Advertisements
Advertisements
प्रश्न
If i2 = −1, then the sum i + i2 + i3 +... upto 1000 terms is equal to
विकल्प
1
−1
i
0
Advertisements
उत्तर
0
\[i + i^2 + i^3 + i^4 . . . i^{1000} \]
\[ i + i^2 + i^3 + i^4 [ \because i^2 = - 1, i^3 = - i \text { and } i^4 = 1]\]
\[ = i - 1 - i + 1 \]
\[ = 0 \]
\[\text { Similarly, the sum of the next four terms of the series will be equal to 0 . This is because the powers of i follow a cyclicity of 4 } . \]
\[\text { Hence, the sum of all terms, till 1000, will be zero } . \]
\[i + i^2 + i^3 + i^4 . . . i^{1000} = 0\]
APPEARS IN
संबंधित प्रश्न
Express the given complex number in the form a + ib: i–39
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
i457
Evaluate the following:
(ii) i528
Evaluate the following:
\[i^{30} + i^{40} + i^{60}\]
Find the value of the following expression:
i30 + i80 + i120
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
\[(3x - 2iy)(2 + i )^2 = 10(1 + i)\]
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 real value of x and y, if
\[(1 + i)(x + iy) = 2 - 5i\]
Evaluate the following:
\[x^4 - 4 x^3 + 4 x^2 + 8x + 44,\text { when } x = 3 + 2i\]
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Write (i25)3 in polar form.
Write −1 + i \[\sqrt{3}\] in polar form .
Write the argument of −i.
If n ∈ \[\mathbb{N}\] then find the value of \[i^n + i^{n + 1} + i^{n + 2} + i^{n + 3}\] .
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
The value of (i5 + i6 + i7 + i8 + i9) / (1 + i) is
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
Which of the following is correct for any two complex numbers z1 and z2?
Find a and b if a + 2b + 2ai = 4 + 6i
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:
`((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:
`(3 + 2"i")/(2 - 5"i") + (3 -2"i")/(2 + 5"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:
`(- sqrt(5) + 2sqrt(-4)) + (1 -sqrt(-9)) + (2 + 3"i")(2 - 3"i")`
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)
Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.
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)`.
Match the statements of column A and B.
| Column A | Column B |
| (a) The value of 1 + i2 + i4 + i6 + ... i20 is | (i) purely imaginary complex number |
| (b) The value of `i^(-1097)` is | (ii) purely real complex number |
| (c) Conjugate of 1 + i lies in | (iii) second quadrant |
| (d) `(1 + 2i)/(1 - i)` lies in | (iv) Fourth quadrant |
| (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |
| (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 |
(vi) may occur in conjugate pairs |
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
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 |
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
