1. (a) Find conjugate and magnitude of \( z=\frac{(3+2 i)(1-2 i)}{4+3 i} \) (To be solved in elass) (b) If \( = \) is a complex number then \( |z| \geq 0 \) and \( |z|=0 \) iff \( z=0 \) (To be solved in class) \( z=3(1-2 i)+2 i(1-2 i) \)

(a) To find the conjugate and magnitude of the complex number \( z=\frac{(3+2i)(1-2i)}{4+3i} \), we first need to simplify the expression. Given: \[ z=\frac{(3+2i)(1-2i)}{4+3i} \] To simplify the expression, we can multiply the numerator and denominator by the conjugate of the denominator to eliminate the complex number in the denominator. The conjugate of \( 4+3i \) is \( 4-3i \). Multiplying the numerator and denominator by \( 4-3i \), we get: \[ z=\frac{(3+2i)(1-2i)(4-3i)}{(4+3i)(4-3i)} \] Now, we can simplify the expression further: \[ z=\frac{(3+2i)(1-2i)(4-3i)}{4^2-(3i)^2} \] \[ z=\frac{(3+2i)(1-2i)(4-3i)}{16+9} \] \[ z=\frac{(3+2i)(1-2i)(4-3i)}{25} \] Now, we can find the conjugate and magnitude of \( z \). The conjugate of \( z \) is \( \overline{z} \), which is obtained by changing the sign of the imaginary part. The magnitude of \( z \) is \( |z| \), which is the square root of the sum of the squares of the real and imaginary parts. Let's calculate the conjugate and magnitude of \( z \) using the simplified expression. Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(3+2i\right)\left(1-2i\right)\left(4-3i\right)}{\left(4+3i\right)}}{\left(4-3i\right)}\) - step1: Remove the parentheses: \(\frac{\frac{\left(3+2i\right)\left(1-2i\right)\left(4-3i\right)}{4+3i}}{4-3i}\) - step2: Multiply the terms: \(\frac{\frac{16-37i}{4+3i}}{4-3i}\) - step3: Divide the terms: \(\frac{-\frac{47}{25}-\frac{196}{25}i}{4-3i}\) - step4: Multiply by the Conjugate: \(\frac{\left(-\frac{47}{25}-\frac{196}{25}i\right)\left(4+3i\right)}{\left(4-3i\right)\left(4+3i\right)}\) - step5: Calculate: \(\frac{16-37i}{\left(4-3i\right)\left(4+3i\right)}\) - step6: Calculate: \(\frac{16-37i}{25}\) - step7: Simplify: \(\frac{16}{25}-\frac{37}{25}i\) The simplified expression for \( z \) is \( \frac{16}{25} - \frac{37}{25}i \). Now, we can find the conjugate and magnitude of \( z \). The conjugate of \( z \) is \( \overline{z} = \frac{16}{25} + \frac{37}{25}i \). The magnitude of \( z \) is \( |z| = \sqrt{\left(\frac{16}{25}\right)^2 + \left(\frac{37}{25}\right)^2} \). Let's calculate the magnitude of \( z \). Calculate the value by following steps: - step0: Calculate: \(\sqrt{\left(\frac{16}{25}\right)^{2}+\left(\frac{37}{25}\right)^{2}}\) - step1: Add the numbers: \(\sqrt{\frac{13}{5}}\) - step2: Use the properties of radicals: \(\frac{\sqrt{13}}{\sqrt{5}}\) - step3: Multiply by the Conjugate: \(\frac{\sqrt{13}\times \sqrt{5}}{\sqrt{5}\times \sqrt{5}}\) - step4: Multiply the numbers: \(\frac{\sqrt{65}}{\sqrt{5}\times \sqrt{5}}\) - step5: Multiply the numbers: \(\frac{\sqrt{65}}{5}\) The magnitude of \( z \) is \( \frac{\sqrt{65}}{5} \). Therefore, the conjugate of \( z \) is \( \frac{16}{25} + \frac{37}{25}i \) and the magnitude of \( z \) is \( \frac{\sqrt{65}}{5} \). (b) If \( z \) is a complex number, then \( |z| \geq 0 \) and \( |z| = 0 \) if and only if \( z = 0 \). This statement is true because the magnitude of a complex number is always non-negative, and it is zero only when the complex number is zero.