#Stay_Ahead of your Class

Listen NCERT Audio Books to boost your productivity and retention power by 2X.

Evaluate the following:

(i) i^{457}

(ii) i^{528}

(iii)

(iv)

(v)

(vi) (i^{77} + i^{70} + i^{87} + i^{414} )^{3}

(vii) (vii) i^{30} + i^{40} + i^{60}

(viii) i^{49} + i^{68} + i^{89} + i^{118}

Show that 1 + i^{10} + i^{20} + i^{30} is a real number ?

Find the value of following expression:

i^{49} + i^{68} + i^{89} + i^{110}

i^{30} + i^{80} + i^{120}

i + i^{2} + i^{3} + i^{4}

i^{5} + i^{10} + i^{15}

1 + i^{2} + i^{4} + i^{6} + i^{8} + ... + i^{20}

(1 + i)^{6} + (1 – i)^{3}

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

(1 + i) (1 + 2i)

(1 + 2i)^{-3}

Find the real values of x and y, if

(x + i y) (2 – 3i) = 4 + i

(3x – 2i y) (2 + i)^{2} = 10 (1 + i)

(1 + i) (x + i y) = 2 – 5i

Find the conjugates of the following complex numbers:

4 – 5 i

Find the multiplicative inverse of the following complex numbers :

1 – i

(1 + i √3)^{2}

4 – 3 i

√5 + 3i

If z_{1} = 2 – i, z_{2} = 1 + i, find

If z_{1} = 2 – i, z_{2} = -2 + i, find

i.

ii.

Find the modulus of

If prove that x^{2} + y^{2} = 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 :

2x^{3} + 2x^{2} – 7x + 72, when

x^{4} – 4x^{3} + 4x^{2} + 8x + 44, when x = 3 + 2i

x^{4} + 4x^{3} + 6x^{2} + 4x + 9, when x = - 1 + i√2

x6 + x4 + x2 + 1, when .

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(z^{2}) = 0, |z| = 2.

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

If z_{1} is a complex number other than -1 such that |z_{1}| = 1 and then show that the real parts of z_{2} 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 z_{1}, z_{2}, z_{3} are complex numbers such that then find the value of |z_{1} + z_{2} + z_{3}|.

Find the number of solutions of z^{2} + |z|^{2} = 0.

Find the square root of the following complex numbers :

-5 + 12i

-7 – 24_{i}

-8 – 6i

8 – 15i

-11 – 60 √-1

1 + 4 √-3

4i

-i

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

1 + i

√3 + i

sin 120^{o} – i cos 120^{o}

Write (i^{25})^{3} in polar form.

Express the following complex numbers in the form

1 + i tan α

tan α – i

1 – sin α + i cos α

If z_{1} and z_{2} are two complex number such that |z_{1}| = |z_{2}| and arg (z_{1}) + arg (z_{2}) = π, then show that

If z_{1}, z_{2} and z_{3}, z_{4} 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 (x^{2} + y^{2})^{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 x^{2} + y^{2}.

Write the sum of the series i + i^{2} + i^{3} + …. Upto 1000 terms.

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

for any two complex numbers z_{1} and z_{2} and any two real numbers a, b find the value of |az_{1} – bz_{2}|^{2} + |az_{2} + bz_{1}|^{2}.

Write the conjugate of.

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

Find the real value of a for which 3i^{3} – 3ai^{2} + (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 + i^{2}) (1 + i^{3})(1 + i^{4}) is

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

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

If then possible value of is

If then

The polar form of (i^{25})^{3} is

If i^{2} = - 1, then the sum i + i^{2} + i^{3} + …. upto 1000 terms is equal to

If then the value of arg(z) is

If a = cos θ + i sin θ, then

If (1 + i) (1 + 2i) (1 + 3i) … (1 + ni) = a + i b, then 2 . 5. 10 . 17 ……..(1 + n^{2}) =

If then x^{2} + y^{2} is equal to

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

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

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

If a = 1 + i, then a^{2} equals

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

is equal to

The argument of is

If then z^{4} equals

If then arg(z) equals

If s then |z| =

If then |z| =

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

If x + i y = (1 + i) (1 + 2 i) (1 + 3i), then x^{2} + y^{2} =

If then Re (z) =

If then y =

If then a^{2} + b^{2} =

If θ is the amplitude of then tan θ =

The amplitude of is equal to

The amplitude of is

The value of (i^{5} + i^{6} + i^{7} + i^{8} + i^{9})/(1 + i) is

equals

The value of is

The value of (1 + i)^{4} + (1 – i)^{4} is

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

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

A real value of x satisfies the equation if a^{2} + b^{2} =

The complex number z which satisfies the condition lies on

If z is a complex number, then

Which of the following is correct for any two complex numbers z_{1} and z_{2}?

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