Advertisements
Advertisements
प्रश्न
Evaluate the following:
\[\frac{1}{i^{58}}\]
Advertisements
उत्तर
`1/i^58 = 1 /(i^(4 xx 14 +2)`
\[ = \frac{1}{\left( i^4 \right)^{14} \times i^2}\]
\[ = \frac{1}{i^2} \left( \because i^4 = 1 \right)\]
\[ = - 1 \left( \because i^2 = - 1 \right)\]
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: i–39
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
Evaluate the following:
\[i^{37} + \frac{1}{i^{67}}\].
Find the value of the following expression:
i5 + i10 + i15
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:
\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]
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 \[x + iy = \frac{a + ib}{a - ib}\] prove that x2 + y2 = 1.
Find the least positive integral value of n for which \[\left( \frac{1 + i}{1 - i} \right)^n\] is 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:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
For a positive integer n, find the value of \[(1 - i )^n \left( 1 - \frac{1}{i} \right)^n\].
If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Write the argument of −i.
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\[z = \cos\frac{\pi}{4} + i \sin\frac{\pi}{6}\], then
The polar form of (i25)3 is
\[\text { If }z = 1 - \text { cos }\theta + i \text { sin }\theta, \text { then } \left| z \right| =\]
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
Simplify : `4sqrt(-4) + 5sqrt(-9) - 3sqrt(-16)`
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 + i)−3
Express the following in the form of a + ib, a, b∈R i = `sqrt(−1)`. State the values of a and b:
(2 + 3i)(2 – 3i)
Evaluate the following : i116
Evaluate the following : `1/"i"^58`
Evaluate the following : i–888
Evaluate the following : i30 + i40 + i50 + i60
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.
State True or False for the following:
2 is not a complex number.
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 |
