HSC Arts (English Medium) ११ वीं कक्षा - Maharashtra State Board Question Bank Solutions

Write the conjugates of the following complex number:

`sqrt(5) - "i"`

Write the conjugates of the following complex number:

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

Write the conjugates of the following complex number:

cosθ + i sinθ

Find the value of i⁴⁹ + i⁶⁸ + i⁸⁹ + i¹¹⁰ 

Find the value of i + i² + i³ + i⁴ 

Simplify:

`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + 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: `("i"^37 + 1/"i"^67)`

Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16

Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`

If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a² = b and b² = a

If x + iy = (a + ib)³, show that `x/"a" + y/"b"` = 4(a² − b²)

If `("a" + 3"i")/(2+ "ib")` = 1 − i, show that (5a − 7b) = 0

If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x² + y²)² = `("a"^2 + "b"^2)/("c"^2 + "d"^2)` 

If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a² + b²) = 1

Show that `((sqrt(7) + "i"sqrt(3))/(sqrt(7) - "i"sqrt(3)) + (sqrt(7) - "i"sqrt(3))/(sqrt(7) + "i"sqrt(3)))` is real

If (x + iy)³ = y + vi then show that `(y/x + "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