Advertisements
Advertisements
प्रश्न
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
उत्तर
\[\left( i^{77} + i^{70} + i^{87} + i^{414} \right)^3 = \left( i^{4 \times 19 + 1} + i^{4 \times 17 + 2} + i^{4 \times 21 + 3} + i^{4 \times 103 + 2} \right)^3 \]
\[ = \left[ \left\{ \left( i^4 \right)^{19} \times i \right\} + \left\{ \left( i^4 \right)^{17} \times i^2 \right\} + \left\{ \left( i^4 \right)^{21} \times i^3 \right\} + \left\{ \left( i^4 \right)^{103} \times i^2 \right\} \right]\]
\[ = \left( i - 1 - i - 1 \right)^3 \left( \because i^4 = 1, i^3 = - i and i^2 = - 1 \right)\]
\[ = \left( - 2 \right)^3 \]
\[ = - 8\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(5i) (- 3/5 i)`
Express the given complex number in the form a + ib: i9 + i19
Express the given complex number in the form a + ib: `(1/5 + i 2/5) - (4 + i 5/2)`
Express the given complex number in the form a + ib: (1 – i)4
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:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
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 + i b:
\[\frac{2 + 3i}{4 + 5i}\]
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:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
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
\[(1 + i)(x + iy) = 2 - 5i\]
If \[z_1 = 2 - i, z_2 = 1 + i,\text { find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
Write 1 − i in polar form.
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 \[\left| z + 1 \right|\].
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
If \[z = \frac{- 2}{1 + i\sqrt{3}}\],then the value of arg (z) is
The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
If θ is the amplitude of \[\frac{a + ib}{a - ib}\] , than tan θ =
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) 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:
(1 + 2i)(– 2 + i)
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
`((2 + "i"))/((3 - "i")(1 + 2"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))`
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Find the value of `(i^(592) + i^(590) + i^(588) + i^(586) + i^(584))/(i^(582) + i^(580) + i^(578) + i^(576) + i^(574))`
Show that `(-1+sqrt3i)^3` is a real number.
