Advertisements
Advertisements
Question
Evaluate the following:
(ii) i528
Solution
`i^(528) = i^(4 xx 132)`
\[ = \left( i^4 \right)^{132} \]
\[ = 1 \left( \because i^4 = 1 \right)\]
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: i9 + i19
Evaluate: `[i^18 + (1/i)^25]^3`
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
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:
\[\frac{(2 + i )^3}{2 + 3i}\]
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:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
Find the real value of x and y, if
\[\frac{(1 + i)x - 2i}{3 + i} + \frac{(2 - 3i)y + i}{3 - i}\]
Find the real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
Evaluate the following:
\[2 x^3 + 2 x^2 - 7x + 72, \text { when } x = \frac{3 - 5i}{2}\]
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
Evaluate the following:
\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
Write −1 + i \[\sqrt{3}\] in polar form .
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
\[\text { If } z = \frac{1}{(2 + 3i )^2}, \text { than } \left| z \right| =\]
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
The argument of \[\frac{1 - i}{1 + i}\] is
Find a and b if a + 2b + 2ai = 4 + 6i
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:
`("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 + sqrt(-3))/(4 + sqrt(-3))`
Evaluate the following : i93
Evaluate the following : i–888
If z1 = 3 – 2i and z2 = –1 + 3i, then Im(z1z2) = ______.
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 |
State True or False for the following:
The order relation is defined on the set of complex numbers.
Show that `(-1+sqrt3i)^3` is a real number.