Solve for $x$: 1.1.1 $2x(x + 3) = 0$ 1.1.2 $x(x + 9) = 12$ 1.1.3 $x(x - y) ext{ } ext{and then represent the solution on a number line}$ Solve for $x$ and $y$ if: 1.2.1 $x = 1 - 2y$ and 1.2.2 $3x^2 = 3 + x + y$ 1.3 The diagram below shows a simple pendulum swinging from point A to point C - NSC Technical Mathematics - Question 1 - 2021 - Paper 1

To solve the equation $x(x + 9) = 12$, we first rewrite it as:

$x^2 + 9x - 12 = 0$

Using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 1$, $b = 9$, and $c = -12$:

1. Calculate the discriminant:

   $b^2 - 4ac = 9^2 - 4(1)(-12) = 81 + 48 = 129$

2. Apply values to the quadratic formula:

   $x = \frac{-9 \pm \sqrt{129}}{2}$

So, the solutions are:

$x_1 = \frac{-9 + 11.36}{2} \approx 1.18$

$x_2 = \frac{-9 - 11.36}{2} \approx -10.18$.

1. Solve for $y$ in terms of $x$ using $x = 1 - 2y$:

   Rearranging gives:

   $2y = 1 - x \Rightarrow y = \frac{1 - x}{2}$

2. Substitute this into the second equation $3x^2 = 3 + x + y$:

   $3x^2 = 3 + x + \frac{1 - x}{2}$

   Solving for $x$ gives further insight into possible $y$ values.

To represent the solutions on a number line, mark the points corresponding to both $x_1$ and $x_2$, as well as additional values obtained.

Starting from:

$T = 2 ext{π} \sqrt{\frac{L}{g}}$

1. Square both sides:

   $T^2 = (2\pi)^2 \frac{L}{g}$

2. Rearranging gives:

   $L = \frac{gT^2}{(2\pi)^2}$.

Substituting the values into the formula:

$L = \frac{9 (1.74)^2}{(2\pi)^2}$

Calculating:

1. $(1.74)^2 = 3.0276$

2. Then, substitute:

   $L = \frac{9 \cdot 3.0276}{39.4784} \approx 0.693 ext{ m}$.

Convert the binary numbers to decimal:

$A = 1101100_2 = 108_{10}$
$B = 11100_2 = 28_{10}$

Perform subtraction:

$108 - 28 = 80_{10}$
Convert back to binary:

$80_{10} = 1010000_2$.

To convert $1010000_2$ to decimal:

• $1 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0$
• Equals $64 + 0 + 16 + 0 + 0 + 0 + 0 = 80_{10}$.