Advertisements
Advertisements
Question
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Advertisements
Solution
i6 = (i2)3 = (–1)3 = – 1
i7 = i6 × i = (i2)3i = (– 1)3i = – i
i11 = i10 × i = (i2)5i = (– 1)5i = – i
`∴(3 + 2/i)(i^6 - i^7)(1 + i^11)`
`= (3 + 2/i)(-1 - (-i))(1 +(-i))`
`= (3 + 2/i)(-1 + i)(1 - i)`
= `(3 + 2/i)(1 - i)(1 - i)`
= `-(3 + 2/i)(1 - i)(1 - i)`
= `-(3 + 2/i)(1 - i)^2`
= `-(3 + 2/i)(1 - 2i + i^2)`
= `-(3 + 2/i)(1 - 2i + i^2)` ...[∵ i2 = −1]
= `-(3 + 2/i)(-2i)`
= 6i + 4
= 4 + 6i
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Show that 1 + i10 + i20 + i30 is a real number.
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:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
Find the real value of x and y, if
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
If π < θ < 2π and z = 1 + cos θ + i sin θ, then write the value of \[\left| z \right|\] .
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
For any two complex numbers z1 and z2 and any two real numbers a, b, find the value of \[\left| a z_1 - b z_2 \right|^2 + \left| a z_2 + b z_1 \right|^2\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =
The polar form of (i25)3 is
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
Find a and b if abi = 3a − b + 12i
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 : i403
Evaluate the following : i30 + i40 + i50 + i60
Show that 1 + i10 + i20 + i30 is a real number
State True or False for the following:
2 is not a complex number.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Show that `(-1+sqrt3i)^3` is a real number.
