Advertisements
Advertisements
प्रश्न
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
उत्तर
`(1/3 + 3i)^3 = (1/3)^3 + 3 (1/3)^2 (3i) + 3(1/3) (3i)^2 + (3i)^3`
= `1/27 + 27i^3 + 9i^2`
= `1/27 + 27 (-i) + i + 9 (-1)`
= `1/27 - 9 + i (1 - 27)`
= `(1 - 243)/27 + i (-26)`
= `(- 242)/27 - 26i`
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib: i9 + i19
Express the given complex number in the form a + ib: i–39
Evaluate: `[i^18 + (1/i)^25]^3`
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
i + i2 + i3 + i4
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
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
\[(x + iy)(2 - 3i) = 4 + i\]
Find the real value of x and y, if
\[\frac{(1 + i)x - 2i}{3 + i} + \frac{(2 - 3i)y + i}{3 - i}\]
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 \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Im `(1/(z_1overlinez_1))`
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
Express the following complex in the form r(cos θ + i sin θ):
\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write 1 − i in polar form.
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
If z is a complex number, then
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Simplify : `sqrt(-16) + 3sqrt(-25) + sqrt(-36) - sqrt(-625)`
Find the value of `(3 + 2/"i")("i"^6 - "i"^7)(1 + "i"^11)`
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 |
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
Show that `(-1 + sqrt3 "i")^3` is a real number.
