हिंदी

If | Z | = 2 and Arg ( Z ) = π 4 ,Find Z.

Advertisements
Advertisements

प्रश्न

If \[\left| z \right| = 2 \text { and } \arg\left( z \right) = \frac{\pi}{4}\],find z.

Advertisements

उत्तर

We know that,

\[z = \left| z \right|\left\{ cos\left[ \arg\left( z \right) \right] + i\sin\left[ \arg\left( z \right) \right] \right\}\] 
 
\[     = 2\left( \cos\frac{\pi}{4} + i\sin\frac{\pi}{4} \right)\] 
 
\[     = 2\left( \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} \right)\] 
 
\[     = \sqrt{2}\left( 1 + i \right)\]
Hence, 
\[z = \sqrt{2}\left( 1 + i \right)\].
shaalaa.com
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 13: Complex Numbers - Exercise 13.5 [पृष्ठ ६३]

APPEARS IN

आर.डी. शर्मा Mathematics [English] Class 11
अध्याय 13 Complex Numbers
Exercise 13.5 | Q 23 | पृष्ठ ६३

वीडियो ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्न

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


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

`[(1/3 + i 7/3) + (4 + i 1/3)] -(-4/3 + i)`


If a + ib  = `(x + i)^2/(2x^2 + 1)` prove that a2 + b2 = `(x^2 + 1)^2/(2x + 1)^2`


Evaluate the following:

\[i^{37} + \frac{1}{i^{67}}\].


Find the value of the following expression:

i49 + i68 + i89 + i110


Find the value of the following expression:

i + i2 + i3 + i4


Express the following complex number in the standard form a + ib:

\[\frac{(2 + i )^3}{2 + 3i}\]


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

\[\frac{2 + 3i}{4 + 5i}\]


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:

\[\frac{3 - 4i}{(4 - 2i)(1 + i)}\]


Find the real value of x and y, if

\[(x + iy)(2 - 3i) = 4 + i\]


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 \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).


If \[\left( \frac{1 - i}{1 + i} \right)^{100} = a + ib\] find (a, b).


If z1, z2, z3 are complex numbers such that \[\left| z_1 \right| = \left| z_2 \right| = \left| z_3 \right| = \left| \frac{1}{z_1} + \frac{1}{z_2} + \frac{1}{z_3} \right| = 1\] then find the value of \[\left| z_1 + z_2 + z_3 \right|\] .


Write (i25)3 in polar form.


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


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

\[\frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}\]


Write 1 − i in polar form.


Write −1 + i \[\sqrt{3}\] in polar form .


Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .


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


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


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 a = cos θ + i sin θ, then \[\frac{1 + a}{1 - a} =\]


The principal value of the amplitude of (1 + i) is


If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]


The amplitude of \[\frac{1}{i}\] is equal to


The value of \[(1 + i )^4 + (1 - i )^4\] is


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


Express the following in the form of a + ib, a, b ∈ R, i = `sqrt(−1)`. State the values of a and b:

`("i"(4 + 3"i"))/((1 - "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 + sqrt(-3))/(4 + sqrt(-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")`


Evaluate the following : i30 + i40 + i50 + i60 


If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.


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


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


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×