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: (1 – i) – (–1 + i6)
Express the given complex number in the form a + ib: (1 – i)4
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Show that 1 + i10 + i20 + i30 is a real number.
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:
\[(1 + i)(1 + 2i)\]
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:
\[(1 + 2i )^{- 3}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Find the multiplicative inverse of the following complex number:
1 − i
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
If \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
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, 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 \[\left| z_1 + z_2 + z_3 \right|\] .
Write −1 + i \[\sqrt{3}\] in polar form .
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
The polar form of (i25)3 is
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
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?
Find a and b if a + 2b + 2ai = 4 + 6i
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
(1 + 2i)(– 2 + i)
Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:
`("i"(4 + 3"i"))/((1 - "i"))`
Evaluate the following : i–888
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
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 |
Show that `(-1+sqrt3i)^3` is a real number.
