In the diagram, the points A, B, C and D are on the circumference of a circle, whose centre O lies on BD - HSC - SSCE Mathematics Extension 1 - Question 12 - 2015 - Paper 1

We know that the tangent at point D is perpendicular to the radius OD. Using the tangent-chord angle theorem where the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment:

$\angle ADX = \angle DCY = 30^\circ.$

To find ( \angle CAB ), we can utilize the triangle angle sum property. We know that: $$\angle CAB + \angle ZACB + \angle ADX = 180^\circ.\n$ Therefore:

( \angle CAB + 50^\circ + 30^\circ = 180^\circ ) ( \angle CAB + 80^\circ = 180^\circ ) ( \angle CAB = 180^\circ - 80^\circ = 100^\circ. )

For the parabola defined by ( x^2 = 4ay ), the focal chord properties state that if P is ( (8a, 16a) ), we have: ( PQ: y - 16a = m(x - 8a) ). Since PQ is a focal chord, it is defined such that the product of the slopes to the focus is negative: ( pq = -1. ) To find Q, considering that P is fixed, we derive Q's coordinates using the defined relationship of focal chords.

From the triangle formed with the angle of elevation: The height ( h ) and base distance can be related through trigonometry:

Using the tangent: ( \tan(15^\circ) = \frac{h}{2000}. ) Rearranging gives: ( h = 2000 \tan(15^\circ). ) Since ( an(15^\circ) = \frac{1}{\cot(15^\circ)} ) Thus we have: ( OA = h \cot(15^\circ). )

Following the calculated relationship:

$OA = h \cot(15^\circ) \Rightarrow h = \frac{OA}{\cot(15^\circ)}$ By substituting the calculated value of OA derived from point A, we obtain the numeric value for height ( h ) depending on the cotangent value.

Utilizing the chord length definition in relation to the radius and central angle:

From the formula for a chord of a circle: $L = 2r \sin(\frac{\theta}{2}) \Rightarrow 160 = 2r \sin(\frac{\theta}{2})$ Squaring this equation results in: $160^2 = 4r^2 \sin^2(\frac{\theta}{2}).$ Then use identity ( \sin^2(\frac{\theta}{2}) = \frac{1 - \cos(θ)}{2} ) to arrive at the desired equation.

Starting from the relationship derived in (i), substituting into the area or arc length relationships can help reach this equality. The substitution of length values and angle equations will yield the quadratic expression needed.

Applying Newton's method requires calculating the function and its derivative at the point ( θ_1 ):

Define the function based on the earlier equality, and calculate: ( θ_{n+1} = θ_n - \frac{f(θ_n)}{f'(θ_n)} ) By substituting ( θ_1 = π ) we will achieve the second approximation iteratively.