Solve for $x$: 1.1.1 $$ \frac{1}{2} \left(2x - 1\right) = 0 $$ 1.1.2 $$ -x(6 - x) = 4 \quad (round \; off \; to \; TWO \; decimal \; places) $$ 1.1.3 $$ \left(2 - x\right)(x + 5) > 0 $$ Given: $y + x - 10 = 0$ and $x^2 - xy + y^2 = 28$ 1.2.1 Express $y + x - 10 = 0$ in the form $y = mx + c$ - NSC Technical Mathematics - Question 1 - 2023 - Paper 1

First, rearrange the equation:

$-x(6 - x) = 4 \rightarrow x^2 - 6x - 4 = 0$

Using the quadratic formula:

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

where $a = 1$, $b = -6$, and $c = -4$:

$b^2 - 4ac = (-6)^2 - 4(1)(-4) = 36 + 16 = 52$

Thus,

$x = \frac{6 \pm \sqrt{52}}{2} = \frac{6 \pm 2\sqrt{13}}{2} = 3 \pm \sqrt{13}$

Calculating the approximate values, we find:

$x \approx 6.61 \; \text{and} \; x \approx -0.61$

Rounded values are $x \approx 6.61$.

To solve the inequality, find the critical values:

Setting each factor to zero gives: $2 - x = 0 \rightarrow x = 2$ $x + 5 = 0 \rightarrow x = -5$

Next, we analyze the intervals based on these critical values:

1. If $x < -5$: both factors are positive.
2. If $-5 < x < 2$: the first factor is positive while the second is negative.
3. If $x > 2$: both factors are positive.

Thus, the solution is: $x < -5 \; \text{or} \; x > 2$

Rearranging the equation gives: $y = -x + 10$

This is in the slope-intercept form $y = mx + c$, where $m = -1$ and $c = 10$.

From the previous part, we have: $y = -x + 10$

Substituting this into the second equation: $x^2 - x(-x + 10) + (-x + 10)^2 = 28$

Simplifying: $x^2 + x^2 - 10x + x^2 - 20x + 100 = 28 \rightarrow 3x^2 - 30x + 72 = 0$

Using the quadratic formula: $x = \frac{30 \pm \sqrt{(-30)^2 - 4 \cdot 3 \cdot 72}}{2 \cdot 3}$

This leads to: $x = 6 \; \text{or} \; x = 4$

Substituting back to find $y$: For $x = 6$: $y = 4$. For $x = 4$: $y = 6$.

Thus, the solutions are $(6, 4)$ and $(4, 6)$.

The given pressure formula is: $P = \frac{F}{A}$

To make $A$ the subject, rearrange the formula: $A = \frac{F}{P}$

Substitute the values into the rearranged formula: $A = \frac{25 \times 10^3}{25 984 480.5}$

Calculating this, we have: $A \approx 0.000962 \; m^2 \approx 9.62 \times 10^{-4} \; m^2$

The binary values are: $A = 1000111 \quad B = 100111$

Performing the subtraction in binary:

  1000111
- 0100111
----------
   0011000

Thus, $A - B = 0011000$.

The binary result $0011000$ converts to decimal as follows:

$2^3 + 2^2 = 8 + 4 = 12$

Thus, the decimal form is $12$.