(a)(i) Write down the binomial expansion of \((1 + x)^{4}\).

(ii) Use the result in (a)(i) to evaluate, correct to three decimal places \((\frac{5}{4})^{4}\).

(b) The first, second and fifth terms of a linear sequence (A.P) are three consecutive terms of an exponential sequence (G.P). If the first term of the linear sequence is 7, find the common difference.

The table shows the distribution of the heights of a group of people.

(a) Draw a histogram to illustrate the distribution.

(b) Using an assumed mean of 1.1m, find, correct to one decimal place, the mean height of the group.

(a) Edem and his wife were invited to a dinner by a family of 5. They all sat in such a way in such a way that Edem sat next to his wife. Find the number of ways of seating them in a row.

(b) A bag contains 4 red and 5 black identical balls. If 5 balls are selected at random, one after the other with replacement, find the probability that :

(i) a red ball was picked 3 times ; (ii) a black ball was picked at most 2 times.

(a) Given that \(\overrightarrow{AB} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}\) and \(\overrightarrow{BC} = \begin{pmatrix} -3 \\ 5 \end{pmatrix}\); find the :

(i) angle between the vectors AB and AC ; (ii) unit vector along \(\overrightarrow{AB} - \overrightarrow{BC}\).

(b) P, Q, R and M are points in the \(O_{XY}\) plane. If \(\overrightarrow{PQ} = 2i + 8j , \overrightarrow{PR} = 11i - 12j\) and M divides QR internally in the ratio 3 : 7, find \(\overrightarrow{PM}\).

(a) A bucket full of water with a mass of 8kg is pulled out of a well with a light inextensible rope. Find its acceleration when tha tension in the rope is 150N. [Take \(g = 10ms^{-2}\)].

(b) A mass of 12kg is acted upon by a force F, changing its speed from 15 m/s to 25 m/s after covering a distance of 50m. Find the :

(i) value of F ; (ii) distance covered when its speed is 35 m/s.