Advertisements
Advertisements
प्रश्न
Express the following complex numbers in polar form and exponential form:
`(1 + 7"i")/(2 - "i")^2`
Advertisements
उत्तर
Let z = `(1 + 7"i")/(2 - "i")^2`
= `(1 + 7"i")/(4 - 4"i" + "i"^2)`
= `(1 + 7"i")/(4 - 4"i" - 1)`
= `(1 + 7"i")/(3 - 4"i")`
= `((1 + 7"i")(3 + 4"i"))/((3 - 4"i")(3 + 4"i"))`
= `(3 + 4"i" + 21"i" + 28"i"^2)/(9 - 16"i"^2)`
= `(25"i" + 3 + 28(-1))/(9 - 16(-1))`
= `(25"i" - 25)/25`
∴ z = – 1 + i
∴ a = – 1, b = 1
∴ |z| = r
= `sqrt("a"^2 + "b"^2)`
= `sqrt((-1)^2 + 1^2)`
= `sqrt(2)`
Here, (–1, 1) lies in 2nd quadrant
θ = amp (z)
= `pi + tan^-1("b"/"a")`
= `pi + tan^-1(1/(-1))`
= π + tan–1(–1)
= π – tan–1(1)
= `pi - pi/4`
= `(3pi)/4`
∴ θ = 135° = `(3pi)/4`
∴ the polar form of z = r (cos θ + i sin θ)
= `sqrt(2) (cos 135^circ + "i" sin 135^circ)`
= `sqrt(2)(cos (3pi)/4 + "i" sin (3pi)/4)`
The exponential form of z = reiθ = `sqrt(2)"e"^((3pi)/4"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.
−3(1 − i)
Find the modulus and amplitude of the following complex number.
−4 − 4i
Find the modulus and amplitude of the following complex numbers.
`sqrt(3) - "i"`
Find the modulus and amplitude of the following complex numbers.
`1 + "i"sqrt(3)`
Find the modulus and amplitude of the following complex numbers.
(1 + 2i)2 (1 − i)
Find real values of θ for which `((4 + 3"i" sintheta)/(1 - 2"i" sin theta))` is purely real.
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 + sqrt(3)"i"`
Express the following complex numbers in polar form and exponential form:
−1
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:
`7(cos(-(5pi)/6) + "i" sin (- (5pi)/6))`
Express the following numbers in the form x + iy:
`"e"^((-4pi)/3"i")`
Find the modulus and argument of the complex number `(1 + 2"i")/(1 - 3"i")`
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)`
Select the correct answer from the given alternatives:
If arg(z) = θ, then arg `bar(("z"))` =
Select the correct answer from the given alternatives:
If z = x + iy and |z − zi| = 1 then
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
8 + 15i
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
6 − i
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
`(-1 - "i")/sqrt(2)`
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
− 3i
Answer the following:
Find the modulus and argument of a complex number and express it in the polar form.
`1/sqrt(2) + 1/sqrt(2)"i"`
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.
z = `-6 + sqrt(2)"i"`
The modulus and amplitude of 4 + 3i are ______
If x + iy = `5/(3 + costheta + isintheta)`, then x2 + y2 is equal to ______
For all complex numbers z1, z2 satisfying |z1| = 12 and |z2 - 3 - 4i| = 5, the minimum value of |z1 - z2| is ______.
