Advertisements
Advertisements
प्रश्न
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
Advertisements
उत्तर
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
संबंधित प्रश्न
Find the multiplicative inverse of the complex number:
4 – 3i
Express the following expression in the form of a + ib.
`((3 + sqrt5)(3 - isqrt5))/((sqrt3 + sqrt2i)-(sqrt3 - isqrt2))`
Find the value of i49 + i68 + i89 + i110
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Find the value of : x3 + 2x2 – 3x + 21, if x = 1 + 2i
Write the conjugates of the following complex number:
3 + i
Write the conjugates of the following complex number:
`-sqrt(-5)`
Write the conjugates of the following complex number:
5i
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Show that 1 + i10 + i100 − i1000 = 0
Evaluate: `("i"^37 + 1/"i"^67)`
Find the value of x and y which satisfy the following equation (x, y∈R).
`(x+ 1)/(1 + "i") + (y - 1)/(1 - "i")` = 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:
(2 + 3i)(1 − 4i)
Answer the following:
Evaluate: (1 − i + i2)−15
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
If (2 + i) (2 + 2i) (2 + 3i) ... (2 + ni) = x + iy, then 5.8.13 ... (4 + n2) = ______.
What is the principal value of amplitude of 1 – i?
1 + i2 + i4 + i6 + ... + i2n is ______.
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.
The equation |z + 1 – i| = |z – 1 + i| represents a ______.
If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
If z = x + iy, then show that `z barz + 2(z + barz) + b` = 0, where b ∈ R, represents a circle.
If |z + 1| = z + 2(1 + i), then find z.
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
Multiplicative inverse of 1 + i is ______.
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
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 ______.
Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.
Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is ______.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals to ______.
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.
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
Evaluate the following:
i35
