Advertisements
Advertisements
प्रश्न
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
Advertisements
उत्तर
\[ \frac{2 + 3i}{4 + 5i}\]
\[ = \frac{2 + 3i}{4 + 5i} \times \frac{4 - 5i}{4 - 5i}\]
\[ = \frac{8 - 10i + 12i - 15 i^2}{16 - 25 i^2} \left( \because i^2 = - 1 \right)\]
\[ = \frac{23 + 2i}{16 + 25}\]
\[ = \frac{23}{41} + \frac{2}{41}i\]
APPEARS IN
संबंधित प्रश्न
Evaluate the following:
i457
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[\left( i^{41} + \frac{1}{i^{257}} \right)^9\]
Find the value of the following expression:
i30 + i80 + i120
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
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:
\[\frac{1}{(2 + i )^2}\]
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
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 real value of x and y, if
\[(x + iy)(2 - 3i) = 4 + i\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
Find the real value of x and y, if `((1+i)x-2i)/(3+i) + ((2-3i)y+i)/(3-i) = i, xy ∈ R, i = sqrt-1`
If \[z_1 = 2 - i, z_2 = - 2 + i,\] find
Re \[\left( \frac{z_1 z_2}{z_1} \right)\]
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
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.
Write (i25)3 in polar form.
Express the following complex in the form r(cos θ + i sin θ):
tan α − i
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
Write −1 + i \[\sqrt{3}\] in polar form .
Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
Which of the following is correct for any two complex numbers z1 and z2?
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 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:
(1 + i)(1 − i)−1
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"))`
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(2 + 3i)(2 – 3i)
Evaluate the following : i888
Evaluate the following : `1/"i"^58`
If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
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 w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
