Advertisements
Advertisements
प्रश्न
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
Advertisements
उत्तर
\[ z = \left( 1 + \sqrt{3}i \right)^2 \]
\[ = 1 + 3 i^2 + 2\sqrt{3}i\]
\[ = - 2 + 2\sqrt{3}i\]
\[\text { Then }, \frac{1}{z} = \frac{1}{- 2 + 2\sqrt{3}i} \times \frac{- 2 - 2\sqrt{3}i}{- 2 - 2\sqrt{3}i}\]
\[ = \frac{- 2 - 2\sqrt{3}i}{4 - 12 i^2}\]
\[ = \frac{- 2 - 2\sqrt{3}i}{16}\]
\[ = \frac{- 1}{8} - \frac{\sqrt{3}}{8}i\]
APPEARS IN
संबंधित प्रश्न
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 + 3i)^3`
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Evaluate the following:
(ii) i528
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Show that 1 + i10 + i20 + i30 is a real number.
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:
\[(1 + 2i )^{- 3}\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the multiplicative inverse of the following complex number:
1 − i
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
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 \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
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 the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
The polar form of (i25)3 is
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
The value of \[(1 + i )^4 + (1 - i )^4\] is
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`("i"(4 + 3"i"))/((1 - "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
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Evaluate the following : i116
If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
State True or False for the following:
The order relation is defined on the set of complex numbers.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
