HCI 9820 2024 H3 Prelim Qn

1 © HCI 2024 9820/01/JC2 Preliminary Examination 2024 [Turn Over HWA CHONG INSTITUTION 2024 JC2 Preliminary Examination MATHEMATICS Higher 3 Paper 1 Wednesday 18 September 2024 3 hours Additional materials: 12-page Answer Booklet List of Formula (MF26) 4-page Additional Answer Booklet (upon request) READ THESE INSTRUCTIONS FIRST Write your name and class on the 12-page Answer Booklet and any other additional 4 -page Answer Booklets you hand in. Write in dark blue or black pen on both sides of the paper. You may use a soft pencil for any diagrams or graphs. Do not use paper clips, highlighters, glue or correction fluid. Do not write anything on the List of Formula (MF26). Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use a graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You are reminded of the need for clear presentation in your answers. The number of marks is given in brackets [ ] at the end of each question or part question. At the end of the examination, slot any additional 4-page Answer Booklets used in your 12-page Answer Booklet and indicate on the 12 -page Answer Booklet the number of additional 4 -page Answer Booklets used (if any). This question paper consists of 5 printed pages and 1 blank page. 9820/01

2 © HCI 2024 9820/01/JC2 Preliminary Examination 2024 [Turn Over 1. A sequence generated from the integer set  1, 2,3,...,n , is denoted as a special sequence if the sequence is strictly increasing and the first term is odd, the second term is even, the third term is odd, the fourth term is even, etc. For example, from the set  1, 2,3, 4,5,6 , some special sequences generated from the set are  1, 2,3,6 ,  1, 4,5 ,  3, 4 ,  5 . Let ()An be the total number of distinct special sequences obtained from  1, 2,3,..., .n (a) Form a recurrence relation involving ( ), ( 1), ( 2).A n A n A n−− [4] (b) Find the number of special sequences that can be formed from  1, 2,...,15 . [2] 2. Let f( )x be a differentiable function such that f( ) f '( ) 1xx+ for all x , and f(0) a= , where a is a constant with 1a . It is given that g( ) e f( )xxx= . (a) Show that g'( ) e xx  , for all x . [2] (b) Hence, or otherwise, find the largest possible value of f(1) and the corresponding expression for f(x) in terms of a. [6] 3. For this question, it can be assumed that all the infinite series converge. Given an infinite sequence of numbers 0u , 1u , 2u , , the generating function, f, for the sequence is defined by 23 0 1 2 3f ( )x u u x u x u x= + + + + . (a) (i) Show that the sequence given by nun= , 0n , has generating function 2f ( ) (1 ) xx x= − , and find the sequence that has the generating function 3f ( ) (1 ) xx x= − . [3] (ii) Find the generating function for the sequence 2 nun= , giving your answer simplified form. [2] (b) The sequence 0u , 1u , 2u , is defined by the recurrence relation 121nnuu −=+ , 1n , with 0 1u = . Find the generating function for this sequence, giving your answer in the form () () Px Qx , where ()Px and ()Qx are polynomials in terms of x to be determined. [3]

3 © HCI 2024 9820/01/JC2 Preliminary Examination 2024 [Turn Over 4. Definition: The function : ++→ is defined as ( ) | 1 dn n = , where |dn denotes that d is a positive divisor of n. (a) Write down the values of ( )1 , ( )3 , ( )12 . [2] (b) Given that the prime decomposition for some n + is 12 12 ... k kn p p p = , where ip is prime and i is a positive integer for every 1 ik , write down ( )n in terms of i . [2] (c) Hence or otherwise, prove that ( )n is odd if and only if n is a perfect square, where a perfect square is an integer that is the square of an integer. [3] (d) Consider the following table: n … Divisors of k 8k = 7k = 6k = 5k = 4k = 1 2 4 3k = 1 3 2k = 1 2 1k = 1 The rows 1k = to 4k = are completely filled. Copy and fill up the table for rows 5k = to 8k = . [1] (e) Hence or otherwise, prove that ( ) 11 nn kk nk k == =  , where x denotes the largest positive integer less than or equals to x. [3] 5. (a) Using integration by parts, find the value of 2 0 e sin dx xx  −  . [3] (b) Explain why 2 0 e sin dx xx  −  exists. [2] (c) Find the value of 2 0 e sin dx xx  −  . [7]

4 © HCI 2024 9820/01/JC2 Preliminary Examination 2024 [Turn Over 6. (a) For n , find the possible values of the positive constant s, when ( ) 23 mod 5ns . [4] (b) Show that for ,,abc  such that 2 2 235a b c+= , ( )0 mod 5a and ( )0 mod 5b . [4] (c) Show that ( )0,0,0 is the only solution for 2 2 235a b c+= from the set ( ) , , : , ,x y z x y z  . [3] (d) Is ( )0,0,0 is the only solution for 2 2 24 4 5a b c+= from the set ( ) , , : , ,x y z x y z  ? Justify your answer briefly. [1] 7. The sequence of non-negative integers 0F , 1F , 2F , is defined by the recurrence relation 21n n nF F F++ =+ for 0n , with 0 0F = and 1 1F = . (a) Show that, for non-negative integers n, 2 5 3 4 3 1 2n n n n n n n nF F F F F F F F+ + + + + + +− = − . [3] (b) Find all possible values of 3 1 2n n n nF F F F+ + +− for 0n . [3] (c) Show that, for r + , 1 1 1 2 2 1 2 2 1 1 1tan tan tan r r rF F F − − − ++      =+            [3] (d) Given that the infinite series 1 1 21 1tan r rF  − = +     converges, find the limit of this infinite series, leaving your answer in exact form. [3]

5 © HCI 2024 9820/01/JC2 Preliminary Examination 2024 [Turn Over 8. (a) Let P be the set of 4-digit numbers wxyz such that w, x, y, z { 1,2,3,4,5,6,7,8,9} and w, x, y, z are pairwise distinct. Find the number of members wxyz in P such that 2w , 4x , 6y and 8.z [4] (b) A school wants to distribute 99 identical notepads to four classes, Class A , Class B , Class C and Class D . (i) Find the number of ways to distribute the notepads if Class A must have at least one notepad and Class B has lesser than nine notepads. [3] (ii) Find the number of ways to distribute the notepads if Class A and C