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:
`(2 + sqrt(-3))/(4 + sqrt(-3))`
Advertisements
उत्तर
`(2 + sqrt(-3))/(4 + sqrt(-3)) = (2 + sqrt(3)"i")/(4 + sqrt(3)"i")`
= `((2 + sqrt(3)"i")(4 -sqrt(3)"i"))/((4 + sqrt(3)"i")(4 - sqrt(3)"i"))`
= `(8 - 2sqrt(3)"i" + 4sqrt(3)"i" - 3"i"^2)/(16 - 3"i"^2)`
= `(8 + 2sqrt(3)"i" - 3(-1))/(16 - 3(-1))` ...[∵ i2 = – 1]
= `(11 + 2sqrt(3)"i")/19`
∴ `(2 + sqrt(-3))/(4 + sqrt(-3)) = 11/19 + (2sqrt(3))/19"i"`
∴ a = `11/19` and b = `(2sqrt(3))/19`
APPEARS IN
संबंधित प्रश्न
Evaluate: `[i^18 + (1/i)^25]^3`
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Find the value of the following expression:
i30 + i80 + i120
Find the value of the following expression:
i + i2 + i3 + i4
Find the real value of x and y, if `((1+i)x-2i)/(3+i) + ((2-3i)y+i)/(3-i) = i, xy ∈ R, i = sqrt-1`
Find the multiplicative inverse of the following complex number:
1 − i
Find the multiplicative inverse of the following complex number:
\[(1 + i\sqrt{3} )^2\]
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
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.
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
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):
1 − sin α + i cos α
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
The principal value of the amplitude of (1 + i) is
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
Which of the following is correct for any two complex numbers z1 and z2?
Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`
Find a and b if abi = 3a − b + 12i
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)−1
Evaluate the following : i35
Evaluate the following : i888
Evaluate the following : i116
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
Show that `(-1 + sqrt3 "i")^3` is a real number.
Show that `(-1+ sqrt(3)i)^3` is a real number.
