Advertisements
Advertisements
प्रश्न
Answer the following:
Convert the complex numbers in polar form and also in exponential form.
z = `(2 + 6sqrt(3)"i")/(5 + sqrt(3)"i")`
उत्तर
z = `(2 + 6sqrt(3)"i")/(5 + sqrt(3)"i")`
= `(2 + 6sqrt(3)"i")/(5 + sqrt(3)"i") xx (5 - sqrt(3)"i")/(5 - sqrt(3)"i")`
= `(10 - 2sqrt(3)"i" + 30sqrt(3)"i" - 18"i"^2)/(25 - 3"i"^2)`
= `(10 + 28sqrt(3)"i" + 18)/(25 + 3)` ...[∵ i2 = – 1]
= `(28 + 28sqrt(3)"i")/28`
∴ z = `1 + sqrt(3)"i"`
This is of the form a + bi, where a = 1, b = `sqrt(3)`
∴ r = `sqrt("a"^2 + "b"^2)`
= `sqrt(1^2 + (sqrt(3))^2`
= `sqrt(1 + 3)`
= 2
If θ is the amplitude, then cos θ = `"a"/"r" = 1/2`
and sin θ = `"b"/"r" = sqrt(3)/2`
∴ θ = `pi/3 ...[because cos pi/3 = 1/2 and sin pi/3 = sqrt(3)/2]`
∴ polar form of z = r(cos θ + i sin θ)
= `2(cos pi/3 + "i" sin pi/3)`
and the exponential form of z = reiθ
= `2"e"^("i"(pi/3))`
= `2"e"^(pi/3"i")`
APPEARS IN
संबंधित प्रश्न
Find the modulus and amplitude of the following complex numbers.
7 − 5i
Find the modulus and amplitude of the following complex numbers.
−8 + 15i
Find the modulus and amplitude of the following complex numbers.
`sqrt(3) - "i"`
Find the modulus and amplitude of the following complex numbers.
3
Find the modulus and amplitude of the following complex numbers.
(1 + 2i)2 (1 − i)
If z = 3 + 5i then represent the `"z", bar("z"), - "z", bar(-"z")` in Argand's diagram
Express the following complex numbers in polar form and exponential form:
−1
Express the following complex numbers in polar form and exponential form:
`1/(1 + "i")`
Express the following numbers in the form x + iy:
`sqrt(3)(cos pi/6 + "i" sin pi/6)`
Express the following numbers in the form x + iy:
`sqrt(2)(cos (7pi)/4 + "i" sin (7pi)/4)`
Express the following numbers in the form x + iy:
`7(cos(-(5pi)/6) + "i" sin (- (5pi)/6))`
Express the following numbers in the form x + iy:
`"e"^((5pi)/6"i")`
Find the modulus and argument of the complex number `(1 + 2"i")/(1 - 3"i")`
Convert the complex number z = `("i" - 1)/(cos pi/3 + "i" sin pi/3)` in the polar form
For z = 2 + 3i verify the following:
`bar((bar"z"))` = z
For z = 2 + 3i verify the following:
`("z" + bar"z")` is real
z1 = 1 + i, z2 = 2 − 3i. Verify the following :
`bar("z"_1 + "z"_2) = bar("z"_1) + bar("z"_2)`
z1 = 1 + i, z2 = 2 − 3i. Verify the following :
`bar("z"_1."z"_2) = bar("z"_1).bar("z"_2)`
z1 = 1 + i, z2 = 2 − 3i. Verify the following :
`bar(("z"_1/"z"_2))=bar("z"_1)/bar("z"_2)`
Select the correct answer from the given alternatives:
If arg(z) = θ, then arg `bar(("z"))` =
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
− 3i
Answer the following:
Represent 1 + 2i, 2 − i, −3 − 2i, −2 + 3i by points in Argand's diagram.
Answer the following:
Convert the complex numbers in polar form and also in exponential form.
`(-3)/2 + (3sqrt(3))/2"i"`
The polar coordinates of the point whose cartesian coordinates are (−2, −2), are given by ____________.
The modulus of z = `sqrt7` + 3i is ______
For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 - 3 - 4i| = 5, the minimum value of |z1 - z2| is ______.
If z = `5i ((-3)/5 i)`, then z is equal to 3 + bi. The value of ‘b’ is ______.
If z = `π/4(1 + i)^4((1 - sqrt(π)i)/(sqrt(π) + i) + (sqrt(π) - i)/(1 + sqrt(π)i))`, then `(|z|/("amp"^((z))))` is equals to ______.
If A, B, C are three points in argand plane representing the complex numbers z1, z2 and z3 such that, z1 = `(λz_2 + z_3)/(λ + 1)`, where λ ∈ R, then find the distance of point A from the line joining points B and C.
