Advertisements
Advertisements
Question
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
Advertisements
Solution
a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2`
∴ a2 = `((-1 + sqrt(3)"i")/2)^2 = (1 - 2sqrt(3)"i" + 3"i"^2)/4`
= `(1 - 2sqrt(3)"i" + 3(-1))/4` ...[∵ i2 = – 1]
= `(-2 - 2sqrt(3)"i")/4`
= `(-1 - sqrt(3)"i")/2` = b
and b2 = `((-1 - sqrt(3)"i")/2)^2 = (1 + 2sqrt(3)"i" + 3"i"^2)/4`
= `(1 + 2sqrt(3)"i" + 3(-1))/4` . ...[∵ i2 = – 1]
= `(-2 + 2sqrt(3)"i")/4`
= `(-1 + sqrt(3)"i")/2` = a
APPEARS IN
RELATED QUESTIONS
Find the multiplicative inverse of the complex number.
`sqrt5 + 3i`
Express the following expression in the form of a + ib.
`((3 + sqrt5)(3 - isqrt5))/((sqrt3 + sqrt2i)-(sqrt3 - isqrt2))`
Reduce `(1/(1-4i) - 2/(1+i))((3-4i)/(5+i))` to the standard form.
If `((1+i)/(1-i))^m` = 1, then find the least positive integral value of m.
Simplify the following and express in the form a + ib:
(2i3)2
Write the conjugates of the following complex number:
cosθ + i sinθ
Find the value of i49 + i68 + i89 + i110
Is (1 + i14 + i18 + i22) a real number? Justify your answer
If x + iy = (a + ib)3, show that `x/"a" + y/"b"` = 4(a2 − b2)
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
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).
`((x + "i"y))/(2 + 3"i") + (2 + "i")/(2 - 3"i") = 9/13(1 + "i")`
Answer the following:
Simplify the following and express in the form a + ib:
`5/2"i"(-4 - 3"i")`
Answer the following:
Simplify the following and express in the form a + ib:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
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:
Simplify `[1/(1 - 2"i") + 3/(1 + "i")] [(3 + 4"i")/(2 - 4"i")]`
The value of (2 + i)3 × (2 – i)3 is ______.
Locate the points for which 3 < |z| < 4.
State true or false for the following:
If three complex numbers z1, z2 and z3 are in A.P., then they lie on a circle in the complex plane.
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
Multiplicative inverse of 1 + i is ______.
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
State True or False for the following:
For any complex number z the minimum value of |z| + |z – 1| is 1.
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
A real value of x satisfies the equation `((3 - 4ix)/(3 + 4ix))` = α − iβ (α, β ∈ R) if α2 + β2 = ______.
If a + ib = c + id, then ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is ______.
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)`
