Suppose a is a complex number such that If m is a positive integer, find the value of Discussion
Use calculus to find the behaviour of the function and sketch the graph of the function for . Show clearly the locations of the maxima, minima and points of inflection in your graph.
Let f(u) be a continuous function and, for any real number u, let [u] denote the greatest integer less than or equal to u. Show that for any x>1,
Show that it is not possible to have a triangle with sides a,b, and c whose medians have length .
Show that for all values of .
Let where n is an odd integer. Let f be a function defined on taking values in S such that
(ii) Show that
Consider a prism with triangular base. The total area of the three faces containing a particular vertex A is K. Show that the maximum possible volume of the prism is and find the height of this largest prism.
The following figure shows a grid divided into subgrids of size . This grid has 81 cells, 9 in each subgrid.
Now consider an grid divided into subgrids of size . Find the number of ways in which you can select n^2 cells from this grid such that there is exactly one cell coming from each subgrid, one from each row and one from each column.
Let X be a set satisfying the following properties:
(i) if and are any two distinct elements in X, then
(ii) there are two elements and in X such that for any ,
(iii) if are two elements of X, then for all
Show that if , then for some
Let A be a set of positive integers satisfying the following properties:
(i) if m and n belong to A, then m+n belong to A;
(ii) there is no prime number that divides all elements of A.(a) Suppose are two integers belonging to A such that . Show that you can find two integers in A such that
(b) Hence show that there are two consecutive integers belonging to A.
(c) Let be two consecutive integers belonging to A. Show that if then n belongs to A.