Advertisements
Advertisements
प्रश्न
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
Advertisements
उत्तर
`((-1 + sqrt(-3))/2)^3 = ((-1 + sqrt(3 xx - 1))/2)^3`
= `((-1 + sqrt(3)"i")/2)^3`
= `(-1 + sqrt(3)"i")^3/8`
= `1/8[(-1)^3 + 3(-1)^2 sqrt(3)"i" + 3(-1)(sqrt(3)"i")^2 + (sqrt(3)"i")^3]`
= `1/8[-1 + 3sqrt(3)"i" - 3 xx 3"i"^2 + 3sqrt(3)"i"^3]`
= `1/8[-1 + 3sqrt(3)"i" + 9 - 3sqrt(3)"i"]` ...[∵ i2 = – 1, i3 = – i]
= `1/8(8)`
= 1, which is a rational number.
APPEARS IN
संबंधित प्रश्न
If `x – iy = sqrt((a-ib)/(c - id))` prove that `(x^2 + y^2) = (a^2 + b^2)/(c^2 + d^2)`
If α and β are different complex numbers with |β| = 1, then find `|(beta - alpha)/(1-baralphabeta)|`
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"`
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Write the conjugates of the following complex number:
cosθ + i sinθ
Find the value of i + i2 + i3 + i4
Is (1 + i14 + i18 + i22) a real number? Justify your answer
Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
Select the correct answer from the given alternatives:
`sqrt(-3) sqrt(-6)` is equal to
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:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
Evaluate: i131 + i49
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
Answer the following:
If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x2 + y2 = 1
Answer the following:
Simplify: `("i"^65 + 1/"i"^145)`
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
The value of (2 + i)3 × (2 – i)3 is ______.
Evaluate: (1 + i)6 + (1 – i)3
Locate the points for which 3 < |z| < 4.
If (2 + i) (2 + 2i) (2 + 3i) ... (2 + ni) = x + iy, then 5.8.13 ... (4 + n2) = ______.
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`?
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.
The value of `(z + 3)(barz + 3)` is equivalent to ______.
Which of the following is correct for any two complex numbers z1 and z2?
If the least and the largest real values of α, for which the equation z + α|z – 1| + 2i = 0 `("z" ∈ "C" and "i" = sqrt(-1))` has a solution, are p and q respectively; then 4(p2 + q2) is equal to ______.
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 ______.
A complex number z is moving on `arg((z - 1)/(z + 1)) = π/2`. If the probability that `arg((z^3 -1)/(z^3 + 1)) = π/2` is `m/n`, where m, n ∈ prime, then (m + n) 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)`
Find the value of `sqrt(-3) xx sqrt(-6)`
