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Chapter 2: Problem 106

If \(Z=[(4+3 i) /(5-3 i)]\) then \(Z^{-1}=\) (a) \((11 / 25)-(27 / 25) \mathrm{i}\) (b) \([(-11) / 25)]-(27 / 25) \mathrm{i}\) (c) \([(-11) / 25)]+(27 / 25) \mathrm{i}\) (d) \((11 / 25)+(27 / 25) \mathrm{i}\)

Short Answer

Expert verified
(a) \(\frac{11}{850} - \frac{27}{850}i\)

Step by step solution

01

To find the conjugate of the denominator, change the sign of the imaginary part. The original denominator is \(5 - 3i\), so its conjugate is \(5 + 3i\). #Step 2: Multiply the numerator and denominator by the conjugate#

Multiply both the numerator and the denominator of the original fraction by the conjugate found in step 1. \[ Z = \frac{(4 + 3i)(5 + 3i)}{(5 - 3i)(5 + 3i)} \] #Step 3: Simplify the numerator#
02

Simplify the numerator by performing arithmetic operations using the distributive property and combining real parts with real parts, and imaginary parts with imaginary parts: \[(4 + 3i)(5 + 3i) = (4*5 + 4*3i + 3i*5 + 3i*3i) = (20 + 12i + 15i + 9i^2)\] Since \(i^2 = -1\), the expression becomes: \(20 + 12i + 15i - 9 = 11 + 27i\) #Step 4: Simplify the denominator#

Perform arithmetic operations on the denominator and combine the real parts: \[(5 - 3i)(5 + 3i) = 5*5 + 5*3i - 3i*5 - 3i*3i = 25 + 15i - 15i + 9 = 34\] #Step 5: Write the simplified fraction#
03

Now that we have simplified the numerator and denominator, write the simplified fraction: \[Z = \frac{11 + 27i}{34}\] #Step 6: Find the inverse of Z#

To find the inverse of Z, write the reciprocal as the ratio of the conjugate of Z to the modulus squared of Z. First, find the conjugate of Z and the modulus squared: - Conjugate of Z: \[ Z^* = 11 - 27i \] - Modulus squared of Z: \[ |Z|^2 = (11+27i)(11-27i) = (11^2 + 27^2) = 850 \] Now, calculate the inverse of Z: \[ Z^{-1} = \frac{Z^*}{|Z|^2} = \frac{11 - 27i}{850} \] Comparing the result with the given options, we see that the correct answer is: \( Z^{-1} = \frac{11}{850} - \frac{27}{850}i \) So, the correct choice is: (a) \(\frac{11}{25} - \frac{27}{25}i\)

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