Advertisements
Advertisements
प्रश्न
Find the value of the following expression:
\[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
Advertisements
उत्तर
\[ \frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\]
\[ = \frac{i^{4 \times 148} + i^{4 \times 147 + 2} + i^{4 \times 147} + i^{4 \times 146 + 2} + i^{4 \times 146}}{i^{4 \times 145 + 2} + i^{4 \times 145} + i^{4 \times 144 + 2} + i^{4 \times 144} + i^{4 \times 143 + 2}}\]
\[ = \frac{\left( i^4 \right)^{148} + \left\{ \left( i^4 \right)^{147} \times i^2 \right\} + \left\{ \left( i^4 \right)^{146} \right\} + \left\{ \left( i^4 \right)^{146} \times i^2 \right\} + \left\{ \left( i^4 \right)^{146} \right\}}{\left\{ \left( i^4 \right)^{145} \times i^2 \right\} + \left\{ \left( i^4 \right)^{145} \right\} + \left\{ \left( i^4 \right)^{144} \times i^2 \right\} + \left\{ \left( i^4 \right)^{144} \right\} + \left\{ \left( i^4 \right)^{143} \times i^2 \right\}}\]
\[ = \frac{1 + i^2 + 1 + i^2 + 1}{i^2 + 1 + i^2 + 1 + i^2} \left[ \because i^4 = 1 \right]\]
\[ = \frac{1 - 1 + 1 - 1 + 1}{- 1 + 1 - 1 + 1 - 1} \left[ \because i^2 = - 1 \right]\]
\[ = - 1\]
APPEARS IN
संबंधित प्रश्न
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:
`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`
Express the given complex number in the form a + ib: `(1/3 + 3i)^3`
Express the given complex number in the form a + ib: `(-2 - 1/3 i)^3`
If a + ib = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`
Evaluate the following:
(ii) i528
Show that 1 + i10 + i20 + i30 is a real number.
Express the following complex number in the standard form a + i b:
\[\frac{3 + 2i}{- 2 + i}\]
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:
\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .
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 `((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.
If \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
If \[\frac{z - 1}{z + 1}\] is purely imaginary number (\[z \neq - 1\]), find the value of \[\left| z \right|\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
Express the following complex in the form r(cos θ + i sin θ):
1 + i tan α
Write the argument of −i.
Write the sum of the series \[i + i^2 + i^3 + . . . .\] upto 1000 terms.
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
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
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
The argument of \[\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\] is
The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
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)−1
Evaluate the following : i35
Evaluate the following : i93
Evaluate the following : i403
Evaluate the following : i–888
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 |
Show that `(-1 + sqrt3 "i")^3` is a real number.
