Advertisements
Advertisements
Question
z1 = 1 + i, z2 = 2 − 3i. Verify the following :
`bar(("z"_1/"z"_2))=bar("z"_1)/bar("z"_2)`
Advertisements
Solution
z1 = 1 + i and z2 = 2 − 3i
∴ `bar("z"_1)` = 1 − i and `bar("z"_2)` = 2 + 3i
`"z"_1/"z"_2 = (1 + "i")/(2 -3"i")`
= `(1 + "i")/(2 - 3"i") xx (2 + 3"i")/(2 + 3"i")`
= `(2 + 3"i" + 2"i" + 3"i"^2)/(4 - 9"i"^2)`
=`(2 + 5"i" - 3)/(4 + 9)` ...[∵ i2 = – 1]
=`(-1 + 5"i")/13`
=`-1/13 + 5/13"i"`
∴ `bar(("z"_1/"z"_2)) = -1/13 - 5/13"i"` .......(1)
`bar("z"_1)/(bar("z")_2) = (1 - "i")/(2 + 3"i")`
= `(1 - "i")/(2 + 3"i") xx (2 - 3"i")/(2 - 3"i")`
= `(2- 3"i" - 2"i" + 3"i"^2)/(4 - 9"i"^2)`
= `(2 - 5"i" - 3)/(4 + 9)` ...[∵ i2 = – 1]
= `(-1 - 5"i")/13`
= `-1/13 - 5/13"i"` ........(2)
From (1) and (2), we get,
`bar(("z"_1/"z"_2))=bar("z"_1)/bar("z"_2)`
APPEARS IN
RELATED QUESTIONS
Find the modulus and amplitude of the following complex numbers.
`sqrt(3) + sqrt(2)"i"`
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.
3
Find the modulus and amplitude of the following complex numbers.
1 + i
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:
− i
Express the following complex numbers in polar form and exponential form:
`1/(1 + "i")`
Express the following complex numbers in polar form and exponential form:
`(1 + 7"i")/(2 - "i")^2`
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")`
Express the following numbers in the form x + iy:
`"e"^((5pi)/6"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:
`"z"bar("z")` = |z|2
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)`
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.
`(1 + sqrt(3)"i")/2`
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.
2i
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 = `(2 + 6sqrt(3)"i")/(5 + sqrt(3)"i")`
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 ______
If z = `5i ((-3)/5 i)`, then z is equal to 3 + bi. The value of ‘b’ is ______.
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.
