If 1 - i z z - i = 1 , then z lies on

a line not passing through origin

MEDIUM

Earn 100

Which index number is used to measure changes in the cost of living?

50% studentsanswered this correctly

Important Questions on Binomial Theorem

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

If the coefficient of the three successive terms in the binomial expansion of

{(1 + x)}^{n}

are in the ratio

1 : 7 : 42

, then the first of these terms in the expansion is

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

For all complex numbers

z

of the form

1 + i α, α \in R,

z^{2} = x + i y,

then

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

Time

(T)

, velocity

(C)

and angular momentum

(h)

are chosen as fundamental quantities instead of mass, length and time. In terms of these, the dimensions of mass would be:

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

The value of sum

\sum_{n = 1}^{13} (i^{n} + i^{n + 1}), where i = \sqrt{- 1},

equals

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

The equation, whose roots are

i, \frac{1 + i}{\sqrt{2}}, \frac{i - 1}{\sqrt{2}},

their conjugates,

- 1 and 1

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

The maximum value of

|z|

where

z

satisfies the condition

|z + \frac{2}{z}| = 2

, is

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

z

is a complex number such that

Re (z) = I m (z)

, then

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

(3 + i) z = (3 - i) \bar{z}

, then the complex number

z

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

The value of

\sum_{n = 0}^{100} i^{n!}

equals (where

i = \sqrt{- 1}

)

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

The relation between the real numbers

a

and

b

, which satisfy the equation

\frac{1 - i x}{1 + i x} = a - i b

, for some real value of

x

, is

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

Let

z

be a complex number such that the imaginary part of

z

is nonzero and

a = z^{2} + z + 1

is real. Then

a

cannot take the value

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

For positive integers

n_{1}, n_{2}

the value of the expression

{(1 + i)}^{n_{1}} + {(1 + i^{3})}^{n_{1}} + {(1 + i^{5})}^{n_{2}} + {(1 + i^{7})}^{n_{2}}, i = \sqrt{- 1}

is a real number if and only if

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

|\frac{1 - i z}{z - i}| = 1,

then

z

lies on

MEDIUM

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

The value of sum

\sum_{n = 1}^{13} (i^{n} + i^{n + 1})

, where

i = \sqrt{- 1}

, equals

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

If the complex number

z

satisfy the equation

(i - z) (1 + 2 i) + (1 - i z) (3 - 4 i) = 1 + 7 i,

then

z + z + z z

is equal to -

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

If $a = \cos α + i \sin α, b = \cos β + i \sin β$ , $c = \cos γ + i \sin γ$ and $\frac{b}{c} + \frac{c}{a} + \frac{a}{b} = 1$ , then $\cos (β - γ) + \cos (γ - α) + \cos (α - β)$ is equal to $(where i = \sqrt{- 1})$

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

Evaluate

{(- \sqrt{- 1})}^{4 n + 3}, n \in N

HARD

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

Dividing

f (z)

z - i

, we obtain the remainder

i

and dividing it by

z + i

, we get the remainder

1 + i

. The remainder upon the division of

f (z)

z^{2} + 1

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

Δ = |\begin{array}{l} \sqrt{6} & 2 i & 3 + \sqrt{6} \\ \sqrt{12} & \sqrt{3} + \sqrt{8} i & 3 \sqrt{2} + \sqrt{6} i \\ \sqrt{18} & \sqrt{2} + \sqrt{12} i & \sqrt{27} + 2 i \end{array}|

, then

Δ

EASY

Mathematics>Algebra>Binomial Theorem>Binomial Theorem for Positive Integral Indices

The complex number which satisfy the equation

z + \sqrt{2} |z + 1| + i = 0

Which index number is used to measure changes in the cost of living?

Important Questions on Binomial Theorem

Learn and Prepare for any exam you want !