Advertisements
Advertisements
प्रश्न
If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.
Advertisements
उत्तर
We have |z1| = |z2| = ... = |zn| = 1
⇒ |z1|2 = |z2|2 = ... = |zn|2 = 1 ......(i)
⇒ `z_1 barz_1 = z_2 barz_2 = ... = z_n barz_n` = 1 .....`[because zbarz = |z|^2]`
⇒ z1 = `1/barz_1, z_2 = 1/barz_2 = ... = z_n = 1/barz_n`
L.H.S. |z1 + z2 + z3 + ... + zn|
= `|(z_1barz_1)/barz_1 + (z_2barz_2)/barz_2 + (z_3barz_3)/barz_3 + ... + (z_nbarz_n)/barz_n|`
= `||z_1|^2/barz_1 + (|z_2|^2)/barz_2 + (|z_3|^2)/barz_3 + ... + (|z_n|^2)/barz_n|` ......`[zbarz = |z|^2]`
= `|1/barz_1 + 1/barz_2 + 1/barz_3 + ... + 1/barz_n|` ......[Using (i)]
= `|bar(1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n)|` .....`[because barz_1 + barz_2 = bar(z_1 + z_2)]`
= `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|` ....`[because |z| = |barz|]`
L.H.S. = R.H.S.
Hence proved.
APPEARS IN
संबंधित प्रश्न
Find the multiplicative inverse of the complex number.
–i
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Show that 1 + i10 + i100 − i1000 = 0
Evaluate: `("i"^37 + 1/"i"^67)`
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
Show that `((sqrt(7) + "i"sqrt(3))/(sqrt(7) - "i"sqrt(3)) + (sqrt(7) - "i"sqrt(3))/(sqrt(7) + "i"sqrt(3)))` is real
If (x + iy)3 = y + vi then show that `(y/x + "v"/y)` = 4(x2 – y2)
Find the value of x and y which satisfy the following equation (x, y ∈ R).
`((x + "i"y))/(2 + 3"i") + (2 + "i")/(2 - 3"i") = 9/13(1 + "i")`
Select the correct answer from the given alternatives:
The value of is `("i"^592 + "i"^590 + "i"^588 + "i"^586 + "i"^584)/("i"^582 + "i"^580 + "i"^578 + "i"^576 + "i"^574)` is equal to:
Answer the following:
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Answer the following:
Simplify the following and express in the form a + ib:
(2 + 3i)(1 − 4i)
Answer the following:
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
Answer the following:
Evaluate: i131 + i49
Answer the following:
show that `((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 2
Answer the following:
Simplify: `("i"^238 + "i"^236 + "i"^234 + "i"^232 + "i"^230)/("i"^228 + "i"^226 + "i"^224 + "i"^222 + "i"^220)`
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
If z1 = 2 – 4i and z2 = 1 + 2i, then `bar"z"_1 + bar"z"_2` = ______.
The real value of ‘a’ for which 3i3 – 2ai2 + (1 – a)i + 5 is real is ______.
State true or false for the following:
The complex number cosθ + isinθ can be zero for some θ.
What is the reciprocal of `3 + sqrt(7)i`.
Solve the equation |z| = z + 1 + 2i.
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
Multiplicative inverse of 1 + i is ______.
State True or False for the following:
Multiplication of a non-zero complex number by –i rotates the point about origin through a right angle in the anti-clockwise direction.
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
The point represented by the complex number 2 – i is rotated about origin through an angle `pi/2` in the clockwise direction, the new position of point is ______.
If z1, z2, z3 are complex numbers such that |z1| = |z2| = |z3| = `|1/z_1 + 1/z_2 + 1/z_3|` = 1, then |z1 + z2 + z3| is ______.
Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is ______.
The complex number z = x + iy which satisfy the equation `|(z - 5i)/(z + 5i)|` = 1, lie on ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals to ______.
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Find the value of `(i^592 + i^590 + i^588 + i^586 + i^584)/ (i^582 + i^580 + i^578 + i^576 + i^574)`
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9) / (i^6 + 2i^8 + 3i^18)`
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
