Advertisements
Advertisements
प्रश्न
For any two complex numbers z1 and z2, prove that Re (z1z2) = Re z1 Re z2 – Imz1 Imz2
उत्तर
Let z1 = a + ib, z2 = c + id
∴ z1z2 = (a + ib)(c + id)
= ac + adi + bci + i2bd
= (ac – ba) + (ad + bc) i [∴ i2 = – 1]
Real part of Re(z1z2) = ac – bd
= Rez1 Rez2 – Imz1 Imz2
Here real part of Rez1 = a, similarly Rez2 = c
Imz1 = imaginary part of z1 = b
Similarly Imz2 = d.
APPEARS IN
संबंधित प्रश्न
Solve the equation x2 + 3 = 0
Solve the equation `sqrt2x^2 + x + sqrt2 = 0`
Solve the equation `3x^2 - 4x + 20/3 = 0`
x2 + 1 = 0
4x2 − 12x + 25 = 0
\[4 x^2 + 1 = 0\]
\[x^2 - 4x + 7 = 0\]
\[5 x^2 - 6x + 2 = 0\]
\[x^2 + x + 1 = 0\]
\[17 x^2 - 8x + 1 = 0\]
\[8 x^2 - 9x + 3 = 0\]
\[\sqrt{3} x^2 - \sqrt{2}x + 3\sqrt{3} = 0\]
\[- x^2 + x - 2 = 0\]
\[x^2 - 2x + \frac{3}{2} = 0\]
\[3 x^2 - 4x + \frac{20}{3} = 0\]
Solving the following quadratic equation by factorization method:
\[x^2 + 10ix - 21 = 0\]
Solving the following quadratic equation by factorization method:
\[6 x^2 - 17ix - 12 = 0\]
Solve the following quadratic equation:
\[x^2 - \left( 3\sqrt{2} + 2i \right) x + 6\sqrt{2i} = 0\]
Solve the following quadratic equation:
\[x^2 - \left( 5 - i \right) x + \left( 18 + i \right) = 0\]
Solve the following quadratic equation:
\[i x^2 - 4 x - 4i = 0\]
Solve the following quadratic equation:
\[x^2 + 4ix - 4 = 0\]
Solve the following quadratic equation:
\[x^2 - \left( 3\sqrt{2} - 2i \right) x - \sqrt{2} i = 0\]
Solve the following quadratic equation:
\[x^2 - \left( \sqrt{2} + i \right) x + \sqrt{2}i = 0\]
Write the number of real roots of the equation \[(x - 1 )^2 + (x - 2 )^2 + (x - 3 )^2 = 0\].
Write roots of the equation \[(a - b) x^2 + (b - c)x + (c - a) = 0\] .
The values of x satisfying log3 \[( x^2 + 4x + 12) = 2\] are
If α, β are the roots of the equation \[x^2 + px + 1 = 0; \gamma, \delta\] the roots of the equation \[x^2 + qx + 1 = 0, \text { then } (\alpha - \gamma)(\alpha + \delta)(\beta - \gamma)(\beta + \delta) =\]
If the roots of \[x^2 - bx + c = 0\] are two consecutive integers, then b2 − 4 c is
The values of k for which the quadratic equation \[k x^2 + 1 = kx + 3x - 11 x^2\] has real and equal roots are
The value of p and q (p ≠ 0, q ≠ 0) for which p, q are the roots of the equation \[x^2 + px + q = 0\] are
The set of all values of m for which both the roots of the equation \[x^2 - (m + 1)x + m + 4 = 0\] are real and negative, is
If α, β are the roots of the equation \[x^2 + px + q = 0 \text { then } - \frac{1}{\alpha} + \frac{1}{\beta}\] are the roots of the equation
The least value of k which makes the roots of the equation \[x^2 + 5x + k = 0\] imaginary is
