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 → show that 2 0 ed 2 x x  − = . [2] (d) Let 22 0 e d ,nx nU x x  −= 0,1, 2,n= . Given that 221 e 0 as nxxx −− → → , prove that 1 21 2 nn nUU − −= for 1n . Hence prove that 22 210 (2 )!ed 2! nx n nxx n  − += . [4]

3 9820/01/PRELIM/2024 [Turn over End of Paper 7 (i) Given that ( ) ( ) ( )1 1 2 2 3 3, , , and ,x y x y x y are collinear points, show that ( ) ( ) ( )3 2 1 1 3 2 2 1 3 0x x y x x y x x y− + − + − = [2] The functions f and g are defined on the real numbers and satisfy the equation g(f ( )) f ( ) (2 )g( )x y x x y y+ = + + (ii) Show that g(f ( )) f ( )xx−= , and hence prove that f ( ) f ( ) (2 )g( ).x y x x y y− − = + + [3] (iii) For real numbers a, b and c, show that f ( ) f ( ) ( )g( ) f ( ) f ( ) ( )g( ) f ( ) f ( ) ( )g( ) a b a b a b b c b c b c c a c a c a − = − + − + − = − + − + − = − + − + [2] (iv) By showing that g( )x is a linear function, prove that 2f ( ) g( ) 0 or f ( ) and g( )x x x x C x x= = = + = where C is a constant. [8] 8 (a) A sequence  nx is defined by 1 1n n m m x m= = . Given that 2 for all 2mmmm , Show that (i) the sequence is bounded by 3 2 , [2] (ii) the sequence converges. [2] (b) The Fibonacci numbers are defined by 01 0, 1FF== and, for 210, n n nn F F F ++ = + . (i) Prove that 1 12r rFF − for all 1r . [2] (ii) Let 1 9 n r n r r FS = = . Show that 111 101 1 1 1 1 1 1 71 99 9 9 9 9 n n n nnr r r nr r r n n r r r FFF F F S F F ++− − − − − = = = − − = − − + +   . [4] (iii) Show that 1 9 9 71 r r r F = = . [3] (iv) Given that 6 7 2 109 r r r F − =  , and using the result in (iii), find the value of 1 71 up to the first six decimal places. [2]