Advertisements
Advertisements
प्रश्न
Answer the following:
Simplify the following and express in the form a + ib:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
उत्तर
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
= `(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i") xx (sqrt(5) + sqrt(3)"i")/(sqrt(5) + sqrt(3)"i")`
= `(sqrt(5) + sqrt(3)"i")^2/((sqrt(5))^2 - (sqrt(3)"i")^2`
= `(5 + 2sqrt(15)"i" + 3"i"^2)/(5 - 3"i"^2)`
= `(5 + 2sqrt(15)"i" - 3)/(5 + 3)` ...[∵ i2 = – 1]
= `(2 + 2sqrt(15)"i")/8`
= `1/4 + sqrt(15)/4"i"`, which is of the form a + bi.
APPEARS IN
संबंधित प्रश्न
Find the multiplicative inverse of the complex number.
–i
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
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Simplify the following and express in the form a + ib:
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
Write the conjugates of the following complex number:
3 + i
Write the conjugates of the following complex number:
3 – i
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
Find the value of i + i2 + i3 + i4
Evaluate: `("i"^37 + 1/"i"^67)`
If (a + ib) = `(1 + "i")/(1 - "i")`, then prove that (a2 + b2) = 1
Find the value of x and y which satisfy the following equation (x, y∈R).
If x + 2i + 15i6y = 7x + i3 (y + 4), find x + y
Select the correct answer from the given alternatives:
The value of is `("i"^592 + "i"^590 + "i"^588 + "i"^586 + "i"^584)/("i"^582 + "i"^580 + "i"^578 + "i"^576 + "i"^574)` is equal to:
Answer the following:
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
Find the real numbers x and y such that `x/(1 + 2"i") + y/(3 + 2"i") = (5 + 6"i")/(-1 + 8"i")`
Answer the following:
show that `((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 2
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| 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:
If three complex numbers z1, z2 and z3 are in A.P., then they lie on a circle in the complex plane.
1 + i2 + i4 + i6 + ... + i2n is ______.
If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.
If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.
For any two complex numbers z1, z2 and any real numbers a, b, |az1 – bz2|2 + |bz1 + az2|2 = ______.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
State True or False for the following:
The inequality |z – 4| < |z – 2| represents the region given by x > 3.
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 = ______.
Which of the following is correct for any two complex numbers z1 and z2?
If `(x + iy)^(1/5)` = a + ib, and u = `x/a - y/b`, then ______.
Let z be a complex number such that `|(z - i)/(z + 2i)|` = 1 and |z| = `5/2`. Then the value of |z + 3i| is ______.
If α and β are the roots of the equation x2 + 2x + 4 = 0, then `1/α^3 + 1/β^3` is equal to ______.
The complex number z = x + iy which satisfy the equation `|(z - 5i)/(z + 5i)|` = 1, lie on ______.
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)`
Show that `(-1 + sqrt3i)^3` is a real number.
