RI H3 Math 2024 Prelim Questions

2024 Raffles Institution H3 Mathematics Preliminary Examinations 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] 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 Raffles Institution H3 Mathematics Preliminary Examinations 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] 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]

2024 Raffles Institution H3 Mathematics Preliminary Examinations 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]