Advertisements
Advertisements
प्रश्न
The value of \[(1 + i )^4 + (1 - i )^4\] is
विकल्प
8
4
-8
-4
Advertisements
उत्तर
-8
\[\text { Using } a^4 + b^4 = \left( a^2 + b^2 \right)^2 - 2 a^2 b^2 \]
\[(1 + i )^4 + (1 - i )^4 \]
\[ = \left( \left( 1 + i \right)^2 + \left( 1 - i \right)^2 \right)^2 - 2 \left( 1 + i \right)^2 \left( 1 - i \right)^2 \]
\[ = \left( 1 + i^2 + 2i + 1 + i^2 - 2i \right)^2 - 2\left( 1 + i^2 + 2i \right)\left( 1 + i^2 - 2i \right) \]
\[ = \left( 1 - 1 + 2i + 1 - 1 - 2i \right)^2 - 2\left( 1 - 1 + 2i \right)\left( 1 - 1 - 2i \right)\]
\[ = \left( 0 \right) - 2\left( 2i \right)\left( - 2i \right) \left( \because i^2 = - 1 \right)\]
\[ = 8 i^2 \]
\[ = - 8\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: i–39
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Express the following complex number in the standard form a + i b:
\[(1 + 2i )^{- 3}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Find the multiplicative inverse of the following complex number:
1 − i
If \[z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write −1 + i \[\sqrt{3}\] in polar form .
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
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:
(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 + 3i)(2 – 3i)
Evaluate the following : i116
Evaluate the following : i403
Evaluate the following : i30 + i40 + i50 + i60
State True or False for the following:
The order relation is defined on the set of complex numbers.
Show that `(-1 + sqrt3 "i")^3` is a real number.
Show that `(-1+sqrt3i)^3` is a real number.
