Advertisements
Advertisements
प्रश्न
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
पर्याय
0
1
−1
none of these
Advertisements
उत्तर
none of these
\[\left( x + iy \right)^\frac{1}{3} = a + ib\]
\[\text { Cubing on both the sides, we get }: \]
\[x + iy = \left( a + ib \right)^3 \]
\[ \Rightarrow x + iy = a^3 + \left( ib \right)^3 + 3 a^2 bi + 3a \left( ib \right)^2 \]
\[ \Rightarrow x + iy = a^3 + i^3 b^3 + 3 a^2 ib + 3 i^2 a b^2 \]
\[ \Rightarrow x + iy = a^3 - i b^3 + 3 a^2 ib - 3a b^2 ( \because i^2 = - 1, i^3 = - i)\]
\[ \Rightarrow x + iy = a^3 - 3a b^2 + i\left( - b^3 + 3 a^2 b \right)\]
\[ \therefore x = a^3 - 3a b^2 \text { and }y = 3 a^2 b - b^3 \]
\[or , \frac{x}{a} = a^2 - 3 b^2\text { and } \frac{y}{b} = 3 a^2 - b^2 \]
\[ \Rightarrow \frac{x}{a} + \frac{y}{b} = a^2 - 3 b^2 + 3 a^2 - b^2 \]
\[ \Rightarrow \frac{x}{a} + \frac{y}{b} = 4 a^2 - 4 b^2\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
Evaluate the following:
i457
Express the following complex number in the standard form a + i b:
\[\frac{3 + 2i}{- 2 + i}\]
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 + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
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`
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_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 \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Solve the equation \[\left| z \right| = z + 1 + 2i\].
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
If z1 and z2 are two complex numbers such that \[\left| z_1 \right| = \left| z_2 \right|\] and arg(z1) + arg(z2) = \[\pi\] then show that \[z_1 = - \bar{{z_2}}\].
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Write 1 − i in polar form.
Find z, if \[\left| z \right| = 4 \text { and }\arg(z) = \frac{5\pi}{6} .\]
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
Find a and b if (a – b) + (a + b)i = a + 5i
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:
`(2 + sqrt(-3))/(4 + sqrt(-3))`
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2)`
Show that `(-1 + sqrt(3)"i")^3` is a real number
Evaluate the following : i888
Evaluate the following : i403
Evaluate the following : `1/"i"^58`
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
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.
Show that `(-1+sqrt3i)^3` is a real number.
