Advertisements
Advertisements
प्रश्न
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
Advertisements
उत्तर
\[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\]
\[ \Rightarrow \frac{z}{\bar{z}} = \frac{1 - i}{1 + i}\]
\[ \Rightarrow \frac{z}{\bar{z}} = \frac{1 - i}{1 + i} \times \frac{1 - i}{1 - i}\]
\[ \Rightarrow \frac{z}{\bar{z}} = \frac{1 + i^2 - 2i}{1 - i^2}\]
\[ \Rightarrow \frac{z}{\bar{z}} = \frac{1 - 1 - 2i}{1 + 1} [ \because i^2 = - 1]\]
\[ \Rightarrow \frac{z}{\bar{z}} = \frac{- 2i}{2}\]
\[ \Rightarrow \frac{z}{\bar{z}} = - i\]
\[ \Rightarrow z = - i \bar{z}\]
Hence,
\[z = - i \bar{z}\].
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 – i) – (–1 + i6)
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:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
Evaluate the following:
i457
Evaluate the following:
(ii) i528
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
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:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
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 \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
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 \[\left( 1 + \sqrt{3} \right)\].
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
Which of the following is correct for any two complex numbers z1 and z2?
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
Find a and b if (a – b) + (a + b)i = a + 5i
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))`
Show that `(-1 + sqrt(3)"i")^3` is a real number
Evaluate the following : i888
Evaluate the following : i–888
Evaluate the following : i30 + i40 + i50 + i60
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
State True or False for the following:
The order relation is defined on the set of complex numbers.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
