Advertisements
Advertisements
प्रश्न
Number of solutions of the equation z2 + |z|2 = 0 is ______.
विकल्प
1
2
3
Infinitely many
Advertisements
उत्तर
Number of solutions of the equation z2 + |z|2 = 0 is infinitely many.
Explanation:
z2 + |z|2 = 0, z ≠ 0
⇒ x2 – y2 + i2xy + x2 + y2 = 0
⇒ 2x2 + i2xy = 0, 2x(x + iy) = 0
⇒ x = 0 or x + iy = 0 ......(not possible)
Therefore, x = 0 and z ≠ 0
So y can have any real value. Hence infinitely many solutions.
APPEARS IN
संबंधित प्रश्न
Find the multiplicative inverse of the complex number.
–i
Find the number of non-zero integral solutions of the equation `|1-i|^x = 2^x`.
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of i + i2 + i3 + i4
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Simplify the following and express in the form a + ib:
`5/2"i"(- 4 - 3 "i")`
Write the conjugates of the following complex number:
cosθ + i sinθ
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
Evaluate: `("i"^37 + 1/"i"^67)`
Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`
Find the value of x and y which satisfy the following equation (x, y ∈ R).
`((x + "i"y))/(2 + 3"i") + (2 + "i")/(2 - 3"i") = 9/13(1 + "i")`
Select the correct answer from the given alternatives:
If n is an odd positive integer then the value of 1 + (i)2n + (i)4n + (i)6n is :
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:
`5/2"i"(-4 - 3"i")`
Answer the following:
Simplify the following and express in the form a + ib:
`(4 + 3"i")/(1 - "i")`
Answer the following:
Evaluate: (1 − i + i2)−15
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
The value of (2 + i)3 × (2 – i)3 is ______.
If `(x + iy)^(1/3)` = a + ib, where x, y, a, b ∈ R, show that `x/a - y/b` = –2(a2 + b2)
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 argument of the complex number z = `(1 + i sqrt(3))(1 + i)(cos theta + i sin theta)` is `(7pi)/12 + theta`.
Evaluate `sum_(n = 1)^13 (i^n + i^(n + 1))`, where n ∈ N.
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.
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
The sum of the series i + i2 + i3 + ... upto 1000 terms is ______.
State True or False for the following:
For any complex number z the minimum value of |z| + |z – 1| is 1.
Which of the following is correct for any two complex numbers z1 and z2?
The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on ______.
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 ______.
If α and β are the roots of the equation x2 + 2x + 4 = 0, then `1/α^3 + 1/β^3` is equal to ______.
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.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
