DHS EJC RVHS 2026 H3 Math Prelim

© DHS 2026 This document consists of 7 printed pages and 1 blank page. [Turn over Name: Centre/Index Number: Class: DUNMAN HIGH SCHOOL Preliminary Examination 2026 Year 6 MATHEMATICS (Higher 3) 9820/01 Paper 1 23 September 2026 3 hours Additional Materials: Printed Answer Booklet List of Formulae (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet . Follow the instructions on the front cover of the answer booklet. 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 an approved 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 must present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question.

2 DHS 2026 Year 6 H3 Mathematics Preliminary Examination 1 Let f, g and h be increasing, continuous functions that map 0 +  to 0 +  such that f (0) g(0) h(0) 0= = = and f ( ) g( ) h( )x x x for all 0x . Here, 0 +  refers to the set of non - negative real numbers. We say that g bisects f and h if, for all points ( , )ab on g( )yx= such that 0a , Area bounded by Area bounded by f ( ), g( ) g( ), h( ) && y x y x y x y x y b x a        = = = = =       ==    . (a) Show that 2g( ) 9xx= bisects 2f ( ) 16xx= and 29h( ) 2xx= . [3] (b) Find a function g that bisects f ( ) 16x x= and h( ) 5x x= . [6] (c) State a function g that bisects f ( ) e 1 xx =− and ( )h( ) ln 1xx=+ . [1] 2 (a) Use the substitution ,y ux= where u is a function of x , show that the differential equation d 2 3 d 3 2 y x y x x y += + has solutions of the form 5 x y C x y− = + , where C is an arbitrary constant. [6] (b) A function f is convex (concave upwards) if for any two points 1x and 2x where 12xx in the domain of the function 0 ,ax  ( ) ( ) ( ) ( )1 2 1 2f (1 ) f 1 ftx t x t x t x+ −  + − for all 0 1.t Convex Function (i) Suppose that f is a convex and differentiable function on a domain 120. x x a   Prove algebraically that 21 1 21 f ( ) f ( )f ( ) . xxx xx−  − [3] (ii) By considering f ( ) lnx x x= or otherwise, show that 2 212 1 e x xxx x −   for 120. xx [3] O y x ( )( )11,fxx 12 (1 )tx t x+− ( ) ( ) ( )12f 1 ft x t x +− ( )( )22,fxx ( )12f (1 )tx t x+−

3 DHS 2026 Year 6 H3 Mathematics Preliminary Examination [Turn over 3 (a) (i) Use the AM-GM inequality to prove that if 1 2 3 4, , ,x x x x are positive numbers, then ( )( )( )( )1 2 3 1 2 4 1 3 4 2 3 4 1 2 3 4 81x x x x x x x x x x x x x x x x+ + + + + + + +  . [3] (ii) Given n positive real numbers 1,..., nxx +  , find, in terms of n, the maximum possible value of 1 1 n i i n j ji i x x  = =     , where the symbol  denotes a product of terms. For example, the expression 1 n i i x =  represents the product 1 2 3 ........... nx x x x    , where the index i takes all integer values from 1 to n inclusive. State the condition on the values of 1,..., nxx for which this maximum value is achieved. [4] (b) Let x, y and z be positive real numbers such that .xy yz zx x y z+ + = + + (i) By using the Cauchy-Schwarz inequality, show ( )( ) ( ) 22 2 .11xy xy+ ++ [1] (ii) By considering ( ) 2 x y z++ or otherwise, prove that 3.x y z+ +  [2] (iii) Hence, or otherwise, show that 2 2 2 1 1 1 1.1 1 1x y y z z x+ + + + + + + + [4]

4 DHS 2026 Year 6 H3 Mathematics Preliminary Examination 4 Consider a unit circle lying on a plane. Let  12, , , nS P P P=  be a set of n distinct points on the circle. If two points iP and jP of S have the property that the length of the chord ijPP is longer than 3 units, we call the pair  ,ijPP a spaced pair. (a) Explain why there do not exist three points 1P , 2P , 3P such that  12,PP ,  31,PP and  23,PP are all spaced pairs. [2] Let M be the set of all spaced pairs, and let mM= be the total number of spaced pairs. For each point iPS , define   f ( ) : ,i j i jP P S P P M=   to be the number of points that form a spaced pair with iP , which also equals the number of spaced pairs involving iP . (b) The diagram below shows an example of a set S with 7 points. Exactly 8 of the possible pairs are spaced pairs, corresponding to the chords drawn in the diagram. Verify that for this example the equation     1, 2 f ( ) f ( ) f ( ) ij n P ik P j kM P P P =  += holds. Show all your calculations clearly in your answer. [2] In fact, the above result is an identity that holds for all configurations of points. (c) (i) Explain why 1 f ( ) 2i n i Pm = = . [1] (ii) Explain why for each  ,ijP P M , we must have f ( ) f ( )i j nPP+  . [2] (iii) Using a combinatorial argument, derive the identity     1, 2 f ( ) f ( ) f ( ) ij n P ik P j kM P P P =  += . [2] (iv) Deduce that   2 1 f ( )k n k P mn =  . [1] (d) Use the results in part (c) to show that 2 4m n . [3]

5 DHS 2026 Year 6 H3 Mathematics Preliminary Examination [Turn over 5 Let n, m be positive integers and let p be an odd prime. (a) Let ( )( ) ( ) 1 12 ma n n n n m= + + +  . By considering the sign of 1mmaa +− , show that the alternating infinite series ( ) ( ) ( )( ) ( )( ) ( )0 11 1 11 ... ... 1 1 2 1 ... r r r r a n n n n n n n n n r  = −− = − + − + ++ + + + + has a value between 1 1n+ and 1 n . [4] (b) Show ( ) ( ) ( ) 2 0 1 ! 1 1 ! e! p r r pp r − = − − − =   , where e denotes the base of the natural logarithm. [5] (c) Explain why, for non-negative integer np , ( ) ( )( ) ( ) ( )! 1 1 2 modulo n n p p p n p − − − −  . [1] (d) Show that for each odd prime p, ( )1!1! 2! 3! ! epp −+ + + + −   is divisible by p. [3]

6 DHS 2026 Year 6 H3 Mathematics Preliminary Examination Use the information in the mathematical text to answer Question 6. You should read the whole mathematical text before you start answering the questions. Factorising Gaussian Integers Gaussian integers were introduced by Carl Friedrich Gauss in 1832. Gauss’ development of this new arithmetic system revolutioni sed number theory and laid the groundwork of algebraic number theory. Today, this field of pure mathematics has many surprising applications such as the optimisation of digital signal processing. In this text, we investigate factorisation and prime elements in the Gaussian integers, usually written as [i]  . The set of Gaussian integers is the subset of complex numbers whose real and imaginary parts are both integers. More formally,  [i] i : ,a b a b= +   . Two examples of Gaussian integers are 1i− and 3 2i+ . If , [i]wz   , we say that w divides z if there exists [i]c  such that wc z= . We denote this by |wz . This is exactly analogous to division in the integers. For instance, 6 7i (2 i)( 1 4i)− + = + − + and so (2 i) ( 6 7i)|+ − + . However, ( 6 7i)(1 i) | −++  . Here, we encounter our first surprising fact. (2 i) 5|+ , and so although 5 is a prime number, it is not a Gaussian prime because it can be further factori sed in [i]  . Two questions then naturally arise: Let [i]z  . How do we know when a given factorisation of z is the ‘prime factorisation’? In the context of [i]  , does ‘prime’ even make sense? To answer these questions