Advertisements
Advertisements
प्रश्न
Express the following complex number in the standard form a + i b:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
Advertisements
उत्तर
\[\frac{\left( 1 + i \right)\left( 1 + \sqrt{3i} \right)}{1 - i}\]
\[ = \frac{\left( 1 + i \right)\left( 1 + \sqrt{3i} \right)}{1 - i} \times \frac{1 + i}{1 + i}\]
\[ = \frac{\left( 1 + \sqrt{3}i \right)\left( 1 + i^2 + 2i \right)}{1 - i^2} \left( \because i^2 = - 1 \right)\]
\[ = \frac{\left( 1 + \sqrt{3}i \right)2i}{2}\]
\[ = i\left( 1 + \sqrt{3}i \right)\]
\[ = i + \sqrt{3} i^2 \]
\[ = - \sqrt{3} + i\]
APPEARS IN
संबंधित प्रश्न
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Find the value of the following expression:
i + i2 + i3 + i4
Express the following complex number in the standard form a + i b:
\[(1 + i)(1 + 2i)\]
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:
\[\frac{(1 - i )^3}{1 - i^3}\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
If \[z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
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.
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
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.
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
If z is a complex number, then
Find a and b if abi = 3a − b + 12i
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:
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"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)−3
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)
Show that `(-1 + sqrt(3)"i")^3` is a real number
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Evaluate the following : i888
Evaluate the following : i403
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
Show that `(-1 + sqrt3 "i")^3` is a real number.
