6(-1)) / (2 + 3i)

= (2 + 11i) / (2 + 3i)

[multiply and divide with (2-3i)]= (2 + 11i)/(2 + 3i) Ã— (2-3i)/(2-3i)

= [2(2-3i) + 11i(2-3i)] / (2^{2} â€“ (3i)^{2})

= (4 â€“ 6i + 22i â€“ 33i^{2}) / (4 â€“ 9i^{2})

= (4 + 16i â€“ 33(-1)) / (4 â€“ 9(-1)) [since, i^{2 }= -1]

= (37 + 16i) / 13

âˆ´Â The values of a, b are 37/13, 16/13

**(vi) **[(1 + i) (1 +âˆš3i)] / (1 – i)

Let us simplify and express in the standard form of (a + ib),

[(1 + i) (1 +âˆš3i)] / (1 – i) = [1(1+âˆš3i) + i(1+âˆš3i)] / (1-i)= (1 + âˆš3i + i + âˆš3i^{2}) / (1 – i)

= (1 + (âˆš3+1)i + âˆš3(-1)) / (1-i) [since, i^{2 }= -1]

= [(1-âˆš3) + (1+âˆš3)i] / (1-i)

[multiply and divide with (1+i)]= [(1-âˆš3) + (1+âˆš3)i] / (1-i) Ã— (1+i)/(1+i)

= [(1-âˆš3) (1+i) + (1+âˆš3)i(1+i)] / (1^{2} â€“ i^{2})

= [1-âˆš3+ (1-âˆš3)i + (1+âˆš3)i + (1+âˆš3)i^{2}] / (1-(-1)) [since, i^{2 }= -1]

= [(1-âˆš3)+(1-âˆš3+1+âˆš3)i+(1+âˆš3)(-1)] / 2

= (-2âˆš3 + 2i) / 2

= -âˆš3 + i

âˆ´Â The values of a, b are -âˆš3, 1

**(vii) **(2 + 3i) / (4 + 5i)

Let us simplify and express in the standard form of (a + ib),

(2 + 3i) / (4 + 5i) = [multiply and divide with (4-5i)]

= (2 + 3i) / (4 + 5i) Ã— (4-5i)/(4-5i)

= [2(4-5i) + 3i(4-5i)] / (4^{2} â€“ (5i)^{2})

= [8 â€“ 10i + 12i â€“ 15i^{2}] / (16 â€“ 25i^{2})

= [8+2i-15(-1)] / (16 â€“ 25(-1)) [since, i^{2 }= -1]

= (23 + 2i) / 41

âˆ´Â The values of a, b are 23/41, 2/41

**(viii) **(1 – i)^{3} / (1 â€“ i^{3})

Let us simplify and express in the standard form of (a + ib),

(1 – i)^{3} / (1 â€“ i^{3}) = [1^{3} â€“ 3(1)^{2}i + 3(1)(i)^{2} â€“ i^{3}] / (1-i^{2}.i)

= [1 â€“ 3i + 3(-1)-i^{2}.i] / (1 â€“ (-1)i) [since, i^{2 }= -1]

= [-2 â€“ 3i â€“ (-1)i] / (1+i)

= [-2-4i] / (1+i)

[Multiply and divide with (1-i)]= [-2-4i] / (1+i) Ã— (1-i)/(1-i)

= [-2(1-i)-4i(1-i)] / (1^{2} â€“ i^{2})

= [-2+2i-4i+4i^{2}] / (1 â€“ (-1))

= [-2-2i+4(-1)] /2

= (-6-2i)/2

= -3 â€“ i

âˆ´Â The values of a, b are -3, -1

**(ix) **(1 + 2i)^{-3}

Let us simplify and express in the standard form of (a + ib),

(1 + 2i)^{-3} = 1/(1 + 2i)^{3}

= 1/(1^{3}+3(1)^{2} (2i)+2(1)(2i)^{2} + (2i)^{3})

= 1/(1+6i+4i^{2}+8i^{3})

= 1/(1+6i+4(-1)+8i^{2}.i) [since, i^{2 }= -1]

= 1/(-3+6i+8(-1)i) [since, i^{2 }= -1]

= 1/(-3-2i)

= -1/(3+2i)

[Multiply and divide with (3-2i)]= -1/(3+2i) Ã— (3-2i)/(3-2i)

= (-3+2i)/(3^{2} â€“ (2i)^{2})

= (-3+2i) / (9-4i^{2})

= (-3+2i) / (9-4(-1))

= (-3+2i) /13

âˆ´Â The values of a, b are -3/13, 2/13

**(x) **(3 â€“ 4i) / [(4 â€“ 2i) (1 + i)]

Let us simplify and express in the standard form of (a + ib),

(3 â€“ 4i) / [(4 â€“ 2i) (1 + i)] = (3-4i)/ [4(1+i)-2i(1+i)]

= (3-4i)/ [4+4i-2i-2i^{2}]

= (3-4i)/ [4+2i-2(-1)] [since, i^{2 }= -1]

= (3-4i)/ (6+2i)

[Multiply and divide with (6-2i)]= (3-4i)/ (6+2i) Ã— (6-2i)/(6-2i)

= [3(6-2i)-4i(6-2i)] / (6^{2} â€“ (2i)^{2})

= [18 â€“ 6i â€“ 24i + 8i^{2}] / (36 â€“ 4i^{2})

= [18 â€“ 30i + 8 (-1)] / (36 â€“ 4 (-1)) [since, i^{2 }= -1]

= [10-30i] / 40

= (1 â€“ 3i) / 4

âˆ´Â The values of a, b are 1/4, -3/4

**(xi)**

**(xii) **(5 +âˆš2i) / (1-âˆš2i)

Let us simplify and express in the standard form of (a + ib),

(5 +âˆš2i) / (1-âˆš2i) = [Multiply and divide with (1+**âˆš**2i)]

= (5 +âˆš2i) / (1-âˆš2i) Ã— (1+**âˆš**2i)/(1+**âˆš**2i)

= [5(1+**âˆš**2i) + **âˆš**2i(1+**âˆš**2i)] / (1^{2} â€“ (**âˆš**2)^{2})

= [5+5**âˆš**2i + **âˆš**2i + 2i^{2}] / (1 â€“ 2i^{2})

= [5 + 6**âˆš**2i + 2(-1)] / (1-2(-1)) [since, i^{2 }= -1]

= [3+6**âˆš**2i]/3

= 1+ 2**âˆš**2i

âˆ´Â The values of a, b are 1, 2**âˆš**2

**2. Find the real values of x and y, if**

**(i) (x + iy) (2 â€“ 3i) = 4 + i**

**(ii) (3x â€“ 2i y) (2 + i) ^{2} = 10(1 + i)**

**(iv) (1 + i) (x + iy) = 2 â€“ 5i**

**Solution:**

**(i) **(x + iy) (2 â€“ 3i) = 4 + i

Given:

(x + iy) (2 – 3i) = 4 + i

Let us simplify the expression we get,

x(2 – 3i) + iy(2 – 3i) = 4 + i

2x – 3xi + 2yi – 3yi^{2 }= 4 + i

2x + (-3x+2y)i – 3y (-1) = 4 + i [since, i^{2 }= -1]

2x + (-3x+2y)i + 3y = 4 + i [since, i^{2 }= -1]

(2x+3y) + i(-3x+2y) = 4 + i

Equating Real and Imaginary parts on both sides, we get

2x+3y = 4â€¦ (i)

And -3x+2y = 1â€¦ (ii)

Multiply (i) by 3 and (ii) by 2 and add

On solving we get,

6x â€“ 6x â€“ 9y + 4y = 12 + 2

13y = 14

y = 14/13

Substitute the value of y in (i) we get,

2x+3y = 4

2x + 3(14/13) = 4

2x = 4 â€“ (42/13)

= (52-42)/13

2x = 10/13

x = 5/13

x = 5/13, y = 14/13

âˆ´Â The real values of x and y are 5/13, 14/13

**(ii) **(3x â€“ 2i y) (2 + i)^{2} = 10(1 + i)

Given:

(3x – 2iy) (2+i)^{2 }= 10(1+i)

(3x – 2yi) (2^{2}+i^{2}+2(2)(i)) = 10+10i

(3x – 2yi) (4 + (-1)+4i) = 10+10i [since, i^{2 }= -1]

(3x – 2yi) (3+4i) = 10+10i

Let us divide with 3+4i on both sides we get,

(3x – 2yi) = (10+10i)/(3+4i)

= Now multiply and divide with (3-4i)

= [10(3-4i) + 10i(3-4i)] / (3^{2} â€“ (4i)^{2})

= [30-40i+30i-40i^{2}] / (9 â€“ 16i^{2})

= [30-10i-40(-1)] / (9-16(-1))

= [70-10i]/25

Now, equating Real and Imaginary parts on both sides we get

3x = 70/25 and -2y = -10/25

x = 70/75 and y = 1/5

x = 14/15 and y = 1/5

âˆ´Â The real values of x and y are 14/15, 1/5

(4+2i) x-3i-3 + (9-7i)y = 10i

(4x+9y-3) + i(2x-7y-3) = 10i

Now, equating Real and Imaginary parts on both sides we get,

4x+9y-3 = 0 â€¦ (i)

And 2x-7y-3 = 10

2x-7y = 13 â€¦ (ii)

Multiply (i) by 7 and (ii) by 9 and add

On solving these equations we get

28x + 18x + 63y â€“ 63y = 117 + 21

46x = 117 + 21

46x = 138

x = 138/46

= 3

Substitute the value of x in (i) we get,

4x+9y-3 = 0

9y = -9

y = -9/9

= -1

x = 3 and y = -1

âˆ´Â The real values of x and y are 3 and -1

**(iv) **(1 + i) (x + iy) = 2 â€“ 5i

Given:

(1 + i) (x + iy) = 2 â€“ 5i

Divide with (1+i) on both the sides we get,

(x + iy) = (2 â€“ 5i)/(1+i)

Multiply and divide by (1-i)

= (2 â€“ 5i)/(1+i) Ã— (1-i)/(1-i)

= [2(1-i) â€“ 5i (1-i)] / (1^{2} â€“ i^{2})

= [2 â€“ 7i + 5(-1)] / 2 [since, i^{2 }= -1]

= (-3-7i)/2

Now, equating Real and Imaginary parts on both sides we get

x = -3/2 and y = -7/2

âˆ´Â Thee real values of x and y are -3/2, -7/2

**3. Find the conjugates of the following complex numbers:**

**(i) 4 â€“ 5i**

**(ii) 1 / (3 + 5i)**

**(iii) 1 / (1 + i)**

**(iv) (3 – i) ^{2} / (2 + i)**

**(v) [(1 + i) (2 + i)] / (3 + i)**

**(vi) [(3 â€“ 2i) (2 + 3i)] / [(1 + 2i) (2 – i)]**

**Solution:**

**(i) **4 â€“ 5i

Given:

4 â€“ 5i

We know the conjugate of a complex number (a + ib) is (a – ib)

So,

âˆ´ The conjugate of (4 â€“ 5i) is (4 + 5i)

**(ii) **1 / (3 + 5i)

Given:

1 / (3 + 5i)

Since the given complex number is not in the standard form of (a + ib)

Let us convert to standard form by multiplying and dividing with (3 â€“ 5i)

We get,

We know the conjugate of a complex number (a + ib) is (a – ib)

So,

âˆ´ The conjugate of (3 â€“ 5i)/34 is (3 + 5i)/34

**(iii) **1 / (1 + i)

Given:

1 / (1 + i)

Since the given complex number is not in the standard form of (a + ib)

Let us convert to standard form by multiplying and dividing with (1 â€“ i)

We get,

We know the conjugate of a complex number (a + ib) is (a – ib)

So,

âˆ´ The conjugate of (1-i)/2 is (1+i)/2

**(iv) **(3 – i)^{2} / (2 + i)

Given:

(3 – i)^{2} / (2 + i)

Since the given complex number is not in the standard form of (a + ib)

Let us convert to standard form,

We know the conjugate of a complex number (a + ib) is (a – ib)

So,

âˆ´ The conjugate of (2 â€“ 4i) is (2 + 4i)

**(v) **[(1 + i) (2 + i)] / (3 + i)

Given:

[(1 + i) (2 + i)] / (3 + i)Since the given complex number is not in the standard form of (a + ib)

Let us convert to standard form,

We know the conjugate of a complex number (a + ib) is (a – ib)

So,

âˆ´ The conjugate of (3 + 4i)/5 is (3 â€“ 4i)/5

**(vi) **[(3 â€“ 2i) (2 + 3i)] / [(1 + 2i) (2 – i)]

Given:

[(3 â€“ 2i) (2 + 3i)] / [(1 + 2i) (2 – i)]Since the given complex number is not in the standard form of (a + ib)

Let us convert to standard form,

We know the conjugate of a complex number (a + ib) is (a – ib)

So,

âˆ´ The conjugate of (63 – 16i)/25 is (63 + 16i)/25

**4. Find the multiplicative inverse of the following complex numbers:**

**(i) 1 â€“ i**

**(ii) (1 + i âˆš3) ^{2}**

**(iii) 4 â€“ 3i**

**(iv) âˆš5 + 3i**

**Solution:**

**(i) **1 â€“ i

Given:

1 â€“ i

We know the multiplicative inverse of a complex number (Z) is Z^{-1} or 1/Z

So,

âˆ´ The multiplicative inverse of (1 – i) is (1 + i)/2

**(ii) **(1 + i âˆš3)^{2}

Given:

(1 + i âˆš3)^{2}

Z = (1 + i âˆš3)^{2}

= 1^{2} + (i âˆš3)^{2} + 2 (1) (iâˆš3)

= 1 + 3i^{2} + 2 iâˆš3

= 1 + 3(-1) + 2 iâˆš3 [since, i^{2} = -1]

= 1 â€“ 3 + 2 iâˆš3

= -2 + 2 iâˆš3

We know the multiplicative inverse of a complex number (Z) is Z^{-1} or 1/Z

So,

Z = -2 + 2 iâˆš3

âˆ´ The multiplicative inverse of (1 + iâˆš3)^{2} is (-1-iâˆš3)/8

**(iii) **4 â€“ 3i

Given:

4 â€“ 3i

We know the multiplicative inverse of a complex number (Z) is Z^{-1} or 1/Z

So,

Z = 4 â€“ 3i

âˆ´ The multiplicative inverse of (4 â€“ 3i) is (4 + 3i)/25

**(iv) **âˆš5 + 3i

Given:

âˆš5 + 3i

We know the multiplicative inverse of a complex number (Z) is Z^{-1} or 1/Z

So,

Z = âˆš5 + 3i

âˆ´ The multiplicative inverse of (âˆš5 + 3i) is (âˆš5 – 3i)/14

**6. If z _{1} = (2 – i), z_{2} = (-2 + i), find**

**Solution:**

Given:

z_{1} = (2 – i) and z_{2} = (-2 + i)

**7. Find the modulus of [(1 + i)/(1 – i)] â€“ [(1 – i)/(1 + i)]**

**Solution:**

Given:

[(1 + i)/(1 – i)] â€“ [(1 – i)/(1 + i)]So,

Z = [(1 + i)/(1 – i)] â€“ [(1 – i)/(1 + i)]

Let us simplify, we get

= [(1+i) (1+i) â€“ (1-i) (1-i)] / (1^{2} â€“ i^{2})

= [1^{2} + i^{2} + 2(1)(i) â€“ (1^{2} + i^{2} â€“ 2(1)(i))] / (1 â€“ (-1)) [Since, i^{2} = -1]

= 4i/2

= 2i

We know that for a complex numberÂ Z = (a+ib) itâ€™s magnitude is given by |z| = **âˆš**(a^{2} + b^{2})

So,

|Z| = **âˆš**(0^{2} + 2^{2})

= 2

âˆ´ The modulus of [(1 + i)/(1 – i)] â€“ [(1 – i)/(1 + i)] is 2.

**8. If x + iy = (a+ib)/(a-ib), prove that x ^{2} + y^{2} = 1**

**Solution:**

Given:

x + iy = (a+ib)/(a-ib)

We know that for a complex numberÂ Z = (a+ib) itâ€™s magnitude is given by |z| = **âˆš**(a^{2} + b^{2})

So,

|a/b| is |a| / |b|

Applying Modulus on both sides we get,

**9. Find the least positive integral value of n for which [(1+i)/(1-i)] ^{n} is real.**

**Solution:**

Given:

[(1+i)/(1-i)]^{n}

Z = [(1+i)/(1-i)]^{n}

Now let us multiply and divide by (1+i), we get

= i [which is not real]

For n = 2, we have

[(1+i)/(1-i)]^{2}= i

^{2}

= -1 [which is real]

So, the smallest positive integral â€˜nâ€™ that can makeÂ [(1+i)/(1-i)]^{n}Â real is 2.

âˆ´Â The smallest positive integral value of â€˜nâ€™ is 2.

**10. Find the real values of Î¸ for which the complex number (1 + i cos Î¸) / (1 â€“ 2i cos Î¸) is purely real.**

**Solution:**

Given:

(1 + i cos Î¸) / (1 â€“ 2i cos Î¸)

Z = (1 + i cos Î¸) / (1 â€“ 2i cos Î¸)

Let us multiply and divide by (1 + 2i cos Î¸)

For a complex number to be purely real, the imaginary part should be equal to zero.

So,

3cos Î¸ = 0 (since,Â 1 + 4cos^{2}Î¸ â‰¥ 1)

cos Î¸ = 0

cos Î¸ = cos Ï€/2

Î¸ = [(2n+1)Ï€] / 2, for n âˆˆ Z

= 2nÏ€ **Â± **Ï€/2, for n âˆˆ Z

âˆ´Â The values of Î¸ to get the complex number to be purely real is 2nÏ€ **Â± **Ï€/2, for n âˆˆ Z

**11. Find the smallest positive integer value ofÂ nÂ for which (1+i) ^{ n} / (1-i)^{ n-2} is a real number.**

**Solution:**

Given:

(1+i)^{ n} / (1-i)^{ n-2}

Z = (1+i)^{ n} / (1-i)^{ n-2}

Let us multiply and divide by (1 – i)^{2}

For n = 1,

Z = -2i^{1+1}

= -2i^{2}

= 2, which is a real number.

âˆ´Â The smallest positive integer value of n is 1.

**12. If [(1+i)/(1-i)] ^{3} â€“ [(1-i)/(1+i)]^{3} = x + iy, find (x, y)**

**Solution:**

Given:

[(1+i)/(1-i)]^{3}â€“ [(1-i)/(1+i)]

^{3}= x + iy

Let us rationalize the denominator, we get

i^{3}â€“(-i)^{3 }= x + iy

2i^{3 }= x + iy

2i^{2}.i = x + iy

2(-1)I = x + iy

-2i = x + iy

Equating Real and Imaginary parts on both sides we get

x = 0 and y = -2

âˆ´Â The values of x and y are 0 and -2.

**13. If (1+i) ^{2} / (2-i) = x + iy, find x + y**

**Solution:**

Given:

(1+i)^{2} / (2-i) = x + iy

Upon expansion we get,

Let us equate real and imaginary parts on both sides we get,

x = -2/5 and y = 4/5

so,

x + y = -2/5 + 4/5

= (-2+4)/5

= 2/5

âˆ´Â The value of (x + y) is 2/5

EXERCISE 13.3 PAGE NO: 13.39

**1. Find the square root of the following complex numbers.**

**(i) â€“ 5 + 12i**

**(ii) -7 – 24i**

**(iii) 1 – i**

**(iv) – 8 – 6i**

**(v) 8 â€“ 15i**

**(vi) -11 – 60âˆš-1**

**(vii) 1 + 4âˆš-3**

**(viii) 4i**

**(ix) -i**

**Solution:**

**(i)** â€“ 5 + 12i

Given:

â€“ 5 + 12i

We know, Z = a + ib

So, **âˆš**(a + ib) = **âˆš**(-5+12i)

Here, b > 0

Let us simplify now,

âˆ´ Square root of (â€“ 5 + 12i) is **Â±**[2 + 3i]

**(ii)** -7 â€“ 24i

Given:

-7 â€“ 24i

We know, Z = -7 â€“ 24i

So, **âˆš**(a + ib) = âˆš(-7 â€“ 24i)

Here, b < 0

Let us simplify now,

âˆ´ Square root of (-7 â€“ 24i) is **Â± **[3 – 4i]

**(iii)** 1 – i

Given:

1 â€“ i

We know, Z = (1 – i)

So, **âˆš**(a + ib) = **âˆš**(1 – i)

Here, b < 0

Let us simplify now,

âˆ´ Square root of (1 – i) is **Â± **[(**âˆš**(**âˆš**2+1)/2) â€“ i (**âˆš**(**âˆš**2-1)/2)]

**(iv)** -8 -6i

Given:

-8 -6i

We know, Z = -8 -6i

So, **âˆš**(a + ib) = -8 -6i

Here, b < 0

Let us simplify now,

= [1^{1/2} â€“ i 9^{1/2}]

= **Â±** [1 â€“ 3i]

âˆ´ Square root of (-8 -6i) is **Â±** [1 â€“ 3i]

**(v)** 8 â€“ 15i

Given:

8 â€“ 15i

We know, Z = 8 â€“ 15i

So, **âˆš**(a + ib) = 8 â€“ 15i

Here, b < 0

Let us simplify now,

âˆ´ Square root of (8 â€“ 15i) is **Â±** 1/**âˆš**2 (5 â€“ 3i)

**(vi)** -11 – 60**âˆš**-1

Given:

-11 – 60**âˆš**-1

We know, Z = -11 – 60**âˆš**-1

So, **âˆš**(a + ib) = -11 – 60**âˆš**-1

= -11 â€“ 60i

Here, b < 0

Let us simplify now,

âˆ´ Square root of (-11 – 60**âˆš**-1) is **Â±** (5 â€“ 6i)

**(vii)** 1 + 4**âˆš**-3

Given:

1 + 4**âˆš**-3

We know, Z = 1 + 4**âˆš**-3

So, **âˆš**(a + ib) = 1 + 4**âˆš**-3

= 1 + 4(**âˆš**3) (**âˆš**-1)

= 1 + 4**âˆš**3i

Here, b > 0

Let us simplify now,

âˆ´ Square root of (1 + 4**âˆš**-3) is **Â±** (2 + **âˆš**3i)

**(viii)** 4i

Given:

4i

We know, Z = 4i

So, **âˆš**(a + ib) = 4i

Here, b > 0

Let us simplify now,

âˆ´ Square root of 4i is **Â±** **âˆš**2 (1 + i)

**(ix)** â€“i

Given:

-i

We know, Z = -i

So, **âˆš**(a + ib) = -i

Here, b < 0

Let us simplify now,

âˆ´ Square root of â€“i is **Â±** 1/**âˆš**2 (1 – i)

EXERCISE 13.4 PAGE NO: 13.57

**1. Find the modulus and arguments of the following complex numbers and hence express each of them in the polar form:**

**(i) 1 + i**

**(ii) âˆš3 + i**

**(iii) 1 – i**

**(iv) (1 – i) / (1 + i)**

**(v) 1/(1 + i)**

**(vi) (1 + 2i) / (1 â€“ 3i)**

**(vii) sin 120 ^{o} â€“ i cos 120^{o}**

**(viii) -16 / (1 + iâˆš3)**

**Solution:**

We know that the polar form of a complex number Z = x + iy is given by Z = |Z| (cos Î¸ + i sin Î¸)

Where,

|Z| = modulus of complex number = **âˆš**(x^{2} + y^{2})

Î¸ = arg (z) = argument of complex number = tan^{-1} (|y| / |x|)

**(i) **1 + i

Given: Z = 1 + i

So now,

|Z| = **âˆš**(x^{2} + y^{2})

= **âˆš**(1^{2} + 1^{2})

= **âˆš**(1 + 1)

= **âˆš**2

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} (1 / 1)

= tan^{-1} 1

Since x > 0, y > 0 complex number lies in 1^{st}Â quadrant and the value of Î¸ is 0^{0}â‰¤Î¸â‰¤90^{0}.

Î¸ = Ï€/4

Z = **âˆš**2 (cos (Ï€/4) + i sin (Ï€/4))

âˆ´ Polar form of (1 + i) is **âˆš**2 (cos (Ï€/4) + i sin (Ï€/4))

**(ii)** âˆš3 + i

Given: Z = âˆš3 + i

So now,

|Z| = **âˆš**(x^{2} + y^{2})

= **âˆš**((âˆš3)^{2} + 1^{2})

= **âˆš**(3 + 1)

= **âˆš**4

= 2

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} (1 / âˆš3)

Since x > 0, y > 0 complex number lies in 1^{st}Â quadrant and the value of Î¸ is 0^{0}â‰¤Î¸â‰¤90^{0}.

Î¸ = Ï€/6

Z = 2 (cos (Ï€/6) + i sin (Ï€/6))

âˆ´ Polar form of (âˆš3 + i) is 2 (cos (Ï€/6) + i sin (Ï€/6))

**(iii)** 1 – i

Given: Z = 1 – i

So now,

|Z| = **âˆš**(x^{2} + y^{2})

= **âˆš**(1^{2} + (-1)^{2})

= **âˆš**(1 + 1)

= **âˆš**2

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} (1 / 1)

= tan^{-1} 1

Since x > 0, y < 0 complex number lies in 4^{th}Â quadrant and the value of Î¸ is -90^{0}â‰¤Î¸â‰¤0^{0}.

Î¸ = -Ï€/4

Z = **âˆš**2 (cos (-Ï€/4) + i sin (-Ï€/4))

= **âˆš**2 (cos (Ï€/4) – i sin (Ï€/4))

âˆ´ Polar form of (1 – i) is **âˆš**2 (cos (Ï€/4) – i sin (Ï€/4))

**(iv) **(1 – i) / (1 + i)

Given: Z = (1 – i) / (1 + i)

Let us multiply and divide by (1 – i), we get

= 0 â€“ i

So now,

|Z| = **âˆš**(x^{2} + y^{2})

= **âˆš**(0^{2} + (-1)^{2})

= **âˆš**(0 + 1)

= **âˆš**1

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} (1 / 0)

= tan^{-1} âˆž

Since x â‰¥ 0, y < 0 complex number lies in 4^{th}Â quadrant and the value of Î¸ is -90^{0}â‰¤Î¸â‰¤0^{0}.

Î¸ = -Ï€/2

Z = 1 (cos (-Ï€/2) + i sin (-Ï€/2))

= 1 (cos (Ï€/2) – i sin (Ï€/2))

âˆ´ Polar form of (1 – i) / (1 + i) is 1 (cos (Ï€/2) – i sin (Ï€/2))

**(v) **1/(1 + i)

Given: Z = 1 / (1 + i)

Let us multiply and divide by (1 – i), we get

So now,

|Z| = **âˆš**(x^{2} + y^{2})

= **âˆš**((1/2)^{2} + (-1/2)^{2})

= **âˆš**(1/4 + 1/4)

= **âˆš**(2/4)

= 1/**âˆš**2

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} ((1/2) / (1/2))

= tan^{-1} 1

Since x > 0, y < 0 complex number lies in 4^{th}Â quadrant and the value of Î¸ is -90^{0}â‰¤Î¸â‰¤0^{0}.

Î¸ = -Ï€/4

Z = 1/**âˆš**2 (cos (-Ï€/4) + i sin (-Ï€/4))

= 1/**âˆš**2 (cos (Ï€/4) – i sin (Ï€/4))

âˆ´ Polar form of 1/(1 + i) is 1/**âˆš**2 (cos (Ï€/4) – i sin (Ï€/4))

**(vi) **(1 + 2i) / (1 â€“ 3i)

Given: Z = (1 + 2i) / (1 â€“ 3i)

Let us multiply and divide by (1 + 3i), we get

So now,

|Z| = **âˆš**(x^{2} + y^{2})

= **âˆš**((-1/2)^{2} + (1/2)^{2})

= **âˆš**(1/4 + 1/4)

= **âˆš**(2/4)

= 1/**âˆš**2

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} ((1/2) / (1/2))

= tan^{-1} 1

Since x < 0, y > 0 complex number lies in 2^{nd}Â quadrant and the value of Î¸ is 90^{0}â‰¤Î¸â‰¤180^{0}.

Î¸ = 3Ï€/4

Z = 1/**âˆš**2 (cos (3Ï€/4) + i sin (3Ï€/4))

âˆ´ Polar form of (1 + 2i) / (1 â€“ 3i) is 1/**âˆš**2 (cos (3Ï€/4) + i sin (3Ï€/4))

**(vii) **sin 120^{o} â€“ i cos 120^{o}

Given: Z = sin 120^{o} â€“ i cos 120^{o}

= **âˆš**3/2 â€“ i (-1/2)

= **âˆš**3/2 + i (1/2)

So now,

|Z| = **âˆš**(x^{2} + y^{2})

= **âˆš**((**âˆš**3/2)^{2} + (1/2)^{2})

= **âˆš**(3/4 + 1/4)

= **âˆš**(4/4)

= **âˆš**1

= 1

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} ((1/2) / (**âˆš**3/2))

= tan^{-1} (1/**âˆš**3)

Since x > 0, y > 0 complex number lies in 1^{st}Â quadrant and the value of Î¸ is 0^{0}â‰¤Î¸â‰¤90^{0}.

Î¸ = Ï€/6

Z = 1 (cos (Ï€/6) + i sin (Ï€/6))

âˆ´ Polar form of **âˆš**3/2 + i (1/2) is 1 (cos (Ï€/6) + i sin (Ï€/6))

**(viii) **-16 / (1 + iâˆš3)

Given: Z = -16 / (1 + iâˆš3)

Let us multiply and divide by (1 – i**âˆš**3), we get

So now,

|Z| = **âˆš**(x^{2} + y^{2})

= **âˆš**((-4)^{2} + (4**âˆš**3)^{2})

= **âˆš**(16 + 48)

= **âˆš**(64)

= 8

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} ((4**âˆš**3) / 4)

= tan^{-1} (**âˆš**3)

Since x < 0, y > 0 complex number lies in 2^{nd}Â quadrant and the value of Î¸ is 90^{0}â‰¤Î¸â‰¤180^{0}.

Î¸ = 2Ï€/3

Z = 8 (cos (2Ï€/3) + i sin (2Ï€/3))

âˆ´ Polar form of -16 / (1 + iâˆš3) is 8 (cos (2Ï€/3) + i sin (2Ï€/3))

**2. Write (i ^{25})^{3}Â in polar form.**

**Solution:**

Given: Z = (i^{25})^{3}

= i^{75}

= i^{74}. i

= (i^{2})^{37}. i

= (-1)^{37}. i

= (-1). i

= – i

= 0 â€“ i

So now,

|Z| = **âˆš**(x^{2} + y^{2})

= **âˆš**(0^{2} + (-1)^{2})

= **âˆš**(0 + 1)

= **âˆš**1

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} (1 / 0)

= tan^{-1} âˆž

Since x â‰¥ 0, y < 0 complex number lies in 4^{th}Â quadrant and the value of Î¸ is -90^{0}â‰¤Î¸â‰¤0^{0}.

Î¸ = -Ï€/2

Z = 1 (cos (-Ï€/2) + i sin (-Ï€/2))

= 1 (cos (Ï€/2) – i sin (Ï€/2))

âˆ´ Polar form of (i^{25})^{3}Â is 1 (cos (Ï€/2) – i sin (Ï€/2))

**3.** **Express the following complex numbers in the formÂ r (cos Î¸ + i sin Î¸):**

**(i) 1 + i tan Î±**

**(ii) tan Î± â€“ i **

**(iii) 1 â€“ sin Î± + i cos Î±**

**(iv) (1 – i) / (cos Ï€/3 + i sin Ï€/3)**

**Solution:**

**(i) **1 + i tan Î±

Given: Z = 1 + i tan Î±

We know that the polar form of a complex number Z = x + iy is given by Z = |Z| (cos Î¸ + i sin Î¸)

Where,

|Z| = modulus of complex number = **âˆš**(x^{2} + y^{2})

Î¸ = arg (z) = argument of complex number = tan^{-1} (|y| / |x|)

We also know that tan Î± is a periodic function with period Ï€.

So Î± is lying in the interval [0, Ï€/2) âˆª (Ï€/2, Ï€].

Let us consider case 1:

Î± âˆˆ [0, Ï€/2)

So now,

|Z| = r = **âˆš**(x^{2} + y^{2})

= **âˆš**(1^{2} + tan^{2} Î±)

= **âˆš**( sec^{2} Î±)

= |sec Î±| since, sec Î± is positive in the interval [0, Ï€/2)

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} (tan Î± / 1)

= tan^{-1} (tan Î±)

= Î± since, tan Î± is positive in the interval [0, Ï€/2)

âˆ´ Polar form is Z = sec Î± (cos Î± + i sin Î±)

Let us consider case 2:

Î± âˆˆ (Ï€/2, Ï€]

So now,

|Z| = r = **âˆš**(x^{2} + y^{2})

= **âˆš**(1^{2} + tan^{2} Î±)

= **âˆš**( sec^{2} Î±)

= |sec Î±|

= – sec Î± since, sec Î± is negative in the interval (Ï€/2, Ï€]

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} (tan Î± / 1)

= tan^{-1} (tan Î±)

= -Ï€ + Î± since, tan Î± is negative in the interval (Ï€/2, Ï€]

Î¸ = -Ï€ + Î± [since, Î¸ lies in 4^{th} quadrant]

Z = -sec Î± (cos (Î± – Ï€) + i sin (Î± – Ï€))

âˆ´ Polar form is Z = -sec Î± (cos (Î± – Ï€) + i sin (Î± – Ï€))

**(ii) **tan Î± â€“ i

Given: Z = tan Î± â€“ i

We know that the polar form of a complex number Z = x + iy is given by Z = |Z| (cos Î¸ + i sin Î¸)

Where,

|Z| = modulus of complex number = **âˆš**(x^{2} + y^{2})

Î¸ = arg (z) = argument of complex number = tan^{-1} (|y| / |x|)

We also know that tan Î± is a periodic function with period Ï€.

So Î± is lying in the interval [0, Ï€/2) âˆª (Ï€/2, Ï€].

Let us consider case 1:

Î± âˆˆ [0, Ï€/2)

So now,

|Z| = r = **âˆš**(x^{2} + y^{2})

= **âˆš**(tan^{2} Î± + 1^{2})

= **âˆš**( sec^{2} Î±)

= |sec Î±| since, sec Î± is positive in the interval [0, Ï€/2)

= sec Î±

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} (1/tan Î±)

= tan^{-1} (cot Î±) since, cot Î± is positive in the interval [0, Ï€/2)

= Î± – Ï€/2 [since, Î¸ lies in 4^{th} quadrant]

Z = sec Î± (cos (Î± – Ï€/2) + i sin (Î± – Ï€/2))

âˆ´ Polar form is Z = sec Î± (cos (Î± – Ï€/2) + i sin (Î± – Ï€/2))

Let us consider case 2:

Î± âˆˆ (Ï€/2, Ï€]

So now,

|Z| = r = **âˆš**(x^{2} + y^{2})

= **âˆš**(tan^{2} Î± + 1^{2})

= **âˆš**( sec^{2} Î±)

= |sec Î±|

= – sec Î± since, sec Î± is negative in the interval (Ï€/2, Ï€]

Î¸ = tan^{-1} (|y| / |x|)

= tan^{-1} (1/tan Î±)

= tan^{-1} (cot Î±)

= Ï€/2 + Î± since, cot Î± is negative in the interval (Ï€/2, Ï€]

Î¸ = Ï€/2 + Î± [since, Î¸ lies in 3^{th} quadrant]

Z = -sec Î± (cos (Ï€/2 + Î±) + i sin (Ï€/2 + Î±))

âˆ´ Polar form is Z = -sec Î± (cos (Ï€/2 + Î±) + i sin (Ï€/2 + Î±))

**(iii)** 1 â€“ sin Î± + i cos Î±

Given: Z = 1 â€“ sin Î± + i cos Î±

By using the formulas,

Sin^{2} Î¸ + cos^{2} Î¸ = 1

Sin 2Î¸ = 2 sin Î¸ cos Î¸

Cos 2Î¸ = cos^{2} Î¸ â€“ sin^{2} Î¸

So,

z= (sin^{2}(Î±/2) + cos^{2}(Î±/2) – 2 sin(Î±/2) cos(Î±/2)) + *i* (cos^{2}(Î±/2) – sin^{2}(Î±/2))

=Â (cos(Î±/2) – sin(Î±/2))^{2}Â + *i* (cos^{2}(Î±/2) – sin^{2}(Î±/2))

We know that the polar form of a complex number Z = x + iy is given by Z = |Z| (cos Î¸ + i sin Î¸)

Where,

|Z| = modulus of complex number = **âˆš**(x^{2} + y^{2})

Î¸ = arg (z) = argument of complex number = tan^{-1} (|y| / |x|)

Now,

We know that sine and cosine functions are periodic with period 2Ï€

Here we have 3 intervals:

0 â‰¤ Î± â‰¤ Ï€/2

Ï€/2 â‰¤ Î± â‰¤ 3Ï€/2

3Ï€/2 â‰¤ Î± â‰¤ 2Ï€

Let us consider case 1:

In the interval 0 â‰¤ Î± â‰¤ Ï€/2

Cos (Î±/2) > sin (Î±/2) and also 0 < Ï€/4 + Î±/2 < Ï€/2

So,

âˆ´ Polar form is Z = **âˆš**2 (cos (Î±/2) â€“ sin (Î±/2)) (cos (Ï€/4 + Î±/2) + i sin (Ï€/4 + Î±/2))

Let us consider case 2:

In the interval Ï€/2 â‰¤ Î± â‰¤ 3Ï€/2

Cos (Î±/2) < sin (Î±/2) and also Ï€/2 < Ï€/4 + Î±/2 < Ï€

So,

Since, (1 – sin Î±) > 0 and cos Î± < 0 [Z lies in 4^{th} quadrant]

= Î±/2 – 3Ï€/4

âˆ´ Polar form is Z = –**âˆš**2 (cos (Î±/2) â€“ sin (Î±/2)) (cos (Î±/2 – 3Ï€/4) + i sin (Î±/2 – 3Ï€/4))

Let us consider case 3:

In the interval 3Ï€/2 â‰¤ Î± â‰¤ 2Ï€

Cos (Î±/2) < sin (Î±/2) and also Ï€ < Ï€/4 + Î±/2 < 5Ï€/4

So,

Î¸ = tan^{-1} (tan (Ï€/4 + Î±/2))

= Ï€ â€“ (Ï€/4 + Î±/2) [since, Î¸ lies in 1st quadrant and tanâ€™s period is Ï€]

= Î±/2 – 3Ï€/4

âˆ´ Polar form is Z = –**âˆš**2 (cos (Î±/2) â€“ sin (Î±/2)) (cos (Î±/2 – 3Ï€/4) + i sin (Î±/2 – 3Ï€/4))

**(iv) **(1 – i) / (cos Ï€/3 + i sin Ï€/3)

Given: Z = (1 – i) / (cos Ï€/3 + i sin Ï€/3)

Let us multiply and divide by (1 – i**âˆš**3), we get

We know that the polar form of a complex number Z = x + iy is given by Z = |Z| (cos Î¸ + i sin Î¸)

Where,

|Z| = modulus of complex number = **âˆš**(x^{2} + y^{2})

Î¸ = arg (z) = argument of complex number = tan^{-1} (|y| / |x|)

Now,

Since x < 0, y < 0 complex number lies in 3^{rd}Â quadrant and the value of Î¸ is 180^{0}â‰¤Î¸â‰¤-90^{0}.

= tan^{-1} (2 + **âˆš**3)

= -7Ï€/12

Z = **âˆš**2 (cos (-7Ï€/12) + i sin (-7Ï€/12))

= **âˆš**2 (cos (7Ï€/12) – i sin (7Ï€/12))

âˆ´ Polar form of (1 – i) / (cos Ï€/3 + i sin Ï€/3) is **âˆš**2 (cos (7Ï€/12) – i sin (7Ï€/12))

**4. If z _{1}Â and z_{2}Â are two complex number such that |z_{1}| = |z_{2}| and arg (z_{1}) + arg (z_{2}) = Ï€, then show that **

**Solution:**

Given:

|z_{1}| = |z_{2}| and arg (z_{1}) + arg (z_{2}) = Ï€

Let us assume arg (z_{1}) = Î¸

arg (z_{2}) = Ï€ – Î¸

We know that in the polar form, z = |z| (cos Î¸ + i sin Î¸)

z_{1 }= |z_{1}| (cos Î¸ + i sin Î¸) â€¦â€¦â€¦â€¦. (i)

z_{2 }= |z_{2}| (cos (Ï€ – Î¸) + i sin (Ï€ – Î¸))

= |z_{2}| (-cos Î¸ + i sin Î¸)

= – |z_{2}| (cos Î¸ â€“ i sin Î¸)

Now let us find the conjugate of

= – |z_{2}| (cos Î¸ + i sin Î¸) â€¦â€¦ (ii)Â (since, \(|\overline{{Z}_{2}}|=|Z_{2}|\))

Now,

z_{1} /

= [|z_{1}| (cos Î¸ + i sin Î¸)] / [-|z_{2}| (cos Î¸ + i sin Î¸)]

= – |z_{1}| / |z_{2}| [since, |z_{1}| = |z_{2}|]

= -1

When we cross multiply we get,

z_{1} = –

Hence proved.

**5. If z _{1}, z_{2}Â and z_{3}, z_{4}Â are two pairs of conjugate complex numbers, prove that arg (z_{1}/z_{4}) + arg (z_{2}/z_{3}) = 0**

**Solution:**

Given:

Hence proved.

**6. Express sin Ï€/5 + i (1 â€“ cos Ï€/5) in polar form.**

**Solution:**

Given:

Z = sin Ï€/5 + i (1 â€“ cos Ï€/5)

By using the formula,

sin 2Î¸ = 2 sin Î¸ cos Î¸

1- cos 2Î¸ = 2 sin^{2} Î¸

So,

Z = 2 sin Ï€/10 cos Ï€/10 + i (2 sin^{2} Ï€/10)

= 2 sin Ï€/10 (cos Ï€/10 + i sin Ï€/10)

âˆ´ The polar form of sin Ï€/5 + i (1 â€“ cos Ï€/5) is 2 sin Ï€/10 (cos Ï€/10 + i sin Ï€/10)