Advertisements
Advertisements
प्रश्न
If (x + iy)3 = y + vi then show that `(y/x + "v"/y)` = 4(x2 – y2)
Advertisements
उत्तर
(x + yi)3 = y + vi
∴ x3 + 3x2yi + 3xy2i2 + y3i3 = y + vi
∴ x3 + 3x2yi + 3xy2 (–1) – y3i = y + vi ...[∵ i2 = – 1, i3 = – i]
∴ x3 + 3x2yi - 3xy2 – iy3 = y + vi
∴ (x3 – 3xy2) + (3x2y – y3)i = y + vi
Equating real and imaginary parts, we get
y = x3 – 3xy2 and v = 3x2y – y3
∴ `y/x` = x2 – 3y2 and `"v"/y` = 3x2 – y2
Consider,
L.H.S = `y/x + "v"/y`
= x2 – 3y2 + 3x2 – y2
= 4x2 – 4y2
= 4(x2 – y2)
= R.H.S.
Hence proved.
APPEARS IN
संबंधित प्रश्न
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Simplify the following and express in the form a + ib:
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Write the conjugates of the following complex number:
3 + i
Write the conjugates of the following complex number:
`-sqrt(-5)`
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
If x + iy = (a + ib)3, show that `x/"a" + y/"b"` = 4(a2 − b2)
Show that `((sqrt(7) + "i"sqrt(3))/(sqrt(7) - "i"sqrt(3)) + (sqrt(7) - "i"sqrt(3))/(sqrt(7) + "i"sqrt(3)))` is real
Find the value of x and y which satisfy the following equation (x, y∈R).
If x(1 + 3i) + y(2 − i) − 5 + i3 = 0, find x + y
Select the correct answer from the given alternatives:
If n is an odd positive integer then the value of 1 + (i)2n + (i)4n + (i)6n is :
Answer the following:
Simplify the following and express in the form a + ib:
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
Answer the following:
Simplify: `("i"^29 + "i"^39 + "i"^49)/("i"^30 + "i"^40 + "i"^50)`
Answer the following:
Simplify: `("i"^238 + "i"^236 + "i"^234 + "i"^232 + "i"^230)/("i"^228 + "i"^226 + "i"^224 + "i"^222 + "i"^220)`
Answer the following:
Simplify `[1/(1 - 2"i") + 3/(1 + "i")] [(3 + 4"i")/(2 - 4"i")]`
If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
Find the value of k if for the complex numbers z1 and z2, `|1 - barz_1z_2|^2 - |z_1 - z_2|^2 = k(1 - |z_1|^2)(1 - |"z"_2|^2)`
State true or false for the following:
Multiplication of a non-zero complex number by i rotates it through a right angle in the anti-clockwise direction.
State true or false for the following:
The argument of the complex number z = `(1 + i sqrt(3))(1 + i)(cos theta + i sin theta)` is `(7pi)/12 + theta`.
State true or false for the following:
The points representing the complex number z for which |z + 1| < |z − 1| lies in the interior of a circle.
If the complex number z = x + iy satisfies the condition |z + 1| = 1, then z lies on ______.
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.
If the real part of `(barz + 2)/(barz - 1)` is 4, then show that the locus of the point representing z in the complex plane is a circle.
Solve the equation |z| = z + 1 + 2i.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
If a + ib = c + id, then ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal to ______.
If α, β, γ and a, b, c are complex numbers such that `α/a + β/b + γ/c` = 1 + i and `a/α + b/β + c/γ` = 0, then the value of `α^2/a^2 + β^2/b^2 + γ^2/c^2` is equal to ______.
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)`
i2 + i3 + ... + i4000 =
