This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to questions 1, 2, and 3. 1a. Represent the following complex numbers on an Argand diagram. To represent a complex number Z = x + yi on an Argand diagram, plot the point (x, y) in the Cartesian plane, where the x-axis represents the real part and the y-axis represents the imaginary part. i) Z = 1 + 2i The real part is 1 and the imaginary part is 2. Plot the point (1, 2). ii) Z = 3 + 2i The real part is 3 and the imaginary part is 2. Plot the point (3, 2). iii) Z = -2 + 4i The real part is -2 and the imaginary part is 4. Plot the point (-2, 4). 1b. Evaluate i) sqrt(-400) Step 1: Rewrite the expression using i = sqrt(-1). sqrt(-400) = sqrt(400 × -1) Step 2: Separate the square roots. sqrt(400 × -1) = sqrt(400) × sqrt(-1) Step 3: Evaluate the square roots. sqrt(400) × sqrt(-1) = 20i The value is 20i. ii) sqrt(-81) Step 1: Rewrite the expression using i = sqrt(-1). sqrt(-81) = sqrt(81 × -1) Step 2: Separate the square roots. sqrt(81 × -1) = sqrt(81) × sqrt(-1) Step 3: Evaluate the square roots. sqrt(81) × sqrt(-1) = 9i The value is 9i. iii) sqrt(-225) Step 1: Rewrite the expression using i = sqrt(-1). sqrt(-225) = sqrt(225 × -1) Step 2: Separate the square roots. sqrt(225 × -1) = sqrt(225) × sqrt(-1) Step 3: Evaluate the square roots. sqrt(225) × sqrt(-1) = 15i The value is 15i. 2. Find the Product and Division of the following complex numbers. i) Z_1 = 1 - 2i and Z_2 = 1 + 2i Product: Step 1: Multiply the complex numbers. Z_1 Z_2 = (1 - 2i)(1 + 2i) Step 2: Use the difference of squares formula (a-b)(a+b) = a^2 - b^2. (1 - 2i)(1 + 2i) = 1^2 - (2i)^2 Step 3: Simplify using i^2 = -1. 1^2 - (2i)^2 = 1 - 4i^2 = 1 - 4(-1) = 1 + 4 = 5 The product is 5. Division: Step 1: Write the division as a fraction. (Z_1)/(Z_2) = (1 - 2i)/(1 + 2i) Step 2: Multiply the numerator and denominator by the conjugate of the denominator, which is 1 - 2i. (1 - 2i)/(1 + 2i) = ((1 - 2i)(1 - 2i))/((1 + 2i)(1 - 2i)) Step 3: Expand the numerator and denominator. Numerator: (1 - 2i)(1 - 2i) = 1 - 2i - 2i + 4i^2 = 1 - 4i - 4 = -3 - 4i Denominator: (1 + 2i)(1 - 2i) = 1^2 - (2i)^2 = 1 - 4i^2 = 1 - 4(-1) = 1 + 4 = 5 Step 4: Combine the results. (-3 - 4i)/(5) = -(3)/(5) - (4)/(5)i The division is -(3)/(5) - (4)/(5)i. ii) Z_1 = 1 + i and Z_2 = 2 + 4i Product: Step 1: Multiply the complex numbers. Z_1 Z_2 = (1 + i)(2 + 4i) Step 2: Expand the product. (1 + i)(2 + 4i) = 1(2) + 1(4i) + i(2) + i(4i) = 2 + 4i + 2i + 4i^2 Step 3: Simplify using i^2 = -1. 2 + 6i + 4(-1) = 2 + 6i - 4 = -2 + 6i The product is -2 + 6i. Division: Step 1: Write the division as a fraction. (Z_1)/(Z_2) = (1 + i)/(2 + 4i) Step 2: Multiply the numerator and denominator by the conjugate of the denominator, which is 2 - 4i. (1 + i)/(2 + 4i) = ((1 + i)(2 - 4i))/((2 + 4i)(2 - 4i)) Step 3: Expand the numerator and denominator. Numerator: (1 + i)(2 - 4i) = 1(2) + 1(-4i) + i(2) + i(-4i) = 2 - 4i + 2i - 4i^2 = 2 - 2i - 4(-1) = 2 - 2i + 4 = 6 - 2i Denominator: (2 + 4i)(2 - 4i) = 2^2 - (4i)^2 = 4 - 16i^2 = 4 - 16(-1) = 4 + 16 = 20 Step 4: Combine the results and simplify. (6 - 2i)/(20) = (6)/(20) - (2)/(20)i = (3)/(10) - (1)/(10)i The division is (3)/(10) - (1)/(10)i. 3. Determine the Real and Imaginary parts of: For a complex number Z = x + yi, the real part is Re(Z) = x and the imaginary part is Im(Z) = y. i) Z = 5 + 2i Real part: 5 Imaginary part: 2 ii) Z = 2i This can be written as Z = 0 + 2i. Real part: 0 Imaginary part: 2 iii) Z = 12 This can be written as Z = 12 + 0i. Real part: 12 Imaginary part: 0 iv) Z = - + i Real part: - Imaginary part: v) Z = -9 + (-i) This can be written as Z = -9 - i. Real part: -9 Imaginary part: -1