मराठी
महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता ११ वी

Evaluate the following : i30 + i40 + i50 + i60

Advertisements
Advertisements

प्रश्न

Evaluate the following : i30 + i40 + i50 + i60 

बेरीज
Advertisements

उत्तर

i30 + i40 + i50 + i60 

= (i2)15 + (i2)20 + (i2)25 + (i2)30

= (– 1)15 + (– 1)20 + (– 1)25 + (– 1)30

= – 1 + 1 – 1 + 1

= 0.

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 1: Complex Numbers - Exercise 1.1 [पृष्ठ ६]

APPEARS IN

संबंधित प्रश्‍न

Express the given complex number in the form a + ib: i–39


Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)


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)`


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


Evaluate the following:

 \[i^{30} + i^{40} + i^{60}\]


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

\[\frac{(1 - i )^3}{1 - i^3}\]


Find the real value of x and y, if

\[(1 + i)(x + iy) = 2 - 5i\]


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

 

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:

\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]


If \[\left( 1 + i \right)z = \left( 1 - i \right) \bar{z}\],then show that \[z = - i \bar{z}\].


Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].


If z1 is a complex number other than −1 such that \[\left| z_1 \right| = 1\] and \[z_2 = \frac{z_1 - 1}{z_1 + 1}\] then show that the real parts of z2 is zero.


What is the smallest positive integer n for which \[\left( 1 + i \right)^{2n} = \left( 1 - i \right)^{2n}\] ?


Express the following complex in the form r(cos θ + i sin θ):

1 − sin α + i cos α


Express \[\sin\frac{\pi}{5} + i\left( 1 - \cos\frac{\pi}{5} \right)\] in polar form.


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 \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.


Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].

Disclaimer: There is a misprinting in the question. It should be  \[\left( 1 + i\sqrt{3} \right)\]  instead of \[\left( 1 + \sqrt{3} \right)\].


If `(3+2i sintheta)/(1-2 i sin theta)`is a real number and 0 < θ < 2π, then θ =


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


The argument of \[\frac{1 - i}{1 + i}\] is


The amplitude of \[\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\] is 


The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is 


The value of \[(1 + i )^4 + (1 - i )^4\] 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 the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on


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 


Find the value of `(3 + 2/i) (i^6 - i^7) (1 + i^11)`.


Evaluate the following : i93  


Evaluate the following : `1/"i"^58`


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:

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

Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×