The sum of the 2nd and 5th terms of an arithmetic progression (AP) is 42. If the difference between the 6th and 3rd term is 12, find the 

(i) Common difference

(ii) first term

(iii) 20th term.

If \(2^{2x - 3y} = 32\) and \(\log_{y} x = 2\), find the values of x and y.

If (x + 2) and (x - 1) are factors of \(f(x) = 6x^{4} + mx^{3} - 13x^{2} + nx + 14\), find the 

(a) values of m and n.

(b) remainder when f(x) is divided be (x + 1).

Find the equation of the tangent to the curve \(y = \frac{x - 1}{2x + 1}, x \neq -\frac{1}{2}\) at the point (1, 0).

(a) There are 6 points in a plane. How many triangles can be formed with the points?

(b) A family of 6 is to be seated in a row . In how many ways can this be done if the father and mother are not to be seated together?