CJC JPJC SAJC H3 Math 2024 Prelim Questions

1 9820/01/PRELIM/2024 [Turn over Proofs 3 (i) Prove that for any integers x and y and for any prime p , ( ) (mod )p p p pxyxy ++ . [2] (ii) Using induction, show that for any prime p , (mod )pa a p for all positive integers a . [4] (iii) Show that if n is not a multiple of 4, then 4 1 (mod 5)0n i i =  . [4] 1 Let a and b be positive integers. (a) (i) Prove that ( ) ( )gcd , gcd ,a b a b a=− [3] (ii) Using the result from part (i), evaluate ( )gcd 72,120 . [2] (b) Prove that for any integers a, b and c, ( )( ) ( )( )gcd ,gcd , gcd gcd , ,a b c a b c = [5] 2 This question is about the third series of coins in Singapore. This series consists of coins of five different denominations: 5 cents, 10 cents, 20 cents, 50 cents and 1 dollar. (a) Thirteen coins are to be selected. In how many ways can this be done (i) if there are no restrictions? [2] (ii) if at least one coin of each denomination and at most three 5 cent coins are selected? [4] (b) A sequence of thirteen coins is to be formed. In how many ways can this be done (i) if no two adjacent coins have the same denomination, [1] (ii) if at least one coin of each denomination must be used? [3] 4 (a) For , , , , ,a b c p q r + , prove that ( )( )ap bq cr a b c p q r+ +  + + + + . [3] (b) For ,,x y z + , prove that ( )( )( ) 1.8 xyz x y y z x z + + + [5] (c) For ,,x y z + , and using the results of parts (a) and (b), prove that 2 2 2 3.x y z x y y z z x+ + + + + Hint: ( )( ) ( )( )( ) 22 x y z z xx x y x y y z z x ++=+ + + + [7]

5 Eugene runs once every day. Each of his runs is either a threshold run, a tempo run or a recovery run. Over a period of n consecutive days, he does not do a threshold run on two consecutive days and he does not do a recovery run for more than two consecutive days. Let na , nb , nc be the number of possible ways Eugene can run over a period of n consecutive days where the first day is a threshold run, a tempo run or a recovery run respectively. (i) For n + , explain why 1 ,nn na b c+ =+ 1 ,n nnnb a b c+ = + + 2 1 1 .n n n n nc a b a b+ + += + + + [4] (ii) Hence express 4na + in terms of 3 2 1,,n n na a a+ + + and na for n + . [3] (iii) Show that 5 40a = . [2] (iv) Find the number of ways Eugene can run over a period of 5 consecutive days. [1] 6 (a) Use suitable substitution(s), prove that, for 1n (i) ( ) 1 2 21 0 1d n nx x I +−= ; (ii) ( ) 1 2 22 0 1d n nx x I − −+ where ( ) 2 0 cos d , 0 k kIk  = . [2] [2] (b) Using standard series from the List of Formulae (MF26), show that ( ) 2 1221 e 1 xxx −−−   + for 0x . Hence prove that 2 2 1 2 2 0 ed n y nnnI y nI − +− for 1n . [5] (c) Given that 2 kkI → when ,k →