Advertisements
Advertisements
प्रश्न
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Advertisements
उत्तर
\[ i^{30} + i^{40} + i^{60} = i^{4 \times 7 + 2} + i^{4 \times 10} + i^{4 \times 15} \]
\[ = \left[ \left( i^4 \right)^7 \times i^2 \right] + \left[ \left( i^4 \right)^{10} \right] + \left[ \left( i^4 \right)^{15} \right]\]
\[ = - 1 + 1 + 1 \left( \because i^4 = 1, i^2 = - 1 \right)\]
\[ = 1 \]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
Evaluate the following:
\[\frac{1}{i^{58}}\]
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:
\[\frac{1 - i}{1 + i}\]
Express the following complex number in the standard form a + i b:
\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]
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
\[(x + iy)(2 - 3i) = 4 + i\]
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`
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.
Find the real values of θ for which the complex number \[\frac{1 + i cos\theta}{1 - 2i cos\theta}\] is purely real.
Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
If \[\left| z + 1 \right| = z + 2\left( 1 + i \right)\],find z.
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of \[x^2 + y^2\].
If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.
If\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
The principal value of the amplitude of (1 + i) is
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
The complex number z which satisfies the condition \[\left| \frac{i + z}{i - z} \right| = 1\] lies on
If z is a complex number, then
Find a and b if (a+b) (2 + i) = b + 1 + (10 + 2a)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:
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"i")`
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
Answer the following:
Show that z = `5/((1 - "i")(2 - "i")(3 - "i"))` is purely imaginary number.
If z1 and z2 both satisfy `z + barz = 2|z - 1|` arg`(z_1 - z_2) = pi/4`, then find `"Im" (z_1 + z_2)`.
State True or False for the following:
The order relation is defined on the set of complex numbers.
Match the statements of Column A and Column B.
| Column A | Column B |
| (a) The polar form of `i + sqrt(3)` is | (i) Perpendicular bisector of segment joining (–2, 0) and (2, 0). |
| (b) The amplitude of `-1 + sqrt(-3)` is | (ii) On or outside the circle having centre at (0, –4) and radius 3. |
| (c) If |z + 2| = |z − 2|, then locus of z is | (iii) `(2pi)/3` |
| (d) If |z + 2i| = |z − 2i|, then locus of z is | (iv) Perpendicular bisector of segment joining (0, –2) and (0, 2). |
| (e) Region represented by |z + 4i| ≥ 3 is | (v) `2(cos pi/6 + i sin pi/6)` |
| (f) Region represented by |z + 4| ≤ 3 is | (vi) On or inside the circle having centre (–4, 0) and radius 3 units. |
| (g) Conjugate of `(1 + 2i)/(1 - i)` lies in | (vii) First quadrant |
| (h) Reciprocal of 1 – i lies in | (viii) Third quadrant |
If w is a complex cube-root of unity, then prove the following
(w2 + w − 1)3 = −8
Show that `(-1+ sqrt(3)i)^3` is a real number.
Show that `(-1+sqrt3i)^3` is a real number.
