Advertisements
Advertisements
प्रश्न
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
विकल्प
1
\[1/\sqrt{26}\]
\[5/\sqrt{26}\]
none of these
Advertisements
उत्तर
\[1/\sqrt{26}\]
\[\text { Let }z = \frac{1}{\left( 1 - i \right)\left( 2 + 3i \right)}\]
\[ \Rightarrow z = \frac{1}{2 + i - 3 i^2} \]
\[ \Rightarrow z = \frac{1}{2 + i + 3}\]
\[\Rightarrow z=\frac{1}{5 + i}\times\frac{5 - i}{5 - i}\]
\[\Rightarrow z=\frac{5 - i}{25 - i^2}\]
\[ \Rightarrow z=\frac{5 - i}{25 + 1}\]
\[\Rightarrow z=\frac{5 - i}{26}\]
\[\Rightarrow z = \frac{5}{26} - \frac{i}{26}\]
\[\Rightarrow \left| z \right|=\sqrt{\frac{25}{676} + \frac{1}{676}}\]
\[\Rightarrow z = \frac{1}{\sqrt{26}}\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Express the given complex number in the form a + ib:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
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))`
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
i30 + i80 + i120
Express the following complex number in the standard form a + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
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:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
Find the multiplicative inverse of the following complex number:
1 − i
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
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:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
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):
1 − sin α + i cos α
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
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 + 2b + 2ai = 4 + 6i
Find a and b if (a+b) (2 + i) = b + 1 + (10 + 2a)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"))^2`
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
