Discuss the following relations for reflexivity, symmetricity and transitivity :
The relation R defined on the set of all positive integers by "mRn if m divided n".

Resolve the following rational expressions into partial fractions.
\({{1}\over{x^2-a^2}}\)

Show that \(\frac { sin8x\ cosx-sin6x\ cos3x }{ cos2x\ cosx-sin3x\ sin4x } =tan2x\)

How many strings can be formed using the letters of the word LOTUS if the word
(i) either starts with L or ends with S?
(ii) neither starts with L nor ends with S?

Using the mathematical induction, show that for any natural number n,\(\frac { 1 }{ 1.2 } +\frac { 1 }{ 2.3 } +\frac { 1 }{ 3.4 } +...+\frac { 1 }{ n(n+1) } =\frac { n }{ n+1 } \).

Compute the sum of first n terms of the following series 6 + 66 + 666 + .......

Find the equations of straight lines which are perpendicular to the line 3x + 4y - 6 = 0 and are at a distance of 4 units from (2, 1).

In a shopping mall there is a hall of cuboid shape with dimension 800 \(\times\)800 \(\times\)720 units, which needs to be added the facility of an escalator in the path as shown by the dotted line in the figure. Find 
(i) the minimum total length of the escalator.
(ii) the heights at which the escalator changes its direction.
(iii) the slopes of the escalator at the turning points.