Tamil Nadu Board of Secondary EducationHSC Science Class 12th

# Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 2 - Complex Numbers [Latest edition]

## Chapter 2: Complex Numbers

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Exercise 2.5Exercise 2.6Exercise 2.7Exercise 2.8Exercise 2.9
Exercise 2.1 [Page 54]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Complex NumbersExercise 2.1 [Page 54]

Exercise 2.1 | Q 1 | Page 54

Simplify the following:

i1947 + i1950

Exercise 2.1 | Q 2 | Page 54

Simplify the following:

i1948 – i–1869

Exercise 2.1 | Q 3 | Page 54

Simplify the following:

sum_("n" = 1)^12 "i"^"n"

Exercise 2.1 | Q 4 | Page 54

Simplify the following:

"i"^59 + 1/"i"^59

Exercise 2.1 | Q 5 | Page 54

Simplify the following:

i i2 i3 ... i2000

Exercise 2.1 | Q 6 | Page 54

Simplify the following:

sum_("n" = 1)^10 "i"^("n" + 50)

Exercise 2.2 [Page 58]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Complex NumbersExercise 2.2 [Page 58]

Exercise 2.2 | Q 1. (i) | Page 58

Evaluate the following if z = 5 – 2i and w = – 1 + 3i

z + w

Exercise 2.2 | Q 1. (ii) | Page 58

Evaluate the following if z = 5 – 2i and w = – 1 + 3i

z – iw

Exercise 2.2 | Q 1. (iii) | Page 58

Evaluate the following if z = 5 – 2i and w = – 1 + 3i

2z + 3w

Exercise 2.2 | Q 1. (iv) | Page 58

Evaluate the following if z = 5 – 2i and w = – 1 + 3i

zw

Exercise 2.2 | Q 1. (v) | Page 58

Evaluate the following if z = 5 – 2i and w = – 1 + 3i

z2 + 2zw + w2

Exercise 2.2 | Q 1. (vi) | Page 58

Evaluate the following if z = 5 – 2i and w = – 1 + 3i

(z + w)2

Exercise 2.2 | Q 2. (i) | Page 58

Given the complex number z = 2 + 3i, represent the complex numbers in Argand diagram

z, iz and z + iz

Exercise 2.2 | Q 2. (ii) | Page 58

Given the complex number z = 2 + 3i, represent the complex numbers in Argand diagram

z, – iz and z – iz

Exercise 2.2 | Q 3 | Page 58

Find the values of the real numbers x and y, if the complex numbers

(3 – i)x – (2 – i)y + 2i + 5 and 2x + (– 1 + 2i)y + 3 + 2i are equal

Exercise 2.3 [Page 60]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Complex NumbersExercise 2.3 [Page 60]

Exercise 2.3 | Q 1. (i) | Page 60

If z1 = 1 – 3i, z2 = – 4i, and z3 = 5, show that (z1 + z2) + z3 = z1 + (z2 + z3)

Exercise 2.3 | Q 1. (ii) | Page 60

If z1 = 1 – 3i, z2 = – 4i, and z3 = 5, show that (z1 z2)z3 = z1(z2 z3)

Exercise 2.3 | Q 2. (i) | Page 60

If z1 = 3, z2 = 7i, and z3 = 5 + 4i, show that z1(z2 + z3) = z1z2 + z1z3

Exercise 2.3 | Q 2. (ii) | Page 60

If z1 = 3, z2 = 7i, and z3 = 5 + 4i, show that (z1 + z2)z3 = z1z3 + z2z3

Exercise 2.3 | Q 3 | Page 60

If z1 = 2 + 5i, z2 = – 3 – 4i, and z3 = 1 + i, find the additive and multiplicative inverse of z1, z2 and z3

Exercise 2.4 [Page 65]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Complex NumbersExercise 2.4 [Page 65]

Exercise 2.4 | Q 1. (i) | Page 65

Write the following in the rectangular form:

bar((5 + 9"i") + (2 - 4"i"))

Exercise 2.4 | Q 1. (ii) | Page 65

Write the following in the rectangular form:

(10 - 5"i")/(6 + 2"i")

Exercise 2.4 | Q 1. (iii) | Page 65

Write the following in the rectangular form:

bar(3"i") + 1/(2 - "i")

Exercise 2.4 | Q 2. (i) | Page 65

If z = x + iy, find the following in rectangular form:

"Re"(1/z)

Exercise 2.4 | Q 2. (ii) | Page 65

If z = x + iy, find the following in rectangular form:

"Re"("i"barz)

Exercise 2.4 | Q 2. (iii) | Page 65

If z = x + iy, find the following in rectangular form:

"Im"(3z + 4bar(z) - 4"i")

Exercise 2.4 | Q 3 | Page 65

If z1 = 2 – i and z2 = – 4 + 3i, find the inverse of z1, z2 and ("z"_1)/("z"_2)

Exercise 2.4 | Q 4 | Page 65

The complex numbers u v, , and w are related by 1/u = 1/v + 1/w. If v = 3 – 4i and w = 4 + 3i, find u in rectangular form

Exercise 2.4 | Q 5. (i) | Page 65

Prove the following properties:

z is real if and only if z = bar(z)

Exercise 2.4 | Q 5. (ii) | Page 65

Prove the following properties:

Re(z) = (z + bar(z))/2 and Im(z) = (z - bar(z))/(2"i")

Exercise 2.4 | Q 6. (i) | Page 65

Find the least value of the positive integer n for which (sqrt(3) + "i")^"n" real

Exercise 2.4 | Q 6. (ii) | Page 65

Find the least value of the positive integer n for which (sqrt(3) + "i")^"n" purely imaginary

Exercise 2.4 | Q 7. (i) | Page 65

Show that (2 + "i"sqrt(3))^10 - (2 - "i" sqrt(3))^10 is purely imaginary

Exercise 2.4 | Q 7. (ii) | Page 65

Show that ((19 - 7"i")/(9 + "i"))^12 + ((20 - 5"i")/(7 - 6"i"))^12 is real

Exercise 2.5 [Page 72]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Complex NumbersExercise 2.5 [Page 72]

Exercise 2.5 | Q 1. (i) | Page 72

Find the modulus of the following complex numbers

(2"i")/(3 + 4"i")

Exercise 2.5 | Q 1. (ii) | Page 72

Find the modulus of the following complex numbers

(2 - "i")/(1 + "i") + (1 - 2"i")/(1 - "i")

Exercise 2.5 | Q 1. (iii) | Page 72

Find the modulus of the following complex numbers

(1 – i)10

Exercise 2.5 | Q 1. (iv) | Page 72

Find the modulus of the following complex numbers

2i(3 – 4i)(4 – 3i)

Exercise 2.5 | Q 2 | Page 72

For any two complex numbers z1 and z2, such that |z1| = |z2| = 1 and z1 z2 ≠ -1, then show that (z_1 + z_2)/(1 + z_1 z_2) is real number

Exercise 2.5 | Q 3 | Page 72

Which one of the points 10 – 8i, 11 + 6i is closest to 1 + i

Exercise 2.5 | Q 4 | Page 72

If |z| = 3, show that 7 ≤ |z + 6 – 8i| ≤ 13

Exercise 2.5 | Q 5 | Page 72

If |z| = 1, show that 2 ≤ |z2 – 3| ≤ 4

Exercise 2.5 | Q 6 | Page 72

If |z| = 2 show that the 8 ≤ |z + 6 + 8i| ≤ 12

Exercise 2.5 | Q 7 | Page 72

If z1, z2 and z3 are three complex numbers such that |z1| = 1, |z2| = 2, |z3| = 3 and |z1 + z2 + z3| = 1, show that |9z1z+ 4z1z3 + z2z3| = 6

Exercise 2.5 | Q 8 | Page 72

If the area of the triangle formed by the vertices z, iz, and z + iz is 50 square units, find the value of |z|

Exercise 2.5 | Q 9 | Page 72

Show that the equation z^3 + 2bar(z) = 0 has five solutions

Exercise 2.5 | Q 10. (i) | Page 72

Find the square roots of 4 + 3i

Exercise 2.5 | Q 10. (ii) | Page 72

Find the square roots of – 6 + 8i

Exercise 2.5 | Q 10. (iii) | Page 72

Find the square roots of – 5 – 12i

Exercise 2.6 [Page 75]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Complex NumbersExercise 2.6 [Page 75]

Exercise 2.6 | Q 1 | Page 75

If 2 = x + iy is a complex number such that |(z - 4"i")/(z + 4"i")| = 1 show that the locus of z is real axis

Exercise 2.6 | Q 2 | Page 75

If z = x + iy is a complex number such that Im ((2z + 1)/("i"z + 1)) = 0, show that the locus of z is 2x2 + 2y2 + x – 2y = 0

Exercise 2.6 | Q 3. (i) | Page 75

Obtain the Cartesian form of the locus of z = x + iy in the following cases:

[Re(iz)]2 = 3

Exercise 2.6 | Q 3. (ii) | Page 75

Obtain the Cartesian form of the locus of z = x + iy in the following cases:

Im[(1 – i)z + 1] = 0

Exercise 2.6 | Q 3. (iii) | Page 75

Obtain the Cartesian form of the locus of z = x + iy in the following cases:

|z + i| = |z – 1|

Exercise 2.6 | Q 3. (iv) | Page 75

Obtain the Cartesian form of the locus of z = x + iy in the following cases:

bar(z) = z^-1

Exercise 2.6 | Q 4. (i) | Page 75

Show that the following equations represent a circle, and, find its centre and radius.

|z – 2 – i| = 3

Exercise 2.6 | Q 4. (ii) | Page 75

Show that the following equations represent a circle, and, find its centre and radius.

|2z + 2 – 4i| = 2

Exercise 2.6 | Q 4. (iii) | Page 75

Show that the following equations represent a circle, and, find its centre and radius.

|3z – 6 + 12i| = 8

Exercise 2.6 | Q 5. (i) | Page 75

Obtain the Cartesian equation for the locus of z = x + iy in the following cases:

|z – 4| = 16

Exercise 2.6 | Q 5. (ii) | Page 75

Obtain the Cartesian equation for the locus of z = x + iy in the following cases:

|z – 4|2 – |z – 1|2 = 16

Exercise 2.7 [Page 83]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Complex NumbersExercise 2.7 [Page 83]

Exercise 2.7 | Q 1. (i) | Page 83

Write in polar form of the following complex numbers

2 + "i" 2sqrt(3)

Exercise 2.7 | Q 1. (ii) | Page 83

Write in polar form of the following complex numbers

3 - "i"sqrt(3)

Exercise 2.7 | Q 1. (iii) | Page 83

Write in polar form of the following complex numbers

– 2 – i2

Exercise 2.7 | Q 1. (iv) | Page 83

Write in polar form of the following complex numbers

("i" - 1)/(cos  pi/3 + "i" sin  pi/3)

Exercise 2.7 | Q 2. (i) | Page 83

Find the rectangular form of the complex numbers

(cos  pi/6  "i" sin  pi/6)(cos  pi/12 + "i" sin  pi/12)

Exercise 2.7 | Q 2. (ii) | Page 83

Find the rectangular form of the complex numbers

(cos  pi/6 - "i" sin  pi/6)/(2(cos  pi/3 + "i" sin  pi/3))

Exercise 2.7 | Q 3. (i) | Page 83

If (x1 + iy1)(x2 + iy2)(x3 + iy3) ... (xn + iyn) = a + ib, show that (x_1^2 + y_1^2)(x_2^2 + y_2^2)(x_3^2 + y_3^2) ... (x_"n"^2 + y_"n"^2) = a2 + b2

Exercise 2.7 | Q 3. (ii) | Page 83

If (x1 + iy1)(x2 + iy2)(x3 + iy3) ... (xn + iyn) = a + ib, show that sum_("r" = 1)^"n" tan^-1 (y_"r"/x_"r") = tan^-1 ("b"/"a") + 2"k"pi, "k" ∈ "z"

Exercise 2.7 | Q 4 | Page 83

If (1 + z)/(1 - z) = cos 2θ + i sin 2θ, show that z = i tan θ

Exercise 2.7 | Q 5. (i) | Page 83

If cos α + cos β + cos γ = sin α + sin β + sin γ = 0, show that cos 3α + cos 3β + cos 3γ = 3 cos (α + β + γ)

Exercise 2.7 | Q 5. (ii) | Page 83

If cos α + cos β + cos γ = sin α + sin β + sin γ = 0. then show that sin 3α + sin 3β + sin 3γ = 3 sin(α + β + γ)

Exercise 2.7 | Q 6 | Page 83

If z = x + iy and arg ((z - "i")/(z + 2)) = pi/4, show that x2 + y3 + 3x – 3y + 2 = 0

Exercise 2.8 [Page 92]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Complex NumbersExercise 2.8 [Page 92]

Exercise 2.8 | Q 1 | Page 92

If to ω ≠ 1 is a cube root of unity, then show that ("a" + "b"omega + "c"omega^2)/("b" + "c"omega + "a"omega^2) + ("a" + "b"omega + "c"omega^2)/("c" + "a"omega + "a"omega^2) = – 1

Exercise 2.8 | Q 2 | Page 92

Show that (sqrt(3)/2 + "i"/2)^5 + (sqrt(3)/2 - "i"/2)^5 = - sqrt(3)

Exercise 2.8 | Q 3 | Page 92

Find the value of [(1 + sin  pi/10 + "i" cos  pi/10)/(1 + sin  pi/10 - "i" cos  pi/10)]^10

Exercise 2.8 | Q 4. (i) | Page 92

If 2 cos α = x + 1/x and 2 cos β = y + 1/y, show that x/y + y/x = 2cos(alpha − beta)

Exercise 2.8 | Q 4. (ii) | Page 92

If 2 cos α = x + 1/x and 2 cos β = y + 1/y, show that xy - 1/xy = 2"i" sin(alpha + beta)

Exercise 2.8 | Q 4. (iii) | Page 92

If 2cos α = x + 1/x and 2 cos β = y + 1/x, show that x^"m"/y^"n" - y^"n"/x^"m" = 2i sin(mα – nβ)

Exercise 2.8 | Q 4. (iv) | Page 92

If 2 cos α = x + 1/x and 2 cos β = y + 1/y, show that x^"m" y^"n" + 1/(x^"m" y^"n") = 2 cos(mα – nβ)

Exercise 2.8 | Q 5 | Page 92

Solve the equation z3 + 27 = 0

Exercise 2.8 | Q 6 | Page 92

If ω ≠ 1 is a cube root of unity, show that the roots of the equation (z – 1)3 + 8 = 0 are – 1, 1 – 2ω, 1 – 2ω2

Exercise 2.8 | Q 7 | Page 92

Find the value of sum_("k" = 1)^8 (cos  (2"k"pi)/9 + "i" sin  (2"kpi)/9)

Exercise 2.8 | Q 8. (i) | Page 92

If ω ≠ 1 is a cube root of unity, show that (1 – ω + ω2)6 + (1 + ω – ω2)6 = 128

Exercise 2.8 | Q 8. (ii) | Page 92

If ω ≠ 1 is a cube root of unity, show that (1 + ω)(1 + ω2)(1 + ω4)(1 + ω8)….. (1 + ω2n) = 1

Exercise 2.8 | Q 9. (i) | Page 92

If z = 2 – 2i, find the rotation of z by θ radians in the counterclockwise direction about the origin when θ = pi/3

Exercise 2.8 | Q 9. (ii) | Page 92

If z = 2 – 2i, find the rotation of z by θ radians in the counterclockwise direction about the origin when θ = (2pi)/3

Exercise 2.8 | Q 9. (iii) | Page 92

If z = 2 – 2i, find the rotation of z by θ radians in the counterclockwise direction about the origin when θ = (3pi)/3

Exercise 2.9 [Pages 93 - 94]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Complex NumbersExercise 2.9 [Pages 93 - 94]

#### MCQ

Exercise 2.9 | Q 1 | Page 93

Choose the correct alternative:

in + in+1+ in+2 + in+3 is

• 0

• 1

• – 1

• z

Exercise 2.9 | Q 2 | Page 93

Choose the correct alternative:

The value of sum_("n" = 1)^13 ("i"^"n" + "i"^("n" - 1)) is

• 1 + i

• i

• 1

• 0

Exercise 2.9 | Q 3 | Page 93

Choose the correct alternative:

The area of the triangle formed by the complex numbers z, iz, and z + iz in the Argand’s diagram is

• 1/2 |z|^2

• |z|^2

• 3/2 |z|^2

• 2|z|^2

Exercise 2.9 | Q 4 | Page 93

Choose the correct alternative:

The conjugate of a complex number is 1/(" - 2). Then, the complex number is

• 1/("i" + 2)

• (-1)/("i" + 2)

• (-1)/("i" - 2)

• 1/("i" - 2)

Exercise 2.9 | Q 5 | Page 93

Choose the correct alternative:

If = ((sqrt(3) + "i")^3 (3"i" + 4)^2)/(8 + 6"i")^2, then |z| is equal to

• 0

• 1

• 2

• 3

Exercise 2.9 | Q 6 | Page 93

Choose the correct alternative:

If z is a non zero complex number, such that 2iz2 = bar(z) then |z| is

• 1/2

• 1

• 2

• 3

Exercise 2.9 | Q 7 | Page 93

Choose the correct alternative:

If |z – 2 + i| ≤ 2, then the greatest value of |z| is

• sqrt(3) - 2

• sqrt(3) + 2

• sqrt(5)  - 2

• sqrt(5) + 2

Exercise 2.9 | Q 8 | Page 93

Choose the correct alternative:

If  |"z" - 3/2|, then the least value of |z| is

• 1

• 2

• 3

• 5

Exercise 2.9 | Q 9 | Page 93

Choose the correct alternative:

If |z| = 1, then the value of  (1 + "z")/(1 + "z") is

• z

• bar(z)

• 1/z

• 1

Exercise 2.9 | Q 10 | Page 93

Choose the correct alternative:

The solution of the equation |z| – z = 1 + 2i is

• 3/2 - 2"i"

• - 3/2 + 2"i"

• 2 - 3/2 "i"

• 2 + 3/2 "i"

Exercise 2.9 | Q 11 | Page 93

Choose the correct alternative:

If |z1| = 1,|z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is

• 1

• 2

• 3

• 4

Exercise 2.9 | Q 12 | Page 93

Choose the correct alternative:

If z is a complex number such that z ∈ C\R and "z" + 1/"z" ∈ R, then |z| is

• 0

• 1

• 2

• 3

Exercise 2.9 | Q 13 | Page 94

Choose the correct alternative:

z1, z2 and z3 are complex numbers such that z1 + z2 + z3 = 0 and |z1| = |z2| = |z3| = 1 then z_1^2 + z_2^2 + z_3^2 is

• 3

• 2

• 1

• 0

Exercise 2.9 | Q 14 | Page 94

Choose the correct alternative:

If (z - 1)/(z + 1) purely imaginary then |z| is

• 1/2

• 1

• 2

• 3

Exercise 2.9 | Q 15 | Page 94

Choose the correct alternative:

If z = x + iy is a complex number such that |z + 2| = |z – 2|, then the locus of z is

• real axis

• imaginary axis

• ellipse

• circle

Exercise 2.9 | Q 16 | Page 94

Choose the correct alternative:

The principal argument of 3/(-1 + "i") is

• (- 5pi)/6

• (- 2pi)/3

• (- 3pi)/4

• (- pi)/2

Exercise 2.9 | Q 17 | Page 94

Choose the correct alternative:

The principal argument of (sin 40° + i cos 40°)5 is

• – 110°

• – 70°

• 70°

• 110°

Exercise 2.9 | Q 18 | Page 94

Choose the correct alternative:

If (1 + i)(1 + 2i)(1 + 3i) ……. (l + ni) = x + iy, then 2.5.10 …… (1 + n2) is

• 1

• i

• x2 + y2

• 1 + n2

Exercise 2.9 | Q 19 | Page 94

Choose the correct alternative:

If ω ≠ 1 is a cubic root of unity and (1 + ω)7 = A + Bω, then (A, B) equals

• (1, 0)

• (– 1, 1)

• (0, 1)

• (1, 1)

Exercise 2.9 | Q 20 | Page 94

Choose the correct alternative:

The principal argument of the complex number ((1 + "i" sqrt(3))^2)/(4"i"(1 - "i" sqrt(3)) is

• (2pi)/3

• pi/6

• (5pi)/6

• pi/2

Exercise 2.9 | Q 21 | Page 94

Choose the correct alternative:

If α and β are the roots of x² + x + 1 = 0, then α2020 + β2020 is

• – 2

• – 1

• 1

• 2

Exercise 2.9 | Q 22 | Page 94

Choose the correct alternative:

The product of all four values of (cos  pi/3 + "i" sin  pi/3)^(3/4) is

• – 2

• – 1

• 1

• 2

Exercise 2.9 | Q 23 | Page 94

Choose the correct alternative:

If ω ≠ 1 is a cubic root of unity and |(1, 1, 1),(1, - omega^2 - 1, omega^2),(1, omega^2, omega^7)| = 3k, then k is equal to

• 1

• – 1

• sqrt(3)"i"

• - sqrt(3)"i"

Exercise 2.9 | Q 24 | Page 94

Choose the correct alternative:

The value of ((1 + sqrt(3)"i")/(1 - sqrt(3)"i"))^10 is

• cis  (2pi)/3

• cis  (4pi)/3

• - cis  (2pi)/3

• - cis  (4pi)/3

Exercise 2.9 | Q 25 | Page 94

Choose the correct alternative:

If ω = cis  (2pi)/3, then the number of distinct roots of |(z + 1, omega, omega^2),(omega, z + omega^2, 1),(omega^2, 1, z + omega)| = 0

• 1

• 2

• 3

• 4

## Chapter 2: Complex Numbers

Exercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Exercise 2.5Exercise 2.6Exercise 2.7Exercise 2.8Exercise 2.9

## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 2 - Complex Numbers

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Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 2 Complex Numbers are Conjugate of a Complex Number, de Moivre’s Theorem and Its Applications, Complex Numbers, Introduction to Complex Numbers, Modulus of a Complex Number, Geometry and Locus of Complex Numbers, Basic Algebraic Properties of Complex Numbers, Polar and Euler Form of a Complex Number.

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