Evaluate the following:

(i) i⁴⁵⁷

(ii) i⁵²⁸

(iii)

(iv)

(v)

(vi) (i⁷⁷ + i⁷⁰ + i⁸⁷ + i⁴¹⁴ )³

(vii) (vii) i³⁰ + i⁴⁰ + i⁶⁰

(viii) i⁴⁹ + i⁶⁸ + i⁸⁹ + i¹¹⁸

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

Find the value of following expression:

i⁴⁹ + i⁶⁸ + i⁸⁹ + i¹¹⁰

Find the value of following expression:

i³⁰ + i⁸⁰ + i¹²⁰

Find the value of following expression:

i + i² + i³ + i⁴

Find the value of following expression:

i⁵ + i¹⁰ + i¹⁵

Find the value of following expression:

Find the value of following expression:

1 + i² + i⁴ + i⁶ + i⁸ + ... + i²⁰

Find the value of following expression:

(1 + i)⁶ + (1 – i)³

Express the following complex numbers in the standard form a + i b :

(1 + i) (1 + 2i)

Express the following complex numbers in the standard form a + i b :

Express the following complex numbers in the standard form a + i b :

Express the following complex numbers in the standard form a + i b :

Express the following complex numbers in the standard form a + i b :

Express the following complex numbers in the standard form a + i b :

Express the following complex numbers in the standard form a + i b :

Express the following complex numbers in the standard form a + i b :

Express the following complex numbers in the standard form a + i b :

(1 + 2i)^-3

Express the following complex numbers in the standard form a + i b :

Express the following complex numbers in the standard form a + i b :

Express the following complex numbers in the standard form a + i b :

Find the real values of x and y, if

(x + i y) (2 – 3i) = 4 + i

Find the real values of x and y, if

(3x – 2i y) (2 + i)² = 10 (1 + i)

Find the real values of x and y, if

Find the real values of x and y, if

(1 + i) (x + i y) = 2 – 5i

Find the conjugates of the following complex numbers:

4 – 5 i

Find the conjugates of the following complex numbers:

Find the conjugates of the following complex numbers:

Find the conjugates of the following complex numbers:

Find the conjugates of the following complex numbers:

Find the conjugates of the following complex numbers:

Find the multiplicative inverse of the following complex numbers :

1 – i

Find the multiplicative inverse of the following complex numbers :

(1 + i √3)²

Find the multiplicative inverse of the following complex numbers :

4 – 3 i

Find the multiplicative inverse of the following complex numbers :

√5 + 3i

If z₁ = 2 – i, z₂ = 1 + i, find

If z₁ = 2 – i, z₂ = -2 + i, find

i.

ii.

Find the modulus of

If prove that x² + y² = 1

Find the least positive integral value of n for which is real.

Find the real values of θ for which the complex number is purely real.

Find the smallest positive integer value of n for which is a real number.

If find (x, y)

If find x + y.

If find (a, b).

If a = cos θ + i sin θ, find the value of

Evaluate the following :

2x³ + 2x² – 7x + 72, when

Evaluate the following :

x⁴ – 4x³ + 4x² + 8x + 44, when x = 3 + 2i

Evaluate the following :

x⁴ + 4x³ + 6x² + 4x + 9, when x = - 1 + i√2

Evaluate the following :

x6 + x4 + x2 + 1, when .

Evaluate the following :

2x4 + 5x3 + 7x2 – x + 41, when x = - 2 - √3i

For a positive integer n, find the value of

If then show that

Solve the system of equations Re(z²) = 0, |z| = 2.

If is purely imaginary number (z ≠ – 1), find the value of |z|.

If z₁ is a complex number other than -1 such that |z₁| = 1 and then show that the real parts of z₂ is zero.

If |z + 1| = z + 2(1 + i), find z.

Solve the equation |z| = z + 1 + 2i.

What is the smallest positive integer n for which (1 + i)²ⁿ = (1 – i)²ⁿ?

If z₁, z₂, z₃ are complex numbers such that then find the value of |z₁ + z₂ + z₃|.

Find the number of solutions of z² + |z|² = 0.

Find the square root of the following complex numbers :

-5 + 12i

Find the square root of the following complex numbers :

-7 – 24_i

Find the square root of the following complex numbers :

1 – i

Find the square root of the following complex numbers :

-8 – 6i

Find the square root of the following complex numbers :

8 – 15i

Find the square root of the following complex numbers :

-11 – 60 √-1

Find the square root of the following complex numbers :

1 + 4 √-3

Find the square root of the following complex numbers :

4i

Find the square root of the following complex numbers :

-i

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form :

1 + i

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form :

√3 + i

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form :

1 – i

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form :

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form :

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form :

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form :

sin 120^o – i cos 120^o

Find the modulus and argument of the following complex numbers and hence express each of them in the polar form :

Write (i²⁵)³ in polar form.

Express the following complex numbers in the form

1 + i tan α

Express the following complex numbers in the form

tan α – i

Express the following complex numbers in the form

1 – sin α + i cos α

Express the following complex numbers in the form

If z₁ and z₂ are two complex number such that |z₁| = |z₂| and arg (z₁) + arg (z₂) = π, then show that

If z₁, z₂ and z₃, z₄ are two pairs of conjugate complex numbers, prove that

Express in polar form.

Write the value of the square root of i.

Write the values of the square root of –i.

If then write the value of (x² + y²)².

If π< θ < 2π and z = 1 + cos θ + i sin θ, then write the value of |z|.

If n is any positive integer, write the value of

Write the value of

Write 1 – i in polar form.

Write -1 + i√3 in polar form.

Write the argument of –i.

Write the least positive integral value of n for which is real.

Find the principal argument of

Find z, if |z| = 4 and

If |z – 5i| = |z + 5i|, then find the locus of z.

If find the value of x² + y².

Write the value of

Write the sum of the series i + i² + i³ + …. Upto 1000 terms.

Write the value of

If |z + 4| ≤ 3, then find the greatest and least values of |z + 1|.

for any two complex numbers z₁ and z₂ and any two real numbers a, b find the value of |az₁ – bz₂|² + |az₂ + bz₁|².

Write the conjugate of.

If n ∈ N, then find the value of iⁿ + i^{n + 1} + i^{n + 2} + i^{n + 3}.

Find the real value of a for which 3i³ – 3ai² + (1 – a) i + 5 is real.

If |z| = 2 and find z.

Write the argument of

Mark the Correct alternative in the following:

The value of (1 + i) (1 + i²) (1 + i³)(1 + i⁴) is

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If is a real number and 0 < θ < 2π, then θ =

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If (1 + i) (1 + 2i) (1 + 3i) …. (1 + n i) = a + i b, then 2 × 5 × 10 × … × (1 + n²) is equal to

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If then possible value of is

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If then

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The polar form of (i²⁵)³ is

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If i² = - 1, then the sum i + i² + i³ + …. upto 1000 terms is equal to

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If then the value of arg(z) is

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If a = cos θ + i sin θ, then

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If (1 + i) (1 + 2i) (1 + 3i) … (1 + ni) = a + i b, then 2 . 5. 10 . 17 ……..(1 + n²) =

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If then x² + y² is equal to

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The principal value of the amplitude of
(1 + i) is

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The least positive integer n such that is a positive integer, is

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If z is a non-zero complex number, then is equal to

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If a = 1 + i, then a² equals

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If (x + iy)^1/3 = a + ib, then

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is equal to

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The argument of is

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If then z⁴ equals

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If then arg(z) equals

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If s then |z| =

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If then |z| =

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If z = 1 – cos θ + i sin θ, then |z| =

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If x + i y = (1 + i) (1 + 2 i) (1 + 3i), then x² + y² =

Mark the Correct alternative in the following:

If then Re (z) =

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If then y =

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If then a² + b² =

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If θ is the amplitude of then tan θ =

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If then

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The amplitude of is equal to

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The argument of is

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The amplitude of is

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The value of (i⁵ + i⁶ + i⁷ + i⁸ + i⁹)/(1 + i) is

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equals

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The value of is

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The value of (1 + i)⁴ + (1 – i)⁴ is

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If z = a + ib lies in third quadrant, then also lies in the third quadrant if

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If where z = 1 + 2i, then |f(z)| is

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A real value of x satisfies the equation if a² + b² =

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The complex number z which satisfies the condition lies on

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If z is a complex number, then

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Which of the following is correct for any two complex numbers z₁ and z₂?

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If the complex number z = x + iy satisfies the condition |z + 1| = 1, then z lies on