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 multiplicative inverse of the complex number:
4 – 3i
Find the multiplicative inverse of the complex number.
–i
If α and β are different complex numbers with |β| = 1, then find `|(beta - alpha)/(1-baralphabeta)|`
Find the value of i + i2 + i3 + i4
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Simplify the following and express in the form a + ib:
(2 + 3i)(1 – 4i)
Simplify the following and express in the form a + ib:
`5/2"i"(- 4 - 3 "i")`
Simplify the following and express in the form a + ib:
(1 + 3i)2 (3 + i)
Write the conjugates of the following complex number:
3 – i
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Find the value of i + i2 + i3 + i4
Show that `((sqrt(7) + "i"sqrt(3))/(sqrt(7) - "i"sqrt(3)) + (sqrt(7) - "i"sqrt(3))/(sqrt(7) + "i"sqrt(3)))` is real
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:
(2 + 3i)(1 − 4i)
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
Evaluate: (1 − i + i2)−15
Answer the following:
Evaluate: i131 + i49
Answer the following:
If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x2 + y2 = 1
Answer the following:
Simplify: `("i"^29 + "i"^39 + "i"^49)/("i"^30 + "i"^40 + "i"^50)`
If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
Locate the points for which 3 < |z| < 4.
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`.
What is the principal value of amplitude of 1 – i?
Number of solutions of the equation z2 + |z|2 = 0 is ______.
If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.
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 ______.
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
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.
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 ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
Simplify the following and express in the form a+ib.
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
