Advertisements
Advertisements
प्रश्न
Express the following complex number in the standard form a + i b:
\[\left( \frac{1}{1 - 4i} - \frac{2}{1 + i} \right)\left( \frac{3 - 4i}{5 + i} \right)\]
Advertisements
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: i9 + i19
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Find the value of the following expression:
i30 + i80 + i120
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
Find the multiplicative inverse of the following complex number:
1 − i
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
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 \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
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 n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
The polar form of (i25)3 is
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
The argument of \[\frac{1 - i}{1 + i}\] is
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
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))`
Evaluate the following : i116
Evaluate the following : `1/"i"^58`
Evaluate the following : i–888
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)`.
Match the statements of column A and B.
| Column A | Column B |
| (a) The value of 1 + i2 + i4 + i6 + ... i20 is | (i) purely imaginary complex number |
| (b) The value of `i^(-1097)` is | (ii) purely real complex number |
| (c) Conjugate of 1 + i lies in | (iii) second quadrant |
| (d) `(1 + 2i)/(1 - i)` lies in | (iv) Fourth quadrant |
| (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |
| (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 |
(vi) may occur in conjugate pairs |
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
Show that `(-1+ sqrt(3)i)^3` is a real number.
