Advertisements
Advertisements
प्रश्न
Evaluate the following:
\[( i^{77} + i^{70} + i^{87} + i^{414} )^3\]
Advertisements
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: i–39
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Let z1 = 2 – i, z2 = –2 + i. Find Re`((z_1z_2)/barz_1)`
Evaluate the following:
i457
Find the value of the following expression:
i30 + i80 + i120
Find the value of the following expression:
i + i2 + i3 + i4
Express the following complex number in the standard form a + i b:
\[\frac{1 - i}{1 + i}\]
If \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\] find x + y.
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
Express the following complex in the form r(cos θ + i sin θ):
1 − sin α + i cos α
If z1 and z2 are two complex numbers such that \[\left| z_1 \right| = \left| z_2 \right|\] and arg(z1) + arg(z2) = \[\pi\] then show that \[z_1 = - \bar{{z_2}}\].
Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.
Write −1 + i \[\sqrt{3}\] in polar form .
Write the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
If a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]
The principal value of the amplitude of (1 + i) is
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
If \[f\left( z \right) = \frac{7 - z}{1 - z^2}\] , where \[z = 1 + 2i\] then \[\left| f\left( z \right) \right|\] is
A real value of x satisfies the equation \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]
Find a and b if (a + ib) (1 + i) = 2 + i
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:
`("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:
`((1 + "i")/(1 - "i"))^2`
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))`
Show that `(-1 + sqrt(3)"i")^3` is a real number
Evaluate the following : i888
Evaluate the following : i403
Evaluate the following : i–888
Show that 1 + i10 + i20 + i30 is a real number
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.
State true or false for the following:
If a complex number coincides with its conjugate, then the number must lie on imaginary axis.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
