English

Evaluate the following : i888

Advertisements
Advertisements

Question

Evaluate the following : i888 

Sum
Advertisements

Solution

i888 = (i2)444

= (– 1)444

= 1.

shaalaa.com
  Is there an error in this question or solution?
Chapter 1: Complex Numbers - Exercise 1.1 [Page 6]

RELATED QUESTIONS

Express the given complex number in the form a + ib: i9 + i19


Express the given complex number in the form a + ib: `(1/5 + i 2/5) - (4 + i 5/2)`


Evaluate: `[i^18 + (1/i)^25]^3`


Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`


Find the value of the following expression:

i5 + i10 + i15


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:

\[\frac{(1 + i)(1 + \sqrt{3}i)}{1 - i}\] .


Express the following complex number in the standard form a + i b:

\[\frac{5 + \sqrt{2}i}{1 - 2\sqrt{i}}\]


Find the multiplicative inverse of the following complex number:

1 − i


If \[z_1 = 2 - i, z_2 = 1 + i,\text {  find } \left| \frac{z_1 + z_2 + 1}{z_1 - z_2 + i} \right|\]


Find the smallest positive integer value of m for which \[\frac{(1 + i )^n}{(1 - i )^{n - 2}}\] is a real number.

 

If \[\frac{\left( 1 + i \right)^2}{2 - i} = x + iy\]  find x + y.


Evaluate the following:

\[2 x^4 + 5 x^3 + 7 x^2 - x + 41, \text { when } x = - 2 - \sqrt{3}i\]


Solve the equation \[\left| z \right| = z + 1 + 2i\].


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 \[z_1 = - \bar{{z_2}}\].


Write the value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}}\] .


Write the least positive integral value of n for which  \[\left( \frac{1 + i}{1 - i} \right)^n\] is real.


If \[\frac{\left( a^2 + 1 \right)^2}{2a - i} = x + iy\] find the value of  \[x^2 + y^2\].


Write the value of \[\sqrt{- 25} \times \sqrt{- 9}\].


The value of \[(1 + i)(1 + i^2 )(1 + i^3 )(1 + i^4 )\] is.


The polar form of (i25)3 is


The least positive integer n such that \[\left( \frac{2i}{1 + i} \right)^n\] is a positive integer, is.

 

If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals


If \[z = \frac{1 + 2i}{1 - (1 - i )^2}\], then arg (z) equal


\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals


A real value of x satisfies the equation  \[\frac{3 - 4ix}{3 + 4ix} = a - ib (a, b \in \mathbb{R}), if a^2 + b^2 =\]


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")`


Evaluate the following : i116 


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:

If a complex number coincides with its conjugate, then the number must lie on imaginary axis.


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 `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).


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.


Show that `(-1+sqrt3i)^3` is a real number.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×