Advertisements
Advertisements
प्रश्न
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
उत्तर
`(1/(1 - 4i) - 2/(1 + i))((3 - 4i)/(5 + i))`
= `(1 + i - 2(1 - 4i))/((1 - 4i)(1 + i)) xx (3 - 4i)/(5 + i)`
= `(1 + i - 2 + 8i)/(1(1 + i)-4i(1 + i))xx (3 - 4i)/(5 + i)`
= `(-1+9i)/((1 + i - 4i + 4)) xx (3 - 4i)/(5 + i)`
= `(-1 + 9i)/((1 + i - 4i + 4)) xx (3 - 4i)/(5 + i)`
= `(-1(3 - 4i) + 9i(3 - 4i))/((5 - 3i)(5 + i))`
= `(-3 + 4i + 27i + 36)/(5(5 + i)-3i(5 + i))`
= `(33 + 31j)/(25 + 5i - 15i + 3)`
= `(33 + 31j)/(28 - 10i)`
= `(33 + 31j)/(28 - 10i) xx ((28 + 10i))/(28 + 10i)`
= `(33 xx 28 + 33 xx 10i + 31i xx 28 + 31i xx 10i)/(28^2 + 10^2)`
= `(924 + 330i + 868i - 310)/(784 + 100)`
= `(614 + 1198i)/(884)`
= `614/884 + (1198)/884 i`
= `307/442 + 599/442 i`
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
i457
Find the value of the following expression:
i + i2 + i3 + i4
Find the value of the following expression:
(1 + i)6 + (1 − i)3
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 + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
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:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
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 \[\left| z_1 + z_2 + z_3 \right|\] .
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
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 \[\left| z \right|\] .
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
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 \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
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:
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"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)`
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
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.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Show that `(-1 + sqrt3 "i")^3` is a real number.
