HCI 2025 Prelim

1 © HCI 2025 9820/01/JC2 Preliminary Examination 2025 [Turn Over HWA CHONG INSTITUTION 2025 JC2 Preliminary Examination MATHEMATICS Higher 3 Paper 1 Wednesday 22 September 2025 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 6 printed pages and 0 blank page. 9820/01

2 © HCI 2025 9820/01/JC2 Preliminary Examination 2025 [Turn Over 1. For  2,knk + , prove by mathematical induction that ( ) 1 12 12 ... ...n nn x x x x x xn + + +  , where 12, ,... nx x x are positive real numbers. [6] 2. Let Si be the set containing all permutations of n objects where the ith object is in the ith position, i.e. the ith object is in its original position. (a) Find the values of 1 n i i S =  and ij ij SS   in terms of n. [4] A derangement is a permutation of the elements of a set in which no element appears in its original position. (b) Using the principle of inclusion-exclusion, show that the number of derangements of n objects, 12 ...nnD S S S=    is given by ( )1 1 1 1! 1 ... 11! 2! 3! ! n n n − + − + + − [3] (c) State the value of lim ! n n D n→ in exact form. [1] 3. Let 2 2 2 1 1 1f ( ) 2 1 1 ... 1(1 i) (1 2i) (1 i)n n     = + + +    + + +     , where i1=− . (a) By showing ( )    ( ) 22 1 ( 1)i 1 ( 1)i11 1 i 1 i kk kk + − + ++= ++ or otherwise, find an expression of f ( )n , leaving your answer in Cartesian form in terms of n. [6] (b) Using the fact that if ,wz  , then arg( ) arg( ) arg( )wz w z=+ , find the value of n such that 2 0 1 36tan arg 1 (1 i) 37 n k k=  + =−  +  . [3]

3 © HCI 2025 9820/01/JC2 Preliminary Examination 2025 [Turn Over 4. Let w, x, y and z be positive real numbers. (a) Show that 2 2 2 2 2 x y z w w x y z x y y z z w w x + + ++ + + + + + + . [3] (b) Deduce that ( ) 2 2 2 2 1 42x y z w wxyzx y y z z w w x+ + + + + + + [2] (c) Suppose 27wx= and 3yz= , show that 2 2 2 2 6x y z w x y y z z w w x+ + + + + + + . [3] 5. Let x denotes the smallest integer greater than or equal to 𝑥 . Then the function f: → where f( )xx= is called the ceiling function . For example, f( ) 4== , f( 4.5) 4.5 4− = − =− and f(3) 3 3== . (a) Sketch the graph of ( )2sin 2yx= , for 02 x  . [4] (b) By considering the period, show that ( ) 2 0 2sin dnx x k  = for all n + , where k is an integer to be determined. [3] (c) Hence, deduce the value of ( ) 2 0 sin dn nx x   in terms of  . [2] 6. (a) The function f, for 0x , satisfies the equation ( ) 12f f 3xx x −= . (i) Explain why ( )132f f xxx  −= . [1] (ii) Hence find ( )f x . [2] (b) The function g, for 0x , satisfies the equation ( ) ( ) 1g g gx x x x + − − =  . Find ( )g x . [6]

4 © HCI 2025 9820/01/JC2 Preliminary Examination 2025 [Turn Over 7. (a) Factorise x5 + 1. Prove that if 21m+ is prime, then m must be of the form 2k , where k  ℤ. [3] (b) Let p be a prime factor of 221 n nF =+ . By considering the smallest positive integer d such that 2 1 (mod )d p , or otherwise, prove that p must be of the form k2n+1 + 1, where k is an integer. [You may use Fermat's Little Theorem: 1 1(mod )pap−  , where a and p are coprime] [3] (c) Deduce that any factor of 221 n nF =+ must be of the form k2n+1 + 1, where k  ℤ. [2] (d) Using the fact that 641 = 5 4 + 24 = 5(27) + 1, show that 5 4 228 + 232  0 (mod 641) and 54 228 – 1  0 (mod 641). Deduce that 52 5 21F =+ is divisible by 641. [3]

5 © HCI 2025 9820/01/JC2 Preliminary Examination 2025 [Turn Over 8. Differentiation from First Principles and the q -Derivative: A Comparative Perspective The concept of the derivative, as we understand it today, was formalised in the 17th century through the independent work of Isaac Newton and Gottfried Wilhelm Leibniz, marking the birth of calculus. Central to this development is the first principle of differentiation, which defines the derivative of a function f at a point x as: 0 f( ) f( )f '( ) lim h x h xx h→ +−= This expression captures the idea of the instantaneous rate of change or the slope of the tangent line to the curve at a given point. For example, to find the derivative of f ( ) nxx= using the first principle of differentiation, we have 0 1 2 2 0 1 2 1 0 0 1 ()'( ) lim ...12lim lim ...12 nn h n n n n n h n n n h h n x h xfx h nnx x h x h h x h nn x x h h nx → −− → − − − → → − +−=    + + + + −      =    = + + +       = While powerful, this approach assumes the ability to take infinitesimal increments 0h→ , which is inherently tied to the real number continuum. In the 20th century, especially in the study of quantum calculus — calculus without limits — mathematicians explored finite difference analogues. One of the most prominent constructions is the q-derivative, introduced by Frank Hilton Jackson in 1908. The q-derivative of a function f(x), for a real number 1q , is defined as: f( ) f( )f( ) ( 1) q qx xDx qx −= − . Unlike the classical derivative, which relies on an additive increment 0h→ , the q - derivative uses a multiplicative increment by scaling the input. This gives rise to a discrete- like calculus that becomes classical in the limit 1q→ : 1 lim f( ) f '( )qq D x x → = . The q-derivative provides a window into discrete versions of continuous phenomena, playing a critical role in q-series, quantum mechanics, and non-commutative geometry. It exemplifies how foundational ideas — such as the derivative — can be reframed and generalised when the underlying assumptions (like continuity or limits) are re-examined.

6 © HCI 2025 9820/01/JC2 Preliminary Examination 2025 [Turn Over (a) For f ( ) nxx= , show that 1 lim f( ) f '( )qq D x x → = . [3] (b) Using