RI 2024 H3 Math Prelim

2024 RI H3 Math Prelim Functions 1 The Chebyshev polynomial of the first kind, ()nTx , is defined by ( ) 1( ) cos cosnT x n x −= so that ( )(cos ) cosnTn = . (a) Write down 0()Tx and 1()Tx , and show that 2 2( ) 2 1T x x =− . [2] (b) Show that 11( ) 2 ( ) ( )n n nT x xT x T x+− =− for 1n . [3] (c) Hence find (0)nT in terms of n, describing carefully all the possible cases that arise. [3] (d) Write down the roots of the equation ( ) 0nTx = . [2] (e) Hence for positive integers n, evaluate 1 0 21cos 2 n k k n  − = +  , describing carefully all the possible cases that arise. [4] Combinatorics 2 A set of positive integers is called well-spaced if it contains at most one out of any three consecutive integers. For example, the set  2,5,8 is well-spaced while  2,5,7,10 is not. Define nS to be the number of well-spaced subsets, including the empty set, of the set  1, 2,3,...,n . (a) Find 1S , 2S and 3S . [2] (b) Explain why, for any positive integer n, 32n n nS S S++ =+ . Hence obtain 8S . [4] Define ,nkT to be the number of well-spaced subsets of the set  1, 2,3,...,n that has k elements. (c) Use the bijection principle to show that , 22 nk nkT k −+= . [4] (d) Hence express nS in terms of binomial coefficients. [2]

2024 RI H3 Math Prelim Calculus 3 (a) Let g and h be continuous functions defined on the real numbers such that g( ) g( ) 1xx+ − = and h( ) h( )xx=− for all real numbers x. (i) Show that 0 g( )h( ) d h( ) d aa a x x x x x − = for any real number a. [3] (ii) Determine the exact value of 1 2 1 1 d12 x x x − − + . [4] (b) Let f be defined on  such that f ( )xx= for 1x  and f ( ) f ( 2)xx=+ for all real numbers x. Determine 0 f ( )e dxxx  −  . [8] Sequences and Series 4 The sequence of real numbers 1 2 3, , , ,u u u  is defined by ( )1 21 2n n n n n uu u u u + ++ =− where k is a constant. It is given that 1ua= and 2ub= , where a and b are non-zero real numbers. (a) Prove that 2 2 1nn buu a −= 2 1 2nnu cu+ = for positive integers n, where c is a constant to be found in terms of a and b. [6] (b) Determine the range of b a such that the series 1 N n n u =  converges. [5] (c) Find the value(s) of b a such that the sequence 1 2 3, , ,u u u  is periodic with period 4. [3] Inequalities 5 (a) Let : →  be a convex function. Using a sketch and by considering the vector equation of the line segment l joining the two points ( ), ( )xx and ( ), ( )yy , explain why if x and y are any real numbers and [0,1], ( ) ( ) ( )(1 ) (1 )x y x y     − +  − + . [3] (b) Let 1n be a real number, and a and b be nonnegative real numbers. Show that 22 n nna b a b++  and determine when equality holds. [4] (c) Show that if a and b are nonnegative real numbers, and p and q are real numbers greater than 1 such that 11 1pq+= , then . pqabab pq+ [4]

2024 RI H3 Math Prelim Combinatorics 6 A Latin square of order n is an nn array, each cell containing one entry from the set  1, 2,...,n , with the property that each element of  1, 2,...,n occurs once in each row and once in each column. The following is an example of a Latin square of order 4. 1 2 3 4 3 4 2 1 2 1 4 3 4 3 1 2 A Latin rectangle is a kn array, where kn , with entries from  1, 2,...,n , with the property that each element of  1, 2,...,n occurs once in each row and at most once in each column. The following is an example of a 24 Latin rectangle. 1 2 3 4 3 4 2 1 (a) Explain why a Latin square of order n exists for each positive integer n. [1] (b) Determine the number of Latin squares of order 2 and 3. [3] (c) Given an ( 1)nn− Latin rectangle, show that it is possible to add one more row to obtain a Latin square of order n. [3] (d) Use the principle of inclusion and exclusion to show that the number of 2 n Latin rectangles is ( ) ( )2 0 1!. ! in i n i= −  [5]

2024 RI H3 Math Prelim 7 A postman needs to distribute letters to houses situated along a street. The houses along this street are regularly spaced apart and numbered 1, 2, …, n, where 2n is an integer. The postman has exactly one letter to distribute to each house, and he does so by first delivering the letter to house 1, before distributing the subsequent letters randomly and then returning to house 1. The postman thus follows a route, represented by the successive numbers of the houses where he drops off the letters. For example, if n = 5, a possible route is 1, 5, 2, 4, 3, 1. The total distanced travelled, which we call the length of the route, is 12, since 1 5 5 2 2 4 4 3 3 1 12− + − + − + − + − = . Another possible route is 1, 3, 5, 4, 2, 1, whose length is 8. (a) State the total number of possible routes. [1] (b) Show that the minimum length of a route is ( )21n− . [3] (c) Determine the number of routes of minimum length , explaining your reasoning clearly. [2] (d) Determine the maximum length of a route. [5] Numbers and Proof 8 Let n be a positive integer and denote by ()n the sum of all the positive divisors of n. We say that a number n is almost perfect if ( ) 2 1nn =− . (a) Show that if n is a power of 2, it is almost perfect. [1] (b) Show that if a and b are coprime, then ( ) ( ) ( )ab a b  = . [3] (c) Deduce that if n is odd and is almost perfect, then n must be a perfect square. [3] (d) Show that for all prime numbers p and positive integers a, 111 ... . 1a p p p p+ + +  − [1] (e) Let n be an odd integer greater than 1. Show that if n is almost perfect, then n must contain at least 3 distinct prime factors. [3]