Advertisements
Advertisements
प्रश्न
If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.
Advertisements
उत्तर
Given that: `(1 + i)^2/(2 - i)` = x + iy
⇒ `(1 + i^2 + 2i)/(2 - i)` = x + iy
⇒ `(1 - 1 + 2i)/(2 - i)` = x + iy
⇒ `(2i)/(2 - i)` = x + iy
⇒ `(2i(2 + i))/((2 - i)(2 + i))` = x + iy
⇒ `(4i + 2i^2)/(4 - i^2)` = x + iy
⇒ `(4i - 2)/(4 + 1)` = x + iy ......[∵ i2 = –1]
⇒ `(-2 + 4i)/5` = x + iy
⇒ `(-2)/5 + 4/5 i` = x + iy
Comparing the real and imaginary parts,
We get x = `(-2)/5` and y = `4/5`
Hence, x + y = `(-2)/5 + 4/5 = 2/5`.
APPEARS IN
संबंधित प्रश्न
Express the following expression in the form of a + ib.
`((3 + sqrt5)(3 - isqrt5))/((sqrt3 + sqrt2i)-(sqrt3 - isqrt2))`
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
Simplify the following and express in the form a + ib:
(2i3)2
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
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
Evaluate: `("i"^37 + 1/"i"^67)`
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
If x + iy = `sqrt(("a" + "ib")/("c" + "id")`, prove that (x2 + y2)2 = `("a"^2 + "b"^2)/("c"^2 + "d"^2)`
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).
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:
(2i3)2
Answer the following:
Simplify the following and express in the form a + ib:
(2 + 3i)(1 − 4i)
Answer the following:
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Solve the following equation for x, y ∈ R:
(4 − 5i)x + (2 + 3i)y = 10 − 7i
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
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 - 2"i")/(3 - 4"i") + (1 + 2"i")/(3 + 4"i")` is real
If z ≠ 1 and `"z"^2/("z - 1")` is real, then the point represented by the complex number z lies ______.
If z1 = 2 – 4i and z2 = 1 + 2i, then `bar"z"_1 + bar"z"_2` = ______.
Evaluate: (1 + i)6 + (1 – i)3
Locate the points for which 3 < |z| < 4.
State true or false for the following:
The points representing the complex number z for which |z + 1| < |z − 1| lies in the interior of a circle.
What is the value of `(i^(4n + 1) -i^(4n - 1))/2`?
If `((1 + i)/(1 - i))^3 - ((1 - i)/(1 + i))^3` = x + iy, then find (x, y).
If z = x + iy, then show that `z barz + 2(z + barz) + b` = 0, where b ∈ R, represents a circle.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
If z1 and z2 are complex numbers such that z1 + z2 is a real number, then z2 = ______.
State True or False for the following:
Multiplication of a non-zero complex number by –i rotates the point about origin through a right angle in the anti-clockwise direction.
State True or False for the following:
For any complex number z the minimum value of |z| + |z – 1| is 1.
State True or False for the following:
The inequality |z – 4| < |z – 2| represents the region given by x > 3.
If z is a complex number, then ______.
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 ______.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
If `|(6i, -3i, 1),(4, 3i, -1),(20, 3, i)|` = x + iy, then ______.
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18`
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)`
If z = 2 + i, then (z − 1) `(barz − 5) + (barz − 1)` (z − 5) is equal to ______.
