1.1 The picture below shows a rectangular advertising board that consists of a frame and a rectangular area for placing an advertisement poster - NSC Technical Mathematics - Question 1 - 2020 - Paper 1

The outside breadth of the rectangular board is determined similarly by adding the height of the poster (3 meters) and twice the width of the frame (2x meters). Thus, the formula for the outside breadth is:

$B = 3 + 2x$

To find the total area A of the rectangular board, we multiply the outside length and outside breadth:

$A = (12 + 2x)(3 + 2x)$

Expanding this, we have:

$A = 12 \cdot 3 + 12 \cdot 2x + 2x \cdot 3 + 2x \cdot 2x$

This simplifies to:

$A = 36 + 24x + 6x + 4x^2 = 4x^2 + 30x + 36$

To find the outside length of the rectangular board, we set the equation obtained for A equal to 52 and solve:

$4x^2 + 30x + 36 = 52$

This simplifies to:

$4x^2 + 30x - 16 = 0$

Using the quadratic formula, we find:

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

Substituting the coefficients a = 4, b = 30, and c = -16 gives the values for x. The corresponding length is:

$L = 12 + 2x$

Starting with the equation:

$\frac{3}{x} = 7x - 5$

Multiply both sides by x (where x ≠ 0) to eliminate the fraction:

$3 = 7x^2 - 5x$

Rearranging gives:

$7x^2 - 5x - 3 = 0$

Using the quadratic formula:

$x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 7 \cdot (-3)}}{2 \cdot 7}$

Calculating gives two possible roots to 2 decimal places.

The inequality (x^2 + 4 > 0) is always true since (x^2) is non-negative and the minimum value of (x^2) is 0. Therefore, the solution set is:

$x \in \mathbb{R}$

From the first equation, rearranging gives:

$y = x + 3$ Substituting into the second equation:

$3x^2 + x(x + 3) - (x + 3)^2 = -3$

Expanding and rearranging leads to a quadratic in terms of x, which can be solved accordingly.

Starting with:

$X_c = \frac{1}{2\pi f C}$

Rearranging gives:

$f = \frac{1}{2\pi X_c C}$

Substituting the values for (X_c = 63.66 \text{ ohms}) and (C = 50 \times 10^{-6} \text{ farads}):

$f = \frac{1}{2\pi (63.66)(50 \times 10^{-6})}$

Calculating gives approximately 50 hertz.

To sum the binary numbers 110011 and 111010:

  110011
+ 111010
 --------
 1101101

The binary sum 1101101 can be converted to decimal as follows:

$1*2^6 + 1*2^5 + 0*2^4 + 1*2^3 + 1*2^2 + 0*2^1 + 1*2^0 = 64 + 32 + 0 + 8 + 4 + 0 + 1 = 109.$

Thus, the equivalent decimal number notation is 109.