Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
(1 + i)−3 = `1/((1 + "i")^3`
= `1/(1 + 3"i" + 3"i"^2 + "i"^3)`
= `1/(1 + 3"i" - 3 - "i")` ...[∵ i2 = – 1, i3 = – i]
= `1/(-2 + 2"i")`
= `1/(-2 + 2"i") xx (-2 - 2"i")/(-2 - 2"i")`
= `(-2 - 2"i")/(4 - 4"i"^2)`
= `(-2 - 2"i")/(4 + 4)` ...[∵ i2 = – 1]
= `(-2 -2"i")/8`
= `-1/4 - 1/4 "i"`
This is of the form a + bi, where a = `-1/4` and b = `-1/4`
APPEARS IN
संबंधित प्रश्न
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Evaluate the following:
i457
Evaluate the following:
\[\frac{1}{i^{58}}\]
Find the value of the following expression:
i49 + i68 + i89 + i110
Find the value of the following expression:
i5 + i10 + i15
Express the following complex number in the standard form a + i b:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
Find the real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_1))`
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
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):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
Write 1 − i in polar form.
Write the argument of −i.
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
The polar form of (i25)3 is
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
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+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`
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Show that 1 + i10 + i20 + i30 is a real number
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
State True or False for the following:
2 is not a complex number.
Show that `(-1 + sqrt3 "i")^3` is a real number.
Show that `(-1+ sqrt(3)i)^3` is a real number.
