Advertisements
Advertisements
Question
Show that 1 + i10 + i20 + i30 is a real number
Advertisements
Solution
1 + i10 + i20 + i30
= 1 + (i4)2.i2 + (i4)5 + (i4)7.i2
= 1 + (1)2 (–1) + (1)5 + (1)7 (–1) ...[∵ i4 = 1, i2 = – 1]
= 1 – 1 + 1 – 1
= 0, which is a real number.
APPEARS IN
RELATED QUESTIONS
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))`
Evaluate the following:
\[\frac{1}{i^{58}}\]
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
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:
\[\frac{1}{(2 + i )^2}\]
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 + ib:
\[\frac{(2 + i )^3}{2 + 3i}\]
Express the following complex number in the standard form a + i b:
\[\frac{2 + 3i}{4 + 5i}\]
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 \[\left( \frac{z_1 z_2}{z_1} \right)\]
If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).
Evaluate the following:
\[x^6 + x^4 + x^2 + 1, \text { when }x = \frac{1 + i}{\sqrt{2}}\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
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.
Find the number of solutions of \[z^2 + \left| z \right|^2 = 0\].
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write 1 − i in polar form.
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
The argument of \[\frac{1 - i}{1 + i}\] is
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
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
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:
`(- sqrt(5) + 2sqrt(-4)) + (1 -sqrt(-9)) + (2 + 3"i")(2 - 3"i")`
Evaluate the following : i116
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
Show that `(-1+sqrt3i)^3` is a real number.
