Advertisements
Advertisements
प्रश्न
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Advertisements
उत्तर
\[\frac{i^{4n + 1} - i^{4n - 1}}{2}\]
\[ = \frac{i - \frac{1}{i}}{2} \left( \because i^{4n} = 1, i^{- 1} = \frac{1}{i} \right)\]
\[ = \frac{\frac{i^2 - 1}{i}}{2}\]
\[ = \frac{i^2 - 1}{2i}\]
\[ = \frac{- 1 - 1}{2i}\]
\[ = \frac{- 2}{- 2i} \]
\[ = \frac{- 1}{i}\]
\[ = \frac{- i}{i^2} \left( \because i^2 = - 1 \right)\]
\[ = \frac{- i}{- 1}\]
\[ = i\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: (1 – i)4
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Evaluate the following:
\[\frac{1}{i^{58}}\]
Find the value of the following expression:
(1 + i)6 + (1 − i)3
Express the following complex number in the standard form a + i b:
\[\frac{3 + 2i}{- 2 + i}\]
Express the following complex number in the standard form a + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
If \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
If z1 and z2 are two complex numbers such that \[\left| z_1 \right| = \left| z_2 \right|\] and arg(z1) + arg(z2) = \[\pi\] then show that \[z_1 = - \bar{{z_2}}\].
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Write −1 + i \[\sqrt{3}\] in polar form .
Write the argument of −i.
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
The polar form of (i25)3 is
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`
Find a and b if a + 2b + 2ai = 4 + 6i
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:
`((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`
Show that `(-1 + sqrt(3)"i")^3` is a real number
Evaluate the following : i35
Show that 1 + i10 + i20 + i30 is a real number
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
Show that `(-1+ sqrt(3)i)^3` is a real number.
