The table shows the marks obtained by some candidates in Physics (y) and Mathematics (x) tests.

(a)(i) Represent this information on a scatter diagram.

(ii) Find \(\bar{x}\) and \(\bar{y}\), the mean of x and y respectively.

(iii) Draw the line of best fit to pass through (x, y).

(b) Find the equation of the line in a(iii).

(c) Use your equation in (b) to find, correct to one decimal place, the mark in Physics for a candidate who scored 28 in Mathematics.

(a) Find, correct to one decimal place, the angle between  \(p = \begin{pmatrix}  3 \\ -1 \end{pmatrix}\) and \(q = \begin{pmatrix} 3 \\ 4 \end{pmatrix}\).

(b) ABCD is a square with vertices at A(0, 0), B(2, 0), C(2, 2) and D(0, 2). Forces of magnitude 10 N, 15 N, 20 N and 5 N act along \(\overrightarrow{BA}, \overrightarrow{BC}, \overrightarrow{DC}\) and \(\overrightarrow{AD}\) respectively. Find the (i) magnitude (ii) direction; of the resultant.

If \(\begin{vmatrix} x - 3 & -4 & 3 \\ 5 & 2 & 2 \\ 2 & -4 & 6 - x \end{vmatrix} = -24 \), find the values of x.

Given that \(\log_{3} x - 3\log_{x} 3 + 2 = 0\), find the values of x.

(a) Using the substitution \(u = x - 2\), write \(\frac{x^{3} + 5}{(x - 2)^{4}}\) as an expression in terms of u.

(b) Using the answer in (a), express \(\frac{x^{3} + 5}{(x - 2)^{4}}\) in partial fractions.