Advertisements
Advertisements
प्रश्न
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)`
Advertisements
उत्तर
i8 = (i2)4 = (–1)4 = 1
i9 = i8 × i = (i2)4i = (– 1)4i = i
i11 = i10 × i = (i2)5i = (– 1)5i = – i
i10 = (i2)5 = (– 1)5 = – 1
∴ `(4"i"^8 - 3"i"^9 + 3)/(3"i"^11 - 4"i"^10 - 2) = (4(1) - 3"i" + 3)/(3(-"i") - 4(-1) - 2)`
= `(4 - 3"i" + 3)/(-3"i" + 4 - 2)`
= `(7 - 3"i")/(2 - 3"i")`
= `(7 - 3"i")/(2 - 3"i") xx (2 + 3"i")/(2 + 3"i")`
= `(14 + 21"i" - 6"i" - 9"i"^2)/(4 - 9"i"^2)`
= `(14 + 15"i" + 9)/(4 + 9)` ...[∵ i2 = – 1]
= `(23 + 15"i")/13`
= `23/13 + 15/13"i"`
This is of the form a + bi, where a = `23/13` and b = `15/13`.
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: i9 + i19
Express the given complex number in the form a + ib:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
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 + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
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:
\[(1 + 2i )^{- 3}\]
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
Im `(1/(z_1overlinez_1))`
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
If \[a = \cos\theta + i\sin\theta\], find the value of \[\frac{1 + a}{1 - a}\].
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
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 \[\sqrt{3}\] in polar form .
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
The value of \[(1 + i )^4 + (1 - i )^4\] is
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
Find a and b if (a+b) (2 + i) = b + 1 + (10 + 2a)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))`
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Evaluate the following : i35
Evaluate the following : i116
Evaluate the following : i30 + i40 + i50 + i60
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
