Advertisements
Advertisements
प्रश्न
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Advertisements
उत्तर
\[ x = - 1 + \sqrt{2}i\]
\[ \Rightarrow x^2 = \left( - 1 + \sqrt{2}i \right)^2 \]
\[ = 1 + 2 i^2 - 2\sqrt{2}i\]
\[ = - 1 - 2\sqrt{2}i\]
\[ \Rightarrow x^3 = \left( - 1 - 2\sqrt{2}i \right) \times \left( - 1 + \sqrt{2}i \right)\]
\[ = 1 - \sqrt{2}i + 2\sqrt{2}i - 4 i^2 \]
\[ = 5 + \sqrt{2}i\]
\[ \Rightarrow x^4 = \left( - 1 - 2\sqrt{2}i \right)^2 \]
\[ = 1 + 8 i^2 + 4\sqrt{2}i\]
\[ = - 7 + 4\sqrt{2}i\]
\[ \Rightarrow x^4 + 4 x^3 + 6 x^2 + 4x + 9 = - 7 + 4\sqrt{2}i + 4\left( 5 + \sqrt{2}i \right) + 6\left( - 1 - 2\sqrt{2}i \right) + 4\left( - 1 + \sqrt{2}i \right) + 9\]
\[ = - 7 + 4\sqrt{2}i + 20 + 4\sqrt{2}i - 6 - 12\sqrt{2}i - 4 + 4\sqrt{2}i + 9\]
\[ = 12\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
Evaluate: `[i^18 + (1/i)^25]^3`
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
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:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
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 \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
If z1 is a complex number other than −1 such that \[\left| z_1 \right| = 1\] and \[z_2 = \frac{z_1 - 1}{z_1 + 1}\] then show that the real parts of z2 is zero.
If z1, z2, z3 are complex numbers such that \[\left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1\] then find the value of \[\left| z_1 + z_2 + z_3 \right|\] .
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
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 .
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
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 \[\left( 1 + \sqrt{3} \right)\].
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
The polar form of (i25)3 is
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
The argument of \[\frac{1 - i}{1 + i}\] is
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
Find a and b if (a – b) + (a + b)i = a + 5i
Find a and b if abi = 3a − b + 12i
Evaluate the following : i888
Show that 1 + i10 + i20 + i30 is a real number
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
Show that `(-1+ sqrt(3)i)^3` is a real number.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
