Balbharati solutions for Mathematics and Statistics 2 (Arts and Science) [English] 11 Standard Maharashtra State Board chapter 1 - Complex Numbers [Latest edition]

Simplify : ${\displaystyle \sqrt{-16}+3\sqrt{-25}+\sqrt{-36}-\sqrt{-625}}$

Simplify : ${\displaystyle 4\sqrt{-4}+5\sqrt{-9}-3\sqrt{-16}}$

Write the conjugates of the following complex number:

3 + i

Write the conjugates of the following complex number:

3 – i

Write the conjugates of the following complex number:

${\displaystyle -\sqrt{5}-\sqrt{7}\text{i}}$

Write the conjugates of the following complex number:

${\displaystyle -\sqrt{-5}}$

Write the conjugates of the following complex number:

5i

Write the conjugates of the following complex number:

${\displaystyle \sqrt{5}-\text{i}}$

Write the conjugates of the following complex number:

${\displaystyle \sqrt{2}+\sqrt{3}\text{i}}$

Write the conjugates of the following complex number:

cosθ + i sinθ

Find a and b if a + 2b + 2ai = 4 + 6i

Find a and b if (a – b) + (a + b)i = a + 5i

Find a and b if (a+b) (2 + i) = b + 1 + (10 + 2a)i

Find a and b if abi = 3a − b + 12i

Find a and b if ${\displaystyle \frac{1}{\text{a}+\text{ib}}}$ = 3 – 2i

Find a and b if (a + ib) (1 + i) = 2 + i

Express the following in the form of a + ib, a, b ∈ R, i = ${\displaystyle \sqrt{-1}}$. State the values of a and b:

(1 + 2i)(– 2 + i)

Express the following in the form of a + ib, a, b ∈ R, i = ${\displaystyle \sqrt{-1}}$. State the values of a and b:

(1 + i)(1 − i)⁻¹ 

Express the following in the form of a + ib, a, b ∈ R, i = ${\displaystyle \sqrt{-1}}$. State the values of a and b:

${\displaystyle \frac{\text{i}(4+3\text{i})}{(1-\text{i})}}$

Express the following in the form of a + ib, a, b∈R i = ${\displaystyle \sqrt{-1}}$. State the values of a and b:

${\displaystyle \frac{(2+\text{i})}{(3-\text{i})(1+2\text{i})}}$

Express the following in the form of a + ib, a, b∈R i = ${\displaystyle \sqrt{-1}}$. State the values of a and b:

${\displaystyle {\left(\frac{1+\text{i}}{1-\text{i}}\right)}^2}$

Express the following in the form of a + ib, a, b∈R i = ${\displaystyle \sqrt{-1}}$. State the values of a and b:

${\displaystyle \frac{3+2\text{i}}{2-5\text{i}}+\frac{3-2\text{i}}{2+5\text{i}}}$

Express the following in the form of a + ib, a, b∈R i = ${\displaystyle \sqrt{-1}}$. State the values of a and b:

(1 + i)⁻³ 

Express the following in the form of a + ib, a, b∈R i = ${\displaystyle \sqrt{-1}}$. State the values of a and b:

${\displaystyle \frac{2+\sqrt{-3}}{4+\sqrt{-3}}}$

Express the following in the form of a + ib, a, b ∈ R i = ${\displaystyle \sqrt{-1}}$. State the values of a and b:

${\displaystyle (-\sqrt{5}+2\sqrt{-4})+(1-\sqrt{-9})+(2+3\text{i})(2-3\text{i})}$

Express the following in the form of a + ib, a, b∈R i = ${\displaystyle \sqrt{-1}}$. State the values of a and b:

(2 + 3i)(2 – 3i)

Express the following in the form of a + ib, a, b ∈ R, i = ${\displaystyle \sqrt{-1}}$. State the values of a and b:

${\displaystyle \frac{4{\text{i}}^8-3{\text{i}}^9+3}{3{\text{i}}^{11}-4{\text{i}}^{10}-2}}$

Show that ${\displaystyle {(-1+\sqrt{3}\text{i})}^3}$ is a real number

Find the value of ${\displaystyle (3+\frac{2}{\text{i}})({\text{i}}^6-{\text{i}}^7)(1+{\text{i}}^{11})}$

Evaluate the following : i³⁵ 

Evaluate the following : i⁸⁸⁸ 

Evaluate the following : i⁹³  

Evaluate the following : i¹¹⁶ 

Evaluate the following : i⁴⁰³ 

Evaluate the following : ${\displaystyle \frac{1}{{\text{i}}^{58}}}$

Evaluate the following : i^–888 

Evaluate the following : i³⁰ + i⁴⁰ + i⁵⁰ + i⁶⁰ 

Show that 1 + i¹⁰ + i²⁰ + i³⁰ is a real number

Find the value of i⁴⁹ + i⁶⁸ + i⁸⁹ + i¹¹⁰ 

Find the value of i + i² + i³ + i⁴ 

Simplify : ${\displaystyle \frac{{\text{i}}^{592}+{\text{i}}^{590}+{\text{i}}^{588}+{\text{i}}^{586}+{\text{i}}^{584}}{{\text{i}}^{582}+{\text{i}}^{580}+{\text{i}}^{578}+{\text{i}}^{576}+{\text{i}}^{574}}}$

Find the value of 1 + i² + i⁴ + i⁶ + i⁸ + ... + i²⁰

Show that 1 + i¹⁰ + i¹⁰⁰ − i¹⁰⁰⁰ = 0 

Is (1 + i¹⁴ + i¹⁸ + i²²) a real number? Justify your answer

Evaluate: ${\displaystyle ({\text{i}}^{37}+\frac{1}{{\text{i}}^{67}})}$

Prove that ${\displaystyle {(1+\text{i})}^4\times {(1+\frac{1}{\text{i}})}^4}$ = 16

Find the value of ${\displaystyle \frac{{\text{i}}^6+{\text{i}}^7+{\text{i}}^8+{\text{i}}^9}{{\text{i}}^2+{\text{i}}^3}}$

If a = ${\displaystyle \frac{-1+\sqrt{3}\text{i}}{2}}$, b = ${\displaystyle \frac{-1-\sqrt{3}\text{i}}{2}}$ then show that a² = b and b² = a

If x + iy = (a + ib)³, show that ${\displaystyle \frac{x}{\text{a}}+\frac{y}{\text{b}}}$ = 4(a² − b²)

If ${\displaystyle \frac{\text{a}+3\text{i}}{2+\text{ib}}}$ = 1 − i, show that (5a − 7b) = 0

If x + iy = ${\displaystyle \sqrt{\frac{\text{a}+\text{ib}}{\text{c}+\text{id}}}}$, prove that (x² + y²)² = ${\displaystyle \frac{{\text{a}}^2+{\text{b}}^2}{{\text{c}}^2+{\text{d}}^2}}$ 

If (a + ib) = ${\displaystyle \frac{1+\text{i}}{1-\text{i}}}$, then prove that (a² + b²) = 1

Show that ${\displaystyle (\frac{\sqrt{7}+\text{i}\sqrt{3}}{\sqrt{7}-\text{i}\sqrt{3}}+\frac{\sqrt{7}-\text{i}\sqrt{3}}{\sqrt{7}+\text{i}\sqrt{3}})}$ is real

If (x + iy)³ = y + vi then show that ${\displaystyle (\frac{y}{x}+\frac{\text{v}}{y})}$ = 4(x² – y²)

Find the value of x and y which satisfy the following equation (x, y∈R).

(x + 2y) + (2x − 3y)i + 4i = 5

Find the value of x and y which satisfy the following equation (x, y∈R).

${\displaystyle \frac{x+1}{1+\text{i}}+\frac{y-1}{1-\text{i}}}$ = i

Find the value of x and y which satisfy the following equation (x, y ∈ R).

${\displaystyle \frac{(x+\text{i}y)}{2+3\text{i}}+\frac{2+\text{i}}{2-3\text{i}}=\frac{9}{13}(1+\text{i})}$

Find the value of x and y which satisfy the following equation (x, y∈R).

If x(1 + 3i) + y(2 − i) − 5 + i³ = 0, find x + y

Find the value of x and y which satisfy the following equation (x, y∈R).

If x + 2i + 15i⁶y = 7x + i³ (y + 4), find x + y

Find the square root of the following complex number: −8 − 6i

Find the square root of the following complex number:

7 + 24i 

Find the square root of the following complex number:

${\displaystyle 1+4\sqrt{3}\text{i}}$

Find the square root of the following complex number: 

${\displaystyle 3+2\sqrt{10}\text{i}}$

Find the square root of the following complex number: 

${\displaystyle 2(1-\sqrt{3}\text{i})}$

Solve the following quadratic equation.

8x² + 2x + 1 = 0

Solve the following quadratic equation

${\displaystyle 2x^2-\sqrt{3}x+1}$ = 0

Solve the following quadratic equation.

3x² − 7x + 5 = 0

Solve the following quadratic equation.

x² − 4x + 13 = 0

Solve the following quadratic equation.

x² + 3ix + 10 = 0

Solve the following quadratic equation.

2x² + 3ix + 2 = 0

Solve the following quadratic equation.

x² + 4ix − 4 = 0

Solve the following quadratic equation.

ix² − 4x − 4i = 0

Solve the following quadratic equation.

x² − (2 + i)x − (1 − 7i) = 0

Solve the following quadratic equation.

${\displaystyle x^2-(3\sqrt{2}+2\text{i})x+6\sqrt{2}\text{i}}$ = 0

Solve the following quadratic equation.

x² − (5 − i) x + (18 + i) = 0

Solve the following quadratic equation.

(2 + i)x² − (5 − i) x + 2(1 − i) = 0

Find the value of x³ − x² + x + 46, if x = 2 + 3i

Find the value of 2x³ − 11x² + 44x + 27, if x = ${\displaystyle \frac{25}{3-4\text{i}}}$

Find the value of x³ + x² − x + 22, if x = ${\displaystyle \frac{5}{1-2\text{i}}}$

Find the value of x⁴ + 9x³ + 35x² − x + 4, if x = ${\displaystyle -5+\sqrt{-4}}$

Find the value of 2x⁴ + 5x³ + 7x² − x + 41, if x = ${\displaystyle -2-\sqrt{3}\text{i}}$

Find the modulus and amplitude of the following complex numbers.

7 − 5i

Find the modulus and amplitude of the following complex numbers.

${\displaystyle \sqrt{3}+\sqrt{2}\text{i}}$

Find the modulus and amplitude of the following complex numbers.

−8 + 15i

Find the modulus and amplitude of the following complex numbers.

−3(1 − i)

Find the modulus and amplitude of the following complex numbers.

−4 − 4i

Find the modulus and amplitude of the following complex numbers.

${\displaystyle \sqrt{3}-\text{i}}$

Find the modulus and amplitude of the following complex numbers.

3

Find the modulus and amplitude of the following complex numbers.

1 + i

Find the modulus and amplitude of the following complex numbers.

${\displaystyle 1+\text{i}\sqrt{3}}$

Find the modulus and amplitude of the following complex numbers.

(1 + 2i)² (1 − i)

Find real values of θ for which ${\displaystyle \left(\frac{4+3\text{i}\sin \theta }{1-2\text{i}\sin \theta }\right)}$ is purely real.

If z = 3 + 5i then represent the ${\displaystyle \text{z},\overline{\text{z}},-\text{z},\overline{-\text{z}}}$ in Argand's diagram

Express the following complex numbers in polar form and exponential form: 

${\displaystyle -1+\sqrt{3}\text{i}}$

Express the following complex numbers in polar form and exponential form:

− i

Express the following complex numbers in polar form and exponential form:

−1

Express the following complex numbers in polar form and exponential form:

${\displaystyle \frac{1}{1+\text{i}}}$

Express the following complex numbers in polar form and exponential form:

${\displaystyle \frac{1+2\text{i}}{1-3\text{i}}}$

Express the following complex numbers in polar form and exponential form:

${\displaystyle \frac{1+7\text{i}}{{(2-\text{i})}^2}}$

Express the following numbers in the form x + iy: 

${\displaystyle \sqrt{3}(\cos \frac{\pi }{6}+\text{i}\sin \frac{\pi }{6})}$

Express the following numbers in the form x + iy: 

${\displaystyle \sqrt{2}(\cos \frac{7\pi }{4}+\text{i}\sin \frac{7\pi }{4})}$

Express the following numbers in the form x + iy:

${\displaystyle 7(\cos (-\frac{5\pi }{6})+\text{i}\sin (-\frac{5\pi }{6}))}$

Express the following numbers in the form x + iy:

${\displaystyle {\text{e}}^{\frac{\pi }{3}\text{i}}}$

Express the following numbers in the form x + iy:

${\displaystyle {\text{e}}^{\frac{-4\pi }{3}\text{i}}}$

Express the following numbers in the form x + iy:

${\displaystyle {\text{e}}^{\frac{5\pi }{6}\text{i}}}$

Find the modulus and argument of the complex number ${\displaystyle \frac{1+2\text{i}}{1-3\text{i}}}$

Convert the complex number z = ${\displaystyle \frac{\text{i}-1}{\cos \frac{\pi }{3}+\text{i}\sin \frac{\pi }{3}}}$ in the polar form

For z = 2 + 3i verify the following:

${\displaystyle \overline{\left(\overline{\text{z}}\right)}}$ = z

For z = 2 + 3i verify the following:

${\displaystyle \text{z}\overline{\text{z}}}$ = |z|²

For z = 2 + 3i verify the following:

${\displaystyle (\text{z}+\overline{\text{z}})}$ is real

For z = 2 + 3i verify the following:

${\displaystyle \text{z}-\overline{\text{z}}}$ = 6i

z₁ = 1 + i, z₂ = 2 − 3i. Verify the following : 

${\displaystyle \overline{{\text{z}}_1+{\text{z}}_2}=\overline{{\text{z}}_1}+\overline{{\text{z}}_2}}$

z₁ = 1 + i, z₂ = 2 − 3i. Verify the following : 

${\displaystyle \overline{{\text{z}}_1-{\text{z}}_2}=\overline{{\text{z}}_1}-\overline{{\text{z}}_2}}$

z₁ = 1 + i, z₂ = 2 − 3i. Verify the following :

${\displaystyle \overline{{\text{z}}_1.{\text{z}}_2}=\overline{{\text{z}}_1}.\overline{{\text{z}}_2}}$

z₁ = 1 + i, z₂ = 2 − 3i. Verify the following :

${\displaystyle \overline{\left(\frac{{\text{z}}_1}{{\text{z}}_2}\right)}=\frac{\overline{{\text{z}}_1}}{\overline{{\text{z}}_2}}}$

Find the value of ω¹⁸

Find the value of ω²¹

Find the value of ω^–30

Find the value of ω^–105

If ω is a complex cube root of unity, show that (2 − ω)(2 − ω²) = 7

If ω is a complex cube root of unity, show that (1 + ω − ω²)⁶ = 64

If ω is a complex cube root of unity, show that (1 + ω)³ − (1 + ω²)³ = 0

If ω is a complex cube root of unity, show that (2 + ω + ω²)³ − (1 − 3ω + ω²)³ = 65

If ω is a complex cube root of unity, show that (3 + 3ω + 5ω²)⁶ − (2 + 6ω + 2ω²)³ = 0

If ω is a complex cube root of unity, show that ${\displaystyle \frac{\text{a}+\text{b}\omega +\text{c}{\omega }^2}{\text{c}+\text{a}\omega +\text{b}{\omega }^2}}$ = ω²

If ω is a complex cube root of unity, show that (a + b) + (aω + bω²) + (aω² + bω) = 0

If ω is a complex cube root of unity, show that (a − b) (a − bω) (a − bω²) = a³ − b³

If ω is a complex cube root of unity, show that (a + b)² + (aω + bω²)² + (aω² + bω)² = 6ab

If ω is a complex cube root of unity, find the value of ${\displaystyle \omega +\frac{1}{\omega }}$

If ω is a complex cube root of unity, find the value of ω² + ω³ + ω⁴

If ω is a complex cube root of unity, find the value of (1 + ω²)³

If ω is a complex cube root of unity, find the value of (1 − ω − ω²)³ + (1 − ω + ω²)³

If ω is a complex cube root of unity, find the value of (1 + ω)(1 + ω²)(1 + ω⁴)(1 + ω⁸)

If α and β are the complex cube root of unity, show that α² + β² + αβ = 0

If α and β are the complex cube root of unity, show that α⁴ + β⁴ + α⁻¹β⁻¹ = 0

If , where α and β are the complex cube-roots of unity, show that xyz = a³ + b³.

Find the equation in cartesian coordinates of the locus of z if |z| = 10

Find the equation in cartesian coordinates of the locus of z if |z – 3| = 2

Find the equation in cartesian coordinates of the locus of z if |z − 5 + 6i| = 5

Find the equation in cartesian coordinates of the locus of z if |z + 8| = |z – 4|

Find the equation in cartesian coordinates of the locus of z if |z – 2 – 2i| = |z + 2 + 2i|

Find the equation in cartesian coordinates of the locus of z if ${\displaystyle \left|\frac{\text{z}+3\text{i}}{\text{z}-6\text{i}}\right|}$ = 1

Use De Moivres theorem and simplify the following:

${\displaystyle \frac{{(\cos 2\theta +\text{i}\sin 2\theta )}^7}{{(\cos 4\theta +\text{i}\sin 4\theta )}^3}}$

Use De Moivres theorem and simplify the following:

${\displaystyle \frac{\cos 5\theta +\text{i}\sin 5\theta }{{(\cos 3\theta -\text{i}\sin 3\theta )}^2}}$

Use De Moivres theorem and simplify the following:

${\displaystyle \frac{{(\cos \frac{7\pi }{13}+\text{i}\sin \frac{7\pi }{13})}^4}{{(\cos \frac{4\pi }{13}-\text{i}\sin \frac{4\pi }{13})}^6}}$

Express the following in the form a + ib, a, b ∈ R, using De Moivre's theorem:

(1 − i)⁵ 

Express the following in the form a + ib, a, b ∈ R, using De Moivre's theorem:

(1 + i)⁶

Express the following in the form a + ib, a, b ∈ R, using De Moivre's theorem:

${\displaystyle {(1-\sqrt{3}\text{i})}^4}$

Express the following in the form a + ib, a, b ∈ R, using De Moivre's theorem:

${\displaystyle {(-2\sqrt{3}-2\text{i})}^5}$

Select the correct answer from the given alternatives:

If n is an odd positive integer then the value of 1 + (i)²ⁿ + (i)⁴ⁿ + (i)⁶ⁿ is :

Select the correct answer from the given alternatives:

The value of is ${\displaystyle \frac{{\text{i}}^{592}+{\text{i}}^{590}+{\text{i}}^{588}+{\text{i}}^{586}+{\text{i}}^{584}}{{\text{i}}^{582}+{\text{i}}^{580}+{\text{i}}^{578}+{\text{i}}^{576}+{\text{i}}^{574}}}$ is equal to:

Select the correct answer from the given alternatives:

${\displaystyle \sqrt{-3}\sqrt{-6}}$ is equal to

Select the correct answer from the given alternatives:

If ω is a complex cube root of unity, then the value of ω⁹⁹+ ω¹⁰⁰ + ω¹⁰¹ is :

Select the correct answer from the given alternatives:

If z = r(cos θ + i sin θ), then the value of ${\displaystyle \frac{\text{z}}{\overline{\text{z}}}+\frac{\overline{\text{z}}}{\text{z}}}$

If ω(≠1) is a cube root of unity and (1 + ω)⁷ = A + Bω, then A and B are respectively the numbers ______.

Select the correct answer from the given alternatives:

The modulus and argument of ${\displaystyle {(1+\text{i}\sqrt{3})}^8}$ are respectively

Select the correct answer from the given alternatives:

If arg(z) = θ, then arg ${\displaystyle \overline{\left(\text{z}\right)}}$ =

Select the correct answer from the given alternatives:

If ${\displaystyle -1+\sqrt{3}\text{i}}$ = re^iθ , then θ = ................. 

Select the correct answer from the given alternatives:

If z = x + iy and |z − zi| = 1 then

Answer the following:

Simplify the following and express in the form a + ib:

${\displaystyle 3+\sqrt{-64}}$

Answer the following:

Simplify the following and express in the form a + ib:

(2i³)² 

Answer the following:

Simplify the following and express in the form a + ib:

(2 + 3i)(1 − 4i)

Answer the following:

Simplify the following and express in the form a + ib:

${\displaystyle \frac{5}{2}\text{i}(-4-3\text{i})}$

Answer the following:

Simplify the following and express in the form a + ib:

(1 + 3i)²(3 + i)

Answer the following:

Simplify the following and express in the form a + ib:

${\displaystyle \frac{4+3\text{i}}{1-\text{i}}}$

Answer the following:

Simplify the following and express in the form a + ib:

${\displaystyle (1+\frac{2}{\text{i}})(3+\frac{4}{\text{i}}){(5+\text{i})}^{-1}}$

Answer the following:

Simplify the following and express in the form a + ib:

${\displaystyle \frac{\sqrt{5}+\sqrt{3}\text{i}}{\sqrt{5}-\sqrt{3}\text{i}}}$

Answer the following:

Simplify the following and express in the form a + ib:

${\displaystyle \frac{3{\text{i}}^5+2{\text{i}}^7+{\text{i}}^9}{{\text{i}}^6+2{\text{i}}^8+3{\text{i}}^{18}}}$

Answer the following:

Simplify the following and express in the form a + ib:

${\displaystyle \frac{5+7\text{i}}{4+3\text{i}}+\frac{5+7\text{i}}{4-3\text{i}}}$

Answer the following:

Solve the following equation for x, y ∈ R:

(4 − 5i)x + (2 + 3i)y = 10 − 7i

Answer the following:

Solve the following equation for x, y ∈ R:

${\displaystyle \frac{x+\text{i}y}{2+3\text{i}}}$ = 7 – i

Answer the following:

Solve the following equations for x, y ∈ R:

(x + iy) (5 + 6i) = 2 + 3i

Solve the following equation for x, y ∈ R:

2x + i⁹y (2 + i) = xi⁷ + 10i¹⁶

Answer the following:

Evaluate: (1 − i + i²)⁻¹⁵ 

Answer the following:

Evaluate: i¹³¹ + i⁴⁹ 

Answer the following:

Find the value of x³ + 2x² − 3x + 21, if x = 1 + 2i

Answer the following:

Find the value of x⁴ + 9x³ + 35x² − x + 164, if x = −5 + 4i

Answer the following:

Find the square root of −16 + 30i

Answer the following:

Find the square root of 15 – 8i

Answer the following:

Find the square root of ${\displaystyle 2+2\sqrt{3}\text{i}}$

Answer the following:

Find the square root of 18i

Answer the following:

Find the square root of 3 − 4i

Answer the following:

Find the square root of 6 + 8i

Answer the following:

Find the modulus and argument of a complex number and express it in the polar form.

8 + 15i

Answer the following:

Find the modulus and argument of a complex number and express it in the polar form.

6 − i

Answer the following:

Find the modulus and argument of a complex number and express it in the polar form.

${\displaystyle \frac{1+\sqrt{3}\text{i}}{2}}$

Answer the following:

Find the modulus and argument of a complex number and express it in the polar form.

${\displaystyle \frac{-1-\text{i}}{\sqrt{2}}}$

Answer the following:

Find the modulus and argument of a complex number and express it in the polar form.

2i

Answer the following:

Find the modulus and argument of a complex number and express it in the polar form.

− 3i

Answer the following:

Find the modulus and argument of a complex number and express it in the polar form.

${\displaystyle \frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}\text{i}}$

Answer the following:

Represent 1 + 2i, 2 − i, −3 − 2i, −2 + 3i by points in Argand's diagram.

Answer the following:

Show that z = ${\displaystyle \frac{5}{(1-\text{i})(2-\text{i})(3-\text{i})}}$ is purely imaginary number.

Answer the following:

Find the real numbers x and y such that ${\displaystyle \frac{x}{1+2\text{i}}+\frac{y}{3+2\text{i}}=\frac{5+6\text{i}}{-1+8\text{i}}}$

Answer the following:

Show that ${\displaystyle {(\frac{1}{\sqrt{2}}+\frac{\text{i}}{\sqrt{2}})}^{10}+{(\frac{1}{\sqrt{2}}-\frac{\text{i}}{\sqrt{2}})}^{10}}$ = 0

Answer the following:

show that ${\displaystyle {\left(\frac{1+\text{i}}{\sqrt{2}}\right)}^8+{\left(\frac{1-\text{i}}{\sqrt{2}}\right)}^8}$ = 2

Answer the following:

Convert the complex numbers in polar form and also in exponential form.

z = ${\displaystyle \frac{2+6\sqrt{3}\text{i}}{5+\sqrt{3}\text{i}}}$

Answer the following:

Convert the complex numbers in polar form and also in exponential form.

z = ${\displaystyle -6+\sqrt{2}\text{i}}$

Answer the following:

Convert the complex numbers in polar form and also in exponential form.

${\displaystyle \frac{-3}{2}+\frac{3\sqrt{3}}{2}\text{i}}$

Answer the following:

If x + iy = ${\displaystyle \frac{\text{a}+\text{ib}}{\text{a}-\text{ib}}}$, prove that x² + y² = 1

Answer the following:

Show that z = ${\displaystyle {\left(\frac{-1+\sqrt{-3}}{2}\right)}^3}$ is a rational number

Answer the following:

Show that ${\displaystyle \frac{1-2\text{i}}{3-4\text{i}}+\frac{1+2\text{i}}{3+4\text{i}}}$ is real

Answer the following:

Simplify: ${\displaystyle \frac{{\text{i}}^{29}+{\text{i}}^{39}+{\text{i}}^{49}}{{\text{i}}^{30}+{\text{i}}^{40}+{\text{i}}^{50}}}$

Answer the following:

Simplify: ${\displaystyle ({\text{i}}^{65}+\frac{1}{{\text{i}}^{145}})}$

Answer the following:

Simplify: ${\displaystyle \frac{{\text{i}}^{238}+{\text{i}}^{236}+{\text{i}}^{234}+{\text{i}}^{232}+{\text{i}}^{230}}{{\text{i}}^{228}+{\text{i}}^{226}+{\text{i}}^{224}+{\text{i}}^{222}+{\text{i}}^{220}}}$

Answer the following:

Simplify ${\displaystyle [\frac{1}{1-2\text{i}}+\frac{3}{1+\text{i}}]\left[\frac{3+4\text{i}}{2-4\text{i}}\right]}$

Answer the following:

If α and β are complex cube roots of unity, prove that (1 − α)(1 − β) (1 − α²)(1 − β²) = 9

Answer the following:

If ω is a complex cube root of unity, prove that (1 − ω + ω²)⁶ +(1 + ω − ω²)⁶ = 128

If ω is the cube root of unity then find the value of ${\displaystyle {\left(\frac{-1+\text{i}\sqrt{3}}{2}\right)}^{18}+{\left(\frac{-1-\text{i}\sqrt{3}}{2}\right)}^{18}}$