Advertisements
Advertisements
प्रश्न
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
Advertisements
उत्तर
\[ x = \frac{1 + i}{\sqrt{2}}\]
\[ \Rightarrow x^2 = \left( \frac{1 + i}{\sqrt{2}} \right)^2 \]
\[ = \left( \frac{1 + i^2 + 2i}{2} \right)\]
\[ = \frac{2i}{2}\]
\[ = i\]
\[ \Rightarrow x^6 = \left( x^2 \right)^3 \]
\[ = i^3 \]
\[ = - i\]
\[ \Rightarrow x^2 = i\]
\[ \Rightarrow x^4 = \left( x^2 \right)^2 \]
\[ = i^2 \]
\[ = - 1\]
\[\text { Now }, x^6 + x^4 + x^2 + 1 = - i - 1 + i + 1\]
\[ = 0\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: (1 – i)4
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
i457
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Express the following complex number in the standard form a + i b:
\[\frac{1}{(2 + i )^2}\]
Express the following complex number in the standard form a + i b:
\[\frac{1 - i}{1 + i}\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
The amplitude of \[\frac{1}{i}\] is equal to
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
If z is a complex number, then
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Find a and b if (a – b) + (a + b)i = a + 5i
Find a and b if (a + ib) (1 + i) = 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 + 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)(1 − i)−1
Express the following in the form of a + ib, a, b ∈ R i = `sqrt(−1)`. State the values of a and b:
`(2 + sqrt(-3))/(4 + sqrt(-3))`
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`
Evaluate the following : i116
Evaluate the following : `1/"i"^58`
State True or False for the following:
2 is not a complex number.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Show that `(-1 + sqrt3 "i")^3` is a real number.
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
