Advertisements
Advertisements
प्रश्न
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is ______.
पर्याय
`(n + 1) pi/2`
`(2n + 1) pi/2`
nπ
None of these, where n ∈N
Advertisements
उत्तर
The real value of α for which the expression `(1 - i sin alpha)/(1 + 2i sin alpha)` is purely real is nπ.
Explanation:
Let z = `(1 - i sin alpha)/(1 + 2i sin alpha)`
= `((1 - i sin alpha)(1 - 2i sin alpha))/((1 + 2i sin alpha)(1 - 2i sin alpha))`
= `(1 - 2i sin alpha - i sin alpha + 2i^2 sin^2 alpha)/((1)^2 - (2i sin alpha)^2`
= `(1 - 3i sin alpha - 2 sin^2 alpha)/(1 - 4i^2 sin^2 alpha)`
= `((1 - 2 sin^2 alpha) - 3i sin alpha)/(1 + 4 sin^2 alpha)`
= `(1 - 2 sin^2 alpha)/(1 + 4 sin^2 alpha) - (3sin alpha)/(1 + 4 sin^2 alpha) .i`
Since, z is purely real.
Then `(-3 sin alpha)/(1 + 4 sin^2 alpha)` = 0
⇒ sinα = 0
So, α = nπ, n ∈ N.
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 (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
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:
(2 + 3i)(1 – 4i)
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Find the value of i49 + i68 + i89 + i110
Show that 1 + i10 + i100 − i1000 = 0
If a = `(-1 + sqrt(3)"i")/2`, b = `(-1 - sqrt(3)"i")/2` then show that a2 = b and b2 = a
If x + iy = (a + ib)3, show that `x/"a" + y/"b"` = 4(a2 − b2)
If `("a" + 3"i")/(2+ "ib")` = 1 − i, show that (5a − 7b) = 0
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 + 2i + 15i6y = 7x + i3 (y + 4), find x + y
Answer the following:
Simplify the following and express in the form a + ib:
`3 + sqrt(-64)`
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:
(1 + 3i)2(3 + i)
Answer the following:
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
Evaluate: i131 + i49
Answer the following:
Find the value of x3 + 2x2 − 3x + 21, if x = 1 + 2i
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")`
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
Find the value of k if for the complex numbers z1 and z2, `|1 - barz_1z_2|^2 - |z_1 - z_2|^2 = k(1 - |z_1|^2)(1 - |"z"_2|^2)`
State true or false for the following:
The complex number cosθ + isinθ can be zero for some θ.
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.
If the complex number z = x + iy satisfies the condition |z + 1| = 1, then z lies on ______.
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.
If the real part of `(barz + 2)/(barz - 1)` is 4, then show that the locus of the point representing z in the complex plane is a circle.
Solve the equation |z| = z + 1 + 2i.
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.
Let `(-2 - 1/3i)^2 = (x + iy)/9 (i = sqrt(-1))`, where x and y are real numbers, then x – y equals to ______.
Simplify the following and express in the form a + ib.
`(3i^5+2i^7+i^9)/(i^6+2i^8+3i^18)`
Evaluate the following:
i35
