An express train makes a run of 240 km at a certain speed. Another train whose speed is 12 km/h less takes an hour longer to cover the same distance. Find the speed of the express train.

Draw and describe the locus in each of the following cases :

(i) The locus of vertices of all isosceles triangles having a common base.

(ii) The locus of points inside a circle and equidistant from two fixed points on the circle.

(iii) The locus of centres of all circles passing through two fixed points.

(iv) The locus of a point in rhombus ABCD which is equidistant from AB and AD.

The expression 2x³ + ax² + bx - 2 leaves remainders 0 and 7 when divided by (x + 2) and (2x - 3), respectively. Calculate the values of a and b. Factorise the expression completely.

Points (8, 0) and (-3, 0) are invariant points under reflection in the line L₁, points (0, -9) and (0, 5) are invariant points under reflection in the line L₂.

(i) Name or write down equations of the lines L₁ and L₂.

(ii) Write down the images of points P(3, 5) and Q(-8, 3) after reflection in line L₁. Name the images as P' and Q' respectively.

(iii) Write down the images of P and Q on reflection in L₂. Name the images as P" and Q" respectively.

(iv) Describe a single transformation that maps P' to P".