Advertisements
Advertisements
प्रश्न
If ω is a complex cube root of unity, show that (2 − ω)(2 − ω2) = 7
Advertisements
उत्तर
ω is a complex cube root of unity.
∴ ω3 = 1 and 1 + ω + ω2 = 0
∴ ω + ω2 = – 1, 1 + ω = – ω2 and 1 + ω2 = – ω
(2 – ω)(2 – ω2) = 7
= 4 – 2ω2 – 2ω + ω3
= 4 – 2(ω2 + ω) + ω3
= 4 – 2(– 1) + 1
= 4 + 2 + 1
= 7
APPEARS IN
संबंधित प्रश्न
If `omega` is a complex cube root of unity, show that `(2 - omega)(2 - omega^2)` = 7
If α and β are the complex cube roots of unity, show that α2 + β2 + αβ = 0.
If x = a + b, y = αa + βb and z = aβ + bα, where α and β are the complex cube roots of unity, show that xyz = a3 + b3.
If ω is a complex cube root of unity, then prove the following: (a + b) + (aω + bω2) + (aω2 + bω) = 0.
Find the value of ω–30
Find the value of ω–105
If ω is a complex cube root of unity, show that (3 + 3ω + 5ω2)6 − (2 + 6ω + 2ω2)3 = 0
If ω is a complex cube root of unity, show that `("a" + "b"ω + "c"ω^2)/("c" + "a"ω + "b"ω^2)` = ω2
If ω is a complex cube root of unity, show that (a − b) (a − bω) (a − bω2) = a3 − b3
If ω is a complex cube root of unity, find the value of `ω + 1/ω`
If α and β are the complex cube root of unity, show that α2 + β2 + αβ = 0
If , where α and β are the complex cube-roots of unity, show that xyz = a3 + b3.
Find the equation in cartesian coordinates of the locus of z if |z – 3| = 2
Find the equation in cartesian coordinates of the locus of z if |z + 8| = |z – 4|
Select the correct answer from the given alternatives:
If ω is a complex cube root of unity, then the value of ω99+ ω100 + ω101 is :
If ω(≠1) is a cube root of unity and (1 + ω)7 = A + Bω, then A and B are respectively the numbers ______.
If α and β are complex cube roots of unity, prove that (1 − α)(1 − β) (1 − α2)(1 − β2) = 9.
Answer the following:
If ω is a complex cube root of unity, prove that (1 − ω + ω2)6 +(1 + ω − ω2)6 = 128
Let α be a root of the equation 1 + x2 + x4 = 0. Then the value of α1011 + α2022 – α3033 is equal to ______.
Let z = `(1 - isqrt(3))/2`, i = `sqrt(-1)`. Then the value of `21 + (z + 1/z)^3 + (z^2 + 1/z^2) + (z^3 + 1/z^3)^3 + ...... + (z^21 + 1/z^21)^3` is ______.
If 1, α1, α2, ...... αn–1 are the roots of unity, then (1 + α1)(1 + α2) ...... (1 + αn–1) is equal to (when n is even) ______.
If the cube roots of the unity are 1, ω and ω2, then the roots of the equation (x – 1)3 + 8 = 0, are ______.
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
If w is a complex cube root of unity, show that `((a+bw+cw^2))/(c+aw+bw^2) = w^2`
If w is a complex cube root of unity, show that, `((a + bw + cw^2))/(c + aw + bw^2) = w^2`
If w is a complex cube root of unity, show that `((a + bw + cw^2))/(c+aw+bw^2) = w^2`
If w is a complex cube root of unity, show that `((a+bw+cw^2))/(c+aw+bw^2) =w^2`
If ω is a complex cube-root of unity, then prove the following :
(ω2 + ω − 1)3 = − 8
Find the value of `sqrt(-3)xx sqrt (-6)`
If w is a complex cube root of unity, show that `((a+bw+cw^2))/(c+aw+bw^2) = w^2`
If w is a complex cube root of unity, show that `((a+bw+cw^2))/(c+aw+bw^2)=w^2`
If w is a complex cube-root of unity, then prove the following.
(w2 + w - 1)3 = - 8
If ω is a complex cube root of unity, show that `((a + bomega + comega^2))/(c + aomega + bomega^2)=omega^2`
If ω is a complex cube-root of unity, then prove the following.
(ω2 + ω − 1)3 = −8
