Balbharati solutions for मैथमेटिक्स एण्ड स्टैटिस्टिक्स २ (आर्ट्स एण्ड सायन्स) [अंग्रेजी] कक्षा ११ महाराष्ट्र स्टेट बोर्ड chapter 1 - Complex Numbers [Latest edition]

Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`

Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-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:

`-sqrt(5) - sqrt(7)"i"`

Write the conjugates of the following complex number:

`-sqrt(-5)`

Write the conjugates of the following complex number:

5i

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 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 `1/("a" + "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 = `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:

(1 + i)(1 − i)⁻¹ 

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:

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

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:

(1 + i)⁻³ 

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:

`(- sqrt(5) + 2sqrt(-4)) + (1 -sqrt(-9)) + (2 + 3"i")(2 - 3"i")`

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

Find the value of `(3 + 2/i) (i^6 - i^7) (1 + 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 : `1/"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:

`(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

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

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

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

`((x + "i"y))/(2 + 3"i") + (2 + "i")/(2 - 3"i") = 9/13(1 + "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:

`1 + 4sqrt(3)"i"`

Find the square root of the following complex number: 

`3 + 2sqrt(10)"i"`

Find the square root of the following complex number: 

`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

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 value of x³ + x² − x + 22, if x = `5/(1 - 2"i")`

Find the value of x⁴ + 9x³ + 35x² − x + 4, if x = `-5+sqrt(-4)`

Find the value of 2x⁴ + 5x³ + 7x² − x + 41, if x = `-2 - sqrt(3)"i"`

Find the modulus and amplitude of the following complex numbers.

7 − 5i

Find the modulus and amplitude of the following complex numbers.

`sqrt(3) + sqrt(2)"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 number.

−4 − 4i

Find the modulus and amplitude of the following complex numbers.

`sqrt(3) - "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.

`1 + "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 `((4 + 3"i" sintheta)/(1 - 2"i" sin theta))` is purely real.

If z = 3 + 5i then represent the `"z", bar("z"), - "z", bar(-"z")` in Argand's diagram

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

`-1 + sqrt(3)"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:

`1/(1 + "i")`

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

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

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

`(1 + 7"i")/(2 - "i")^2`

Express the following numbers in the form x + iy: 

`sqrt(3)(cos  pi/6 + "i" sin  pi/6)`

Express the following numbers in the form x + iy: 

`sqrt(2)(cos  (7pi)/4 + "i" sin  (7pi)/4)`

Express the following numbers in the form x + iy:

`7(cos(-(5pi)/6) + "i" sin (- (5pi)/6))`

Express the following numbers in the form x + iy:

`"e"^(pi/3"i")`

Express the following numbers in the form x + iy:

`"e"^((-4pi)/3"i")`

Express the following numbers in the form x + iy:

`"e"^((5pi)/6"i")`

Find the modulus and argument of the complex number `(1 + 2"i")/(1 - 3"i")`

Convert the complex number z = `("i" - 1)/(cos  pi/3 + "i" sin  pi/3)` in the polar form

For z = 2 + 3i verify the following:

`bar((bar"z"))` = z

For z = 2 + 3i verify the following:

`"z"bar("z")` = |z|²

For z = 2 + 3i verify the following:

`("z" + bar"z")` is real

For z = 2 + 3i verify the following:

`"z" - bar"z"` = 6i

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

`bar("z"_1 + "z"_2) = bar("z"_1) + bar("z"_2)`

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

`bar("z"_1 - "z"_2) = bar("z"_1) - bar("z"_2)`

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

`bar("z"_1."z"_2) = bar("z"_1).bar("z"_2)`

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

`bar(("z"_1/"z"_2))=bar("z"_1)/bar("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 `("a" + "b"ω + "c"ω^2)/("c" + "a"ω + "b"ω^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 `ω + 1/ω`

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 `|("z" + 3"i")/("z" - 6"i")|` = 1

Use De Moivres theorem and simplify the following:

`(cos2theta + "i"sin2theta)^7/(cos4theta + "i"sin4theta)^3`

Use De Moivres theorem and simplify the following:

`(cos5theta + "i"sin5theta)/((cos3theta - "i"sin3theta)^2`

Use De Moivres theorem and simplify the following:

`(cos  (7pi)/13 + "i"sin  (7pi)/13)^4/(cos  (4pi)/13 - "i"sin  (4pi)/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:

`(1 - sqrt(3)"i")^4`

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

`(-2sqrt(3) - 2"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 `("i"^592 + "i"^590 + "i"^588 + "i"^586 + "i"^584)/("i"^582 + "i"^580 + "i"^578 + "i"^576 + "i"^574)` is equal to:

Select the correct answer from the given alternatives:

`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 `"z"/bar("z") + bar("z")/"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 `(1 + "i"sqrt(3))^8` are respectively

Select the correct answer from the given alternatives:

If arg(z) = θ, then arg `bar(("z"))` =

Select the correct answer from the given alternatives:

If `-1 + sqrt(3)"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:

`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:

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

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

Answer the following:

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

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

Answer the following:

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

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

Answer the following:

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)`

Answer the following:

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

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

`(x + "i"y)/(2 + 3"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 `2 + 2sqrt(3)"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.

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

Answer the following:

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

`(-1 - "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.

`1/sqrt(2) + 1/sqrt(2)"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 = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.

Answer the following:

Find the real numbers x and y such that `x/(1 + 2"i") + y/(3 + 2"i") = (5 + 6"i")/(-1 + 8"i")`

Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0

Answer the following:

show that `((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 2

Answer the following:

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

z = `(2 + 6sqrt(3)"i")/(5 + sqrt(3)"i")`

Answer the following:

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

z = `-6 + sqrt(2)"i"`

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

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

Answer the following:

If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x² + y² = 1

Answer the following:

Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number

Answer the following:

Show that `(1 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real

Answer the following:

Simplify: `("i"^29 + "i"^39 + "i"^49)/("i"^30 + "i"^40 + "i"^50)`

Answer the following:

Simplify: `("i"^65 + 1/"i"^145)`

Answer the following:

Simplify: `("i"^238 + "i"^236 + "i"^234 + "i"^232 + "i"^230)/("i"^228 + "i"^226 + "i"^224 + "i"^222 + "i"^220)`

Answer the following:

Simplify `[1/(1 - 2"i") + 3/(1 + "i")] [(3 + 4"i")/(2 - 4"i")]`

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 `((-1 + "i"sqrt(3))/2)^18 + ((-1 - "i"sqrt(3))/2)^18`