2021 DHS-EJC H3 Math Questions

1 (i) Find the expressions for  d f ( )g( )d n n xxx in terms of the functions and their derivatives for 1, 2.n [2] (ii) Hence, solve (a)   1 2 dsin e , d 1 xyyx x x    given 1y when 1.x [3] (b)   2 2 dd 2 tan .dd yy x y xxx   [4] 2 Let the sequence of numbers 1 2 3, , , ... , , ... ,nT T T T for 1n with 1 1T  satisfy the recurrence relation a b a bT T T ab    for all integers , ab greater than or equal to 1. (i) Show that  1 2 n nnT  for 1.n [3] (ii) Verify that 11 .ab a b a bT T T T T  [1] (iii) Show that  0 or 1 mod3nT  for all positive integers n. [3] Three positive integers x, y, z are collectively denoted as a Pythagorean Triple if 2 2 2 .x y z (iv) Show that 81nT  is a perfect square for all positive integers n. [1] (v) Hence or otherwise, show that there exist Pythagorean Triples with 4 nyT for any positive integer n. [2]

3 Let f be a monotone decreasing function defined on the interval [ ),N  where N is an integer. The integral test for convergence states that the infinite series f ( ) nN n    converges if and only if the integral f ( ) d N nn   is finite. (i) Show 1 1 p n n    converges for 1p and diverges otherwise. [3] (ii) By considering an appropriate comparison with the series in part (i) with an appropriate value of p, show that 4 1 41n n n    converges. [2] (iii) Show that 4 1 41n n n    converges by directly applying the integral test for convergence. [2] (iv) Verify that    2 2 42 2 1 2 2 1 4 1.n n n n n      Hence or otherwise, show that 4 1 1 .4 1 4n n n    [3] 4 A function f is continuous at the point k if limf ( ) f ( ). xk xk   Let f:  be a function satisfying the following conditions:       f f f ,x y x y   for all ,,xy     0 flim 1 h h h  . (i) Find  f 0 . [2] (ii) Suppose that  f 1 1, show that  f, nn for any positive integer n. [2] (iii) Using part (ii), prove that  f, xx for any positive rational number x. [3] (iv) By considering   0 limf , h h  show that f is continuous at any point .x [4]

5 There are three types of flowers: roses, sunflowers and tulips. (i) Twelve flowers are selected to form a bunch. In how many ways can this be done (a) if there are no other restrictions, [2] (b) if at least two of each type of flower are selected? [3] (ii) Twelve flowers are now selected and placed in a row on the table. In how many ways can that be done (a) if there are no other restrictions, [1] (b) if at least one of each type of flower is selected? [4] (iii) Ten sunflowers are distributed among three identical vases such that there is at least one flower in each vase. In how many ways can that be done? [2] (iv) Instead of just the ten sunflowers, one rose and one tulip are added. The twe