Question 17 of 28intermediate🔧 ApplyNumerical2 marks

Reduce (11−4i−21+i)(3−4i5+i)\left( \frac{1}{1-4i} - \frac{2}{1+i} \right) \left( \frac{3-4i}{5+i} \right) to the standard form.

Correct Answer

307442+599442\frac{307}{442} + \frac{599}{442}i

Exercise: Miscellaneous Exercise on Chapter 4 | Q: 3 | (Page 10)
For More Understanding

Explanation

This question tests the student's understanding of operations on complex numbers, specifically division and subtraction, followed by multiplication. The key technique used is rationalizing the denominator by multiplying with the conjugate, as demonstrated in Example 6 of the textbook context. Students must carefully apply the distributive property and remember that i² = -1.

Solution Steps

  1. Step 1: Simplify the first term by multiplying numerator and denominator by the conjugate (1+4i): 11−4i=1+4i1+16=1+4i17\frac{1}{1-4i} = \frac{1+4i}{1+16} = \frac{1+4i}{17}

  2. Step 2: Simplify the second term by multiplying numerator and denominator by the conjugate (1-i): 21+i=2(1−i)1+1=2(1−i)2=1\frac{2}{1+i} = \frac{2(1-i)}{1+1} = \frac{2(1-i)}{2} = 1-i

  3. Step 3: Subtract the two results: 1+4i17\frac{1+4i}{17} - (1-i) = 1+4i−17+17i17=−16+21i17\frac{1+4i-17+17i}{17} = \frac{-16+21i}{17}

  4. Step 4: Simplify the third fraction by multiplying by conjugate (5-i): 3−4i5+i=(3−4i)(5−i)25+1=15−23i−426=11−23i26\frac{3-4i}{5+i} = \frac{(3-4i)(5-i)}{25+1} = \frac{15-23i-4}{26} = \frac{11-23i}{26}

  5. Step 5: Multiply the results: −16+21i17×11−23i26=(−16+21i)(11−23i)442=−176+368i+231i+483442=307+599i442=307442+599442\frac{-16+21i}{17} \times \frac{11-23i}{26} = \frac{(-16+21i)(11-23i)}{442} = \frac{-176+368i+231i+483}{442} = \frac{307+599i}{442} = \frac{307}{442} + \frac{599}{442}i

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