Samacheer Kalvi solutions for Mathematics - Volume 1 and 2 [English] Class 12 TN Board chapter 2 - Complex Numbers [Latest edition]

Simplify the following:

i¹⁹⁴⁷ + i¹⁹⁵⁰

Simplify the following:

i¹⁹⁴⁸ – i^–1869

Simplify the following:

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

Simplify the following:

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

Simplify the following:

i i² i³ ... i²⁰⁰⁰ 

Simplify the following:

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

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

z² + 2zw + w² 

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

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

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

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

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

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

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 z₁ = 2 – i and z₂ = – 4 + 3i, find the inverse of z₁, z₂ 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

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 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

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 ≤ |z² – 3| ≤ 4

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

If z₁, z₂ and z₃ are three complex numbers such that |z₁| = 1, |z₂| = 2, |z₃| = 3 and |z₁ + z₂ + z₃| = 1, show that |9z₁z₂ + 4z₁z₃ + z₂z₃| = 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

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 2x² + 2y² + 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

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 (x₁ + iy₁)(x₂ + iy₂)(x₃ + iy₃) ... (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)` = a² + b² 

If (x₁ + iy₁)(x₂ + iy₂)(x₃ + iy₃) ... (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 x² + y³ + 3x – 3y + 2 = 0 

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 z³ + 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 – 2ω² 

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 + ω²)⁶ + (1 + ω – ω²)⁶ = 128

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

Choose the correct alternative:

iⁿ + iⁿ⁺¹+ iⁿ⁺² + iⁿ⁺³ is

Choose the correct alternative:

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

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

Choose the correct alternative:

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

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

Choose the correct alternative:

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

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