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 given by:

$3 + 2x$

This includes the width of the advertisement poster (3m) plus twice the width of the frame.

The total front area A can be calculated using the formula for the area of a rectangle:

egin{align*} A &= (12 + 2x)(3 + 2x)
&= 12 \cdot 3 + 12 \cdot 2x + 3 \cdot 2x + 2x \cdot 2x
&= 36 + 24x + 6x + 4x^2
&= 4x^2 + 30x + 36 \end{align*}

Thus, the expression for A is confirmed.

To find the outside length, we must first solve the equation for the area:

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

which simplifies to:

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

Using the quadratic formula, we get:

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

Substituting values gives:

$x = \frac{-30 \pm \sqrt{30^2 - 4 \cdot 4 \cdot (-16)}}{2 \cdot 4}$

which leads to:

After finding the values of x, plug the valid value of x into the expression for the outside length:

Outside length = $12 + 2x$

To solve this equation:

$3 = x(7x - 5)$

which simplifies to:

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

Using the quadratic formula:

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

Calculate to find the two possible values of x, rounding to two decimal places.

The inequality ( x^2 + 4 > 0 ) is satisfied for all x since

yields no real roots. Hence, it is true for all real numbers.

From the first equation, we can express y as:

$y = x + 3$

Substituting this into the second equation gives:

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

Expanding and rearranging leads to a quadratic equation in terms of x. Solve this quadratic equation to find the values of x and use them to find y.

Starting from the formula:

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

Rearranging gives:

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

Using the previous result for f:

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

Calculating this value gives the frequency, which is rounded to the nearest integer.

Adding the two binary numbers 1100111 and 1111010:

   1100111
 + 1111010
 ---------
  11010001

This gives a binary sum of 11010001.

To convert the binary number 11010001 to decimal, compute:

$1 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 = 128 + 64 + 16 + 1 = 209$

Thus, the equivalent decimal number is 209.