Problems related to imaginary quantities (the square root of a negative real number is called an imaginary quantity or an imaginary number), addition, subtraction, multiplication, division of complex numbers using its properties, the conjugate of a complex number, modulus and reciprocal of complex numbers are discussed in this exercise. Expert tutors at BYJUâ€™S have formulated the solutions in a step by step manner where every student can understand the concepts clearly. Students can refer to RD Sharma Class 11 Solutions for Maths and can also download the pdf from the links given below.

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**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