Advertisements
Advertisements
प्रश्न
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Advertisements
उत्तर
z1 = 2 – i, z2 = –2 + i
`((z_1z_2)/barz_1) = ((2 - i)(-2 +i))/(2 -i) = (-(2 - i)(2 -i))/(2 + i)`
= `- (2-i)^2/(2 + i) = (- (4 + i^2 - 4i))/(2 + i)`
= `(-(4 - 1 - 4i))/((2 + i)) = -(3 - 4i)/(2 + i)`
= `-(3 - 4i)/(2 + i) xx (2 - i)/(2 - i)`
= `(- 6 - 4i^2 + 3i + 8i)/(4 - i^2) = (- 6 + 4 + 11i)/(4 + 1)`
= `(- 2 + 11i)/5 = - 2/5 + 11/5 i`
Re`((z_1z_2)/barz_1) = - 2/5`
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: i–39
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: `(1/5 + i 2/5) - (4 + i 5/2)`
Express the given complex number in the form a + ib: (1 – i)4
Evaluate the following:
i457
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Find the value of the following expression:
i5 + i10 + i15
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
\[(1 + i)(x + iy) = 2 - 5i\]
If \[z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
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.
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 \[z_1 = - \bar{{z_2}}\].
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Write the argument of −i.
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
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 `(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 \[\frac{1 + a}{1 - a} =\]
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) 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 `1/("a" + "ib")` = 3 – 2i
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((2 + "i"))/((3 - "i")(1 + 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))`
If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
If z1 and z2 both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.
Show that `(-1 + sqrt3 "i")^3` is a real number.
