3.1 Simplify the following without the use of a calculator: 3.1.1 $\log_a a^2$ 3.1.2 $\sqrt{5x \left( \sqrt{45x + 2\sqrt{80x}} \right)}$ 3.1.3 $\left( \frac{4^{2} - 2}{3^{2} + 8} \right) \times 8$ 3.2 Solve for x: $\log(2x - 5) + \log 2 = 1$ 3.3 Given the complex number: $z = 2 + 2i$ 3.3.1 In which quadrant of the complex plane does $z$ lie? 3.3.2 Determine the value of the modulus of z - NSC Technical Mathematics - Question 3 - 2023 - Paper 1

First, simplify the inner expression:\n $\sqrt{45x + 2\sqrt{80x}} = \sqrt{5x(9 + 2\sqrt{16})} = \sqrt{5x(9 + 8)} = \sqrt{5x(17)}.$

Therefore, we have:

$\sqrt{5x \cdot \sqrt{5x(17)}} = \sqrt{5x \cdot \sqrt{85x}} = \sqrt{85x^2} = \sqrt{85} \cdot x.$

Final simplification yields: $55.$

Calculating the numerator and denominator:

$4^2 - 2 = 16 - 2 = 14$ $3^2 + 8 = 9 + 8 = 17.$

Now substituting back into the expression gives us:

$\frac{14}{17} \times 8 = \frac{112}{17}.$

Using the logarithmic property to combine:

$\log((2x - 5) \times 2) = 1.$

This implies:

The complex number $z = 2 + 2i$ has positive real and imaginary parts, indicating it lies in the first quadrant.

Using the modulus formula:

$|z| = \sqrt{(2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}.$

The angle θ can be calculated using:

$\tan(\theta) = \frac{\text{Imaginary}}{\text{Real}} = \frac{2}{2} = 1$ Thus, θ = 45°.

The polar form can be expressed as:

$z = |z| (\cos \theta + i \sin \theta) = 2\sqrt{2}(\cos 45° + i \sin 45°).$

To solve for x and y, separate the real and imaginary parts:

From the equation $x - 3y = 6 + 9i$, We have:

1. $x = 6$ (Real part)
2. $-3y = 9 \Rightarrow y = -3.$

Thus, the solution is $x = 6$, $y = -3$.