The magnitude of the ratio 1+2j1−j1+2𝑗1−𝑗 is 1.58 2.5 5 1.414

The question seems to be asking for the magnitude of the complex number ratio (1+2j)/(1-j). Here are the steps to solve it:

1. Write down the complex number ratio: (1+2j)/(1-j)

2. Multiply the numerator and the denominator by the conjugate of the denominator to get rid of the imaginary part in the denominator. The conjugate of (1-j) is (1+j). So, we have: [(1+2j)(1+j)] / [(1-j)(1+j)]

3. Expand the numerator and the denominator: [1 + j + 2j + 2j^2] / [1 - j + j - j^2]

4. Remember that j^2 = -1. So, replace j^2 with -1 in the equation: [1 + j + 2j - 2] / [1 - j + j + 1]

5. Simplify the numerator and the denominator: [-1 + 3j] / [2]

6. Divide the real part and the imaginary part in the numerator by the real number in the denominator: -1/2 + 3j/2 = -0.5 + 1.5j

7. The magnitude of a complex number a + bj is given by sqrt(a^2 + b^2). So, the magnitude of -0.5 + 1.5j is sqrt((-0.5)^2 + (1.5)^2) = sqrt(0.25 + 2.25) = sqrt(2.5) = 1.58 (approximately).

So, the magnitude of the ratio (1+2j)/(1-j) is approximately 1.58.