(a)
(i) Copy and complete the addition \(\oplus\) and multiplication \(\otimes\) tables in modulo 5 on the set {2, 3, 4}.

(ii) Use the tables to:

(a) solve the equation 4 \(\otimes\)e\(\oplus\) 2 \(\equiv\) 1 (mod 5):
(b) find the value of n, if 4 \(\oplus\)n\(\otimes\) 2 \(\equiv\) (mod 5).

(b) Consider the following statements:

p: Landi has cholera,

q: Landi is in the hospital.

If p = q, state whether or not the following statements are valid:

(i) If Landi is in the hospital, then he has cholera.

(ii) If Landi is not in the hospital, then he does not have cholera.

(iii) If Landi does not have cholera, then he is not in the hospital.

Using ruler and a pair of compasses only, construct:

(a) (i) quadrilateral PQRS with |PQ| = 6 cm, |PS| = 8 cm, < PSR = 90\(^o\), |SR| = 12 cm and |QR| = 11 cm;

(ii) perpendicular from Q to cut \(\over{SR}\) at K.

(b) Measure:

(i) IQRI;

(ii) < QRS.
 

The table shows the distribution of marks scored by students in a test.

(a) Construct a cumulative frequency table for the distribution.

(b) Draw a cumulative frequency curve for the distribution.

(c) Use the curve to estimate the:

(i) median;

(ii) probability that a student selected at random obtained distinction, if the lowest mark for distinction is 75%.

 

A container, in the form of a cone resting on its vertex, is full when 4.158 litres of water is poured into it.

(a) If the radius of its base is 21 cm,

(i) represent the information in a diagram;

(ii) calculate the height of the container.

(b) A certain amount of water is drawn out of the container such that the surface diameter of the water drops to 28 cm. Calculate the volume of the water drawn out. (Take \(\pi\) = \(\frac{22}{7}\))

A man starts from a point X and walk 285 m to Y on a bearing of 078\(^o\). He then walks due South to a point Z which is 307 m from X.

(a) Illustrate the information on a diagram.

(b) Find, correct to the nearest whole number, the:

(i) bearing of X from Z;
(ii) distance between Y and Z.