problem archive
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Express the following in the form of a + bi: (ii) (-i)(2i)(-1/8)³ -
Express \((5 - 3i)^3\) in the form \(a + ib\). -
Express \((\sqrt{-3} + \sqrt{-2})(\sqrt{3} - i)\) in the form \(a + ib\). -
Find the multiplicative inverse of \(2 - 3i\). -
Express the following in the form \(a + ib\): \(\frac{5 + \sqrt{2}i}{1 - \sqrt{2}i}\) and \(i^{-35}\). -
Express the following expression in the form of a + ib: (3 + i√5)/(√3 + √2i) * (3 - i√5)/(√3 - i√2) -
Find the conjugate of \( \frac{(3-2i)(2+3i)}{(1+2i)(2-i)} \) -
If \( x + iy = \frac{a+ib}{a-ib} \), prove that \( x^2 + y^2 = 1 \) -
Evaluate: \[ \left(1^i + \left(\frac{1}{i}\right)^{25}\right)^3 \] -
For any two complex numbers \(z_1\) and \(z_2\), prove that \[ \text{Re} \left(\overline{z_1} z_2\right) = \text{Re} \left(z_1\right) \text{Re} \left(... -
Reduce \left(\frac{1}{1-4i} - \frac{2}{1+i}\right) \left(\frac{3-4i}{5+i}\right) to the standard form. -
If \frac{x-iy}{a-ib} = \frac{c-id}{c-id} prove that \left(\frac{x^2+y^2}{a^2+b^2}\right) = \frac{a^2+b^2}{c^2+d^2}. -
If z_1 = 2 - i, z_2 = 1 + i, find \frac{z_1 + z_2 + 1}{z_1 - z_2 + 1}. -
If a + ib = \frac{(x+i)^2}{2x^2 + 1}, prove that a^2 + b^2 = \left(\frac{x^2 + 1}{2x^2 + 1}\right)^2. -
Let z_1 = 2 - i, z_2 = -2 + i. Find (i) Re\left(\frac{z_2}{z_1}\right), (ii) Im\left(\frac{1}{z_1 - z_2}\right). -
Find the real numbers x and y if (x - iy)(3 + 5i) is the conjugate of -6 - 24i. -
Find the modulus of \frac{1+i}{1-i} \cdot \frac{1-i}{1+i}. -
If (x + iy)^3 = u + iv, then show that \frac{u}{x} + \frac{v}{y} = 4(x^2 - y^2). -
If \alpha and \beta are different complex numbers with |\beta| = 1, then find \left|\frac{\beta - \alpha}{1 - \alpha \beta}\right|. -
Find the number of non-zero integral solutions of the equation |1 - i|^x = 2^x.