NYJC TJC VJC 2024 H3 Math Prelim

2024 NYJC-TJC-VJC H3 Prelim Inequalities 1 Let 1x , 2x ,  , nx be real numbers and 1 , 2 ,  , n be positive real numbers. (a) Prove that 2 2 11 11 nn ii ii xxnn==    . [2] (b) Prove that 2 2 1 1 1 n n n i i i i i i i i xx   = = =                and find a necessary and sufficient condition for the equality to hold. [3] (c) Let 1y , 2y ,  , ny be real numbers, not all zero, and 2t. Prove that 2 2 11 1nn i iii ii yytt==   . [3] Functions 2 For any real number s, the greatest integer less than o r equal to s is denoted by s . For example, 3.7 3= and 5 5= . (a) The functions f and g are defined by 2 f : for , 15 41 11g : for 1.4 4 2 x x x x x x x x − − +    (i) Prove that f is a periodic function with the period of the function as 1. [3] (ii) On the same diagram, s ketch the graph of ( )fyx= for 1.5 3 x−   and the graph of ( )gyx= , labelling the coordinates of all endpoints and the points of intersection of the two graphs clearly. [3] (iii) Find the area bounded by the graphs of ( )fyx= , ( )gyx= and the line 1x= . [3] (b) (i) Solve the equation 3log 2k = , where k +  . [2] (ii) A sequence {xn} is given by the recurrence relation 3 1n nxx   =+ , x0 = 2. Write down the values of 1x , 2x , …, 9x . Hence, write down nx in terms of n, for 1,2,3,...n= . [3]

2 Calculus 3 In this question, you may swap the summation and integration (differentiation) symbols without justification, i.e. ( ) ( ) 000 0 f d f d a a nn nn t t t t  ==  =    and ( ) ( ) 00 ddffdd nn nn tttt  ==  = . (a) Show that ( ) ( ) 12 2! 2! nn nn − = . [1] (b) Use induction and integration by parts to prove that for all 0n +  , π 1 2 21 0 24sin d 21 n n nxx nn − + = +  . [7] (c) Use the result in part (b) to prove that for 1x  , ( ) π 1 2 21 22 0 0 24 sin d 21 1 1 cos n n n n xtxtnn xt − + =  =+ −−    . [3] (d) Using the result in part (c), and a suitable substitution, show that for 1x  , 1 1 21 2 0 24 sin 21 1 n n n n xxnn x − − + =  =+ − . [3] (e) Deduce the exact value of ( ) ( ) 2 0 ! 2!n n n  =  . [4] Numbers 4 Let p be a prime number . It is given that x, y and z are positive integers such that 0 x y z p    , and 3 3 3x y z (mod p). (a) Show that p divides 22x y xy++ . [3] (b) Hence, or otherwise, show that p divides x y z++ . [2] (c) Prove that x y z++ divides 2 2 2x y z++ . [6]

3 Functions 5 The function f: ++→  is such that ( ) ( )( ) ( )f f f fx y x x y =+ for all ,xy +  . (a) Prove by contradiction, or otherwise, that ( )f1 x  for all x +  . [3] (b) Show that if ( )0f1 x = for some 0x +  , then ( )f1 x = for all x +  . [3] (c) Suppose now that ( )f1 x  for all x +  . By replacing x with 1 and y with ( ) 1 1fx x+− in the original equation ( ) ( )( ) ( )f f f fx y x x y =+ , show that ( ) ( ) ( )1f 1 f 1 ffxx x  + − = . [2] (d) Given that ( ) 1f1 2= , show that f is strictly decreasing on +  and find ( )f x . [4] Numbers 6 (a) Suppose that a sequence of prime numbers 1 2 3 4 5, , , ,a a a a a forms an arithmetic progression with positive common difference k and 1 5a  . (i) Show that 0k (mod 10). [3] (ii) State one such sequence of 1 2 3 4 5, , , ,a a a a a . [1] (b) Let 1 2 3, , , ..., pb b b b be an arithmetic progression of prime numbers with positive common difference d, such that p is a prime number and 1bp . (i) Prove that p divides d. [4] (ii) Using a suitable example, show that if 1bp , then p may not divide d. [2] (iii) Show that if 1 2 3 18, , , ...,b b b b is an arithmetic progression of prime numbers with positive common difference m, and 1 18b  , then m is at least 510510. [2]

4 Combinatorics 7 (a) A set S containing four distinct positive integers is called “connected” if for every xS at least one of the numbers 1x− and 1x+ belongs to S. Let nC be the number of connected subsets of the set  1, 2, n  . (i) Evaluate 7C . [3] (ii) Find a general formula for nC . [3] (b) A permutation of a set is called a derangement if none of the objects appear in their “natural” (i.e. ordered) place. For example, the only derangements of the set  1, 2,3 are  2,3,1 and  3,1, 2 . Find the number of derangements of the set  1, 2,3, , n  , showing your workings clearly . Hence show that as n gets large, the number of derangements is approximately ! e n . [5] Numbers 8 (a) Prove that if p is a prime and a is an integer, then np a p a where n is a positive integer. [3] (b) It is given that the real number x satisfies the polynomial equation 1 1 1 0 0nn nx c x c x c− −+ + + + =  , for some integers 0 1 1, , , nc c c −  . By writing x in the form a b for some , , 0a b b  and stating any assumption you need to make, prove that x is either an integer or an irrational number. [7] (c) By expressing ( ) 22 11x − as a polynomial in x where 632x= − − , prove that x is irrational. [4] −−−− END OF PAPER −−−−