#### Online Mock Tests

#### Chapters

▶ Chapter 2: Complex Numbers

Chapter 3: Theory of Equations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Two Dimensional Analytical Geometry-II

Chapter 6: Applications of Vector Algebra

Chapter 7: Applications of Differential Calculus

Chapter 8: Differentials and Partial Derivatives

Chapter 9: Applications of Integration

Chapter 10: Ordinary Differential Equations

Chapter 11: Probability Distributions

Chapter 12: Discrete Mathematics

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## Solutions for Chapter 2: Complex Numbers

Below listed, you can find solutions for Chapter 2 of Tamil Nadu Board of Secondary Education Tamil Nadu Board Samacheer Kalvi for Class 12th Mathematics Volume 1 and 2 Answers Guide.

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

Simplify the following:

i^{1947} + i^{1950}

Simplify the following:

i^{1948} – i^{–1869}

Simplify the following:

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

Simplify the following:

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

Simplify the following:

i i^{2} i^{3} ... i^{2000 }

Simplify the following:

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

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

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

z + w

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

z – iw

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

2z + 3w

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

zw

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

z^{2} + 2zw + w^{2}

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

(z + w)^{2}

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

z, iz and z + iz

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

z, – iz and z – iz

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

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

If z_{1} = 1 – 3i, z_{2} = – 4i, and z_{3} = 5, show that (z_{1} + z_{2}) + z_{3} = z_{1} + (z_{2} + z_{3})

If z_{1} = 1 – 3i, z_{2} = – 4i, and z_{3} = 5, show that (z_{1} z_{2})z_{3} = z_{1}(z_{2} z_{3})

If z_{1} = 3, z_{2} = 7i, and z_{3} = 5 + 4i, show that z_{1}(z_{2} + z_{3}) = z_{1}z_{2} + z_{1}z_{3}

If z_{1} = 3, z_{2} = 7i, and z_{3} = 5 + 4i, show that (z_{1} + z_{2})z_{3} = z_{1}z_{3} + z_{2}z_{3}

If z_{1} = 2 + 5i, z_{2} = – 3 – 4i, and z_{3} = 1 + i, find the additive and multiplicative inverse of z_{1}, z_{2} and z_{3}

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

Write the following in the rectangular form:

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

Write the following in the rectangular form:

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

Write the following in the rectangular form:

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

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

`"Re"(1/z)`

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

`"Re"("i"barz)`

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

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

If z_{1} = 2 – i and z_{2} = – 4 + 3i, find the inverse of z_{1}, z_{2} and `("z"_1)/("z"_2)`

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

Prove the following properties:

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

Prove the following properties:

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

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

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

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

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

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

Find the modulus of the following complex numbers

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

Find the modulus of the following complex numbers

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

Find the modulus of the following complex numbers

(1 – i)^{10}

Find the modulus of the following complex numbers

2i(3 – 4i)(4 – 3i)

For any two complex numbers z_{1} and z_{2}, such that |z_{1}| = |z_{2}| = 1 and z_{1} z_{2} ≠ -1, then show that `(z_1 + z_2)/(1 + z_1 z_2)` is real number

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

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

If |z| = 1, show that 2 ≤ |z^{2} – 3| ≤ 4

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

If z_{1}, z_{2} and z_{3} are three complex numbers such that |z_{1}| = 1, |z_{2}| = 2, |z_{3}| = 3 and |z_{1} + z_{2} + z_{3}| = 1, show that |9z_{1}z_{2 }+ 4z_{1}z_{3} + z_{2}z_{3}| = 6

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

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

Find the square roots of 4 + 3i

Find the square roots of – 6 + 8i

Find the square roots of – 5 – 12i

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Complex Numbers Exercise 2.6 [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

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

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

[Re(iz)]^{2} = 3

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

Im[(1 – i)z + 1] = 0

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

|z + i| = |z – 1|

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

`bar(z) = z^-1`

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

|z – 2 – i| = 3

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

|2z + 2 – 4i| = 2

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

|3z – 6 + 12i| = 8

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

|z – 4| = 16

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

|z – 4|^{2} – |z – 1|^{2} = 16

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

Write in polar form of the following complex numbers

`2 + "i" 2sqrt(3)`

Write in polar form of the following complex numbers

`3 - "i"sqrt(3)`

Write in polar form of the following complex numbers

– 2 – i2

Write in polar form of the following complex numbers

`("i" - 1)/(cos pi/3 + "i" sin pi/3)`

Find the rectangular form of the complex numbers

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

Find the rectangular form of the complex numbers

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

If (x_{1} + iy_{1})(x_{2} + iy_{2})(x_{3} + iy_{3}) ... (x_{n} + iy_{n}) = 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)` = a^{2} + b^{2}

If (x_{1} + iy_{1})(x_{2} + iy_{2})(x_{3} + iy_{3}) ... (x_{n} + iy_{n}) = a + ib, show that `sum_("r" = 1)^"n" tan^-1 (y_"r"/x_"r") = tan^-1 ("b"/"a") + 2"k"pi, "k" ∈ "z"`

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

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)

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)

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

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 2 Complex Numbers Exercise 2.8 [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

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

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

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

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

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)

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)

Solve the equation z^{3} + 27 = 0

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 – 2ω^{2}

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

If ω ≠ 1 is a cube root of unity, show that (1 – ω + ω^{2})^{6} + (1 + ω – ω^{2})^{6} = 128

If ω ≠ 1 is a cube root of unity, show that (1 + ω)(1 + ω^{2})(1 + ω^{4})(1 + ω^{8})….. (1 + ω^{2n}) = 1

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

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

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

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

#### MCQ

Choose the correct alternative:

i^{n} + i^{n+1}+ i^{n+2} + i^{n+3} is

0

1

– 1

z

Choose the correct alternative:

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

1 + i

i

1

0

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`

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)`

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

Choose the correct alternative:

If z is a non zero complex number, such that 2iz^{2} = `bar(z)` then |z| is

`1/2`

1

2

3

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`

Choose the correct alternative:

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

1

2

3

5

Choose the correct alternative:

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

z

`bar(z)`

`1/z`

1

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"`

Choose the correct alternative:

If |z_{1}| = 1,|z_{2}| = 2, |z_{3}| = 3 and |9z_{1}z_{2} + 4z_{1}z_{3} + z_{2}z_{3}| = 12, then the value of |z_{1} + z_{2} + z_{3}| is

1

2

3

4

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

Choose the correct alternative:

z_{1}, z_{2} and z_{3} are complex numbers such that z_{1} + z_{2} + z_{3} = 0 and |z_{1}| = |z_{2}| = |z_{3}| = 1 then `z_1^2 + z_2^2 + z_3^2` is

3

2

1

0

Choose the correct alternative:

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

`1/2`

1

2

3

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

Choose the correct alternative:

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

`(- 5pi)/6`

`(- 2pi)/3`

`(- 3pi)/4`

`(- pi)/2`

Choose the correct alternative:

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

– 110°

– 70°

70°

110°

Choose the correct alternative:

If (1 + i)(1 + 2i)(1 + 3i) ……. (l + ni) = x + iy, then 2.5.10 …… (1 + n^{2}) is

1

i

x

^{2}+ y^{2}1 + n

^{2}

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)

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`

Choose the correct alternative:

If α and β are the roots of x² + x + 1 = 0, then α^{2020} + β^{2020} is

– 2

– 1

1

2

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

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"`

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`

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

## Solutions for Chapter 2: Complex Numbers

## 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 Introduction to Complex Numbers, Complex Numbers, Basic Algebraic Properties of Complex Numbers, Conjugate of a Complex Number, Modulus of a Complex Number, Geometry and Locus of Complex Numbers, Polar and Euler Form of a Complex Number, de Moivreâ€™s Theorem and Its Applications.

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