Advertisements
Advertisements
प्रश्न
If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
Advertisements
उत्तर
`(x + iy)^(1/3)` = a + ib
⇒ x + iy = (a + ib)3
i.e., x + iy = a3 + i3 b3 + 3iab (a + ib)
= a3 – ib3 + i3a2b – 3ab2
= a3 – 3ab2 + i(3a2b – b3)
⇒ x = a3 – 3ab2 and y = 3a2b – b3
Thus `x/a = a^2 - 3b^2` and `y/b = 3a^2 - b^2`
So, `x/a - y/b = a^2 - 3b^2 + b^2`
= `-2a^2 - 2b^2`
= –2(a2 + b2)
APPEARS IN
संबंधित प्रश्न
Find the multiplicative inverse of the complex number.
`sqrt5 + 3i`
Find the number of non-zero integral solutions of the equation `|1-i|^x = 2^x`.
Find the value of i + i2 + i3 + i4
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:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Write the conjugates of the following complex number:
`-sqrt(-5)`
If x + iy = (a + ib)3, show that `x/"a" + y/"b"` = 4(a2 − b2)
If `("a" + 3"i")/(2+ "ib")` = 1 − i, show that (5a − 7b) = 0
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")`
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
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:
(4 − 5i)x + (2 + 3i)y = 10 − 7i
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
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:
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 complex number cosθ + isinθ can be zero for some θ.
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.
State true or false for the following:
If n is a positive integer, then the value of in + (i)n+1 + (i)n+2 + (i)n+3 is 0.
What is the principal value of amplitude of 1 – i?
1 + i2 + i4 + i6 + ... + i2n is ______.
If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).
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.
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
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 ______.
If `((1 + i)/(1 - i))^x` = 1, then ______.
A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α2 + β2 = ______.
The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on ______.
The smallest positive integer n for which `((1 + i)/(1 - i))^n` = –1 is ______.
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Show that `(-1 + sqrt3 i)^3` is a real number.
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 `sqrt(-3) xx sqrt(-6)`
Evaluate the following:
i35
