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: i9 + i19
Express the given complex number in the form a + ib: `(1/5 + i 2/5) - (4 + i 5/2)`
Express the given complex number in the form a + ib: (1 – i)4
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
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:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
Find the real value of x and y, if `((1+i)x-2i)/(3+i) + ((2-3i)y+i)/(3-i) = i, xy ∈ R, i = sqrt-1`
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
If \[z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
If \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
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):
tan α − i
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Evaluate the following : i116
Evaluate the following : `1/"i"^58`
Evaluate the following : i30 + i40 + i50 + i60
Show that 1 + i10 + i20 + i30 is a real number
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
State True or False for the following:
2 is not a complex number.
