Balbharati solutions for Mathematics and Statistics 1 (Commerce) [English] 11 Standard Maharashtra State Board chapter 3 - Complex Numbers [Latest edition]

Write the conjugates of the following complex numbers: 3 + i

Write the conjugates of the following complex numbers: 3 – i

Write the conjugates of the following complex numbers: `-sqrt(5) - sqrt(7)  "i"`

Write the conjugates of the following complex number:

`-sqrt(-5)`

Write the conjugates of the following complex numbers: 5i

Write the conjugates of the following complex numbers: `sqrt(5) - "i"`

Write the conjugates of the following complex numbers: `sqrt(2) + sqrt(3)  "i"`

Express the following in the form of a + ib, a, b ∈ R, i = `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 = `sqrt(-1)`. State the values  of a and b: `("i"(4 + 3"i"))/((1 - "i"))`

Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(-1)`. State the values  of a and b: `((2 + "i"))/((3 - "i")(1 + 2"i"))`

Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(-1)`. State the values of a and b: `(3 + 2"i")/(2 - 5"i") + (3 - 2"i")/(2 + 5"i")`

Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(-1)`. State the values of a and b: `(2 + sqrt(-3))/(4 + sqrt(-3))`

Express the following in the form of a + ib, a, b ∈ R, i = `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 = `sqrt(-1)`. State the values of a and b: `(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`

Show that `(-1 + sqrt(3)"i")^3` is a real number.

Evaluate the following:

i³⁵

Evaluate the following: i⁸⁸⁸ 

Evaluate the following: i⁹³ 

Evaluate the following: i¹¹⁶ 

Evaluate the following: i⁴⁰³ 

Evaluate the following: `1/("i"^58)`

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 

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

Find the values of x and y which satisfy the following equations (x, y ∈ R): (x + 2y) + (2x – 3y i + 4i = 5

Find the values of x and y which satisfy the following equations (x, y ∈ R):

`(x + 1)/(1 + "i") + (y - 1)/(1 - "i")` = i

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

Find the value of: 2x³ – 11x² + 44x + 27, if x = `25/(3 - 4"i")`

Find the square root of the following complex numbers: – 8 – 6i

Find the square root of the following complex numbers: 7 + 24i

Find the square root of the following complex numbers: 1 + 4 `sqrt(3)  "i"`

Find the square root of the following complex numbers: `3 + 2 sqrt(10)  "i"`

Find the square root of the following complex numbers: `2(1 - sqrt(3)  "i")`

Solve the following quadratic equation: 

8x² + 2x + 1 = 0

Solve the following quadratic equation:

`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:

`x^2 - (3 sqrt(2) + 2"i") x + 6 sqrt(2)"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

If `omega` is a complex cube root of unity, show that `(2 - omega)(2 - omega^2)` = 7

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 `(("a" + "b"omega + "c"omega^2))/("c" + "a"omega + "b"omega^2) = omega^2`.

If ω is a complex cube root of unity, find the value of `omega + 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 `omega` is a complex cube root of unity, find the value of `(1 + omega)(1 + omega^2)(1 + omega^4)(1 + omega^8)`

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

If x = a + b, y = αa + βb and z = aβ + bα, where α and β are the complex cube roots of unity, show that xyz = a³ + b³.

If ω is a complex cube root of unity, then prove the following: (ω² + ω - 1)³ = – 8

If ω is a complex cube root of unity, then prove the following:  (a + b) + (aω + bω²) + (aω² + bω) = 0.

Find the value of `("i"^592+ "i"^590 + "i"^588 + "i"^586 + "i"^584)/("i"^582 + "i"^580 + "i"^578 + "i"^576 + "i"^574)`.

Find the value of `sqrt(-3) xx sqrt(-6)`.

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

`3 + sqrt(-64)`

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

(2i³)² 

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

(2 + 3i)(1 – 4i)

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

`5/2"i"(- 4 - 3 "i")`

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

(1 + 3i)² (3 + i)

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

`(4 + 3"i")/(1 - "i")`

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

`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`

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

`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`

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

`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`

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

`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`

Solve the following equation for x, y ∈ R:

(4 – 5i) x + (2 + 3i) y = 10 – 7i

Solve the following equation for x, y ∈ R:

(1 – 3i) x + (2 + 5i) y = 7 + i

Solve the following equation for x, y ∈ R:

`(x + "i"y)/(2 + 3"i")` = 7 – i

Solve the following equation for x, y ∈ R:

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

Solve the following equation for x, y ∈ R:

2x + i⁹ y (2 + i) = x i⁷ + 10 i¹⁶ 

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

Find the value of: x³ – 5x² + 4x + 8, if x = `10/(3 - "i")`.

Find the value of: x³ – 3x² + 19x – 20, if x = 1 – 4i

Find the square root of: – 16 + 30i

Find the square root of 15 – 8i

Find the square root of: `2 + 2 sqrt(3)"i"`

Find the square root of : 18i

Find the square root of: 3 – 4i

Find the square root of 6 + 8i.