2023 TJC VJC NYJC H3 Math Prelim QP

TEMASEK JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 3 H3 MATHEMATICS 9820/01 Paper 1 22 September 2023 3 hours Additional Materials: Answer Booklet List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST An answer booklet will be provided with this question paper. You should follow the instructions on the front cover of the answer booklet. If you need additional answer paper ask the invigilator for a continuation booklet. Write your Civics Group and name on all the work that 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 staples, paper clips, highlighters, glue or correction fluid. 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 graphic calculator. Unsupported answers from a graphic calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphic 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. This document consists of 6 printed pages and 2 blank pages. [Turn over

2 1 For each 1, 2, ...,in= , let ia and ib be non-zero real numbers. (i) By considering 2 1 ( ) 0 n ii i a t b = − for all real t, prove the Cauchy Schwarz’s inequality 2 22 1 1 1 n n n i i i i i i i a b a b = = =                     . [3] (ii) Let p and q be positive real numbers. If for each 1, 2, ...,in= , i i bpq a , show that 22 ( ) 0i i i ipqa p q a b b− + +  and deduce that 22 1 1 1 () n n n i i i i i i i p q a b b pq a = = = +  +   . [3] (iii) Let ,mM + be such that im a M and im b M for each 1, 2, ...,in= . By using the result in part (ii), or otherwise, show that 22 22 1 1 1 1 4 n n n i i i i i i i Mma b a b mM= = =      +                 . [3] 2 A differential equation of the form d f ( , ) d y xy x = is said to be homogeneous if f ( , ) f ( , )tx ty x y= for any real value t. (i) Show that such a differential equation can be written in the form d g d yx xy =   , where g is a function of x y . [2] (ii) Show that 22 2 2 2d2 e ( 2 )e d xx yy xxy y y x y            = + + is a homogeneous differential equation. [2] (iii) By using an appropriate substitution, find the general solution of the differential equation, leaving your answer in the form h xy y =   . [4] A solution curve of this differential equation has a tangent at the point (4, 2)− that is perpendicular to the line y mx= . Find the exact value of m . [3]

3 3 Let ()f k denote the kth derivative of the function f and define (0)ff = . Taylor’s Theorem states the following. If f , f , …, and ( 1)f n+ are all continuous on an interval containing a and x, then 0 () ( 1)f ( ) 1f ( ) + f d !! ( ) ( )( ) n x rn ar r na x a t rn x t x t = +=− −  . (i) Write down the value of f d() x a tt in terms of f ( )a and f ( )x . Hence, show that Taylor’s Theorem holds for the case 0n= . [2] (ii) Prove Taylor’s Theorem by induction on n. [4] For a given positive integer n, the finite series T ()n x is defined by () 0 fT ! ()( ) ( ) n r r n r r ax x a = = − is called the nth degree Taylor polynomial of f ( )x about a. (iii) Use the 2nd degree Taylor polynomial of sin x about 2  to show that 2 1sin1.6 1 1.6 22  − − . [1] An alternative version of Taylor’s Theorem is () 0 ffR ! ()( ) ( ) ( ) n r r n r r ax x a x = =+ − , where ( 1) 1fR ( 1)! ()() n n n x n cx + + = + and c is a real number between a and x. (iv) Let x + be fixed and let ˆx denote an integer greater than x. Show that for a positive integer n large enough such that ˆ2nx , ˆ2 1 !2 nxn x k n −   , where ˆ ˆ ˆ ˆ ˆ1 2 3 2 x x x xk x =   . [2] (v) Deduce from part (iv) the value of lim ! n n x n→   for x . [3] (vi) Use the alternative version of Taylor’s Theorem and part (v) to show that lim R 0()nn x → = and hence, explain why for each x , 0 e ! r x r x r  = = . [3]

4 4 Let p be a prime number. (i) Explain why for any k with 2 kp , there exist integers kx and ky with 0ky  such that ( 1) 1kkpx k y+ − = . [2] (ii) With ky established in part (i), let ( 1)kku k k y=− , for each 2 kp . Explain why 0ku  (mod 1k− ) and kuk (mod p). [2] (iii) Show that in modulo p, each 1 ku k− for 2 kp are all distinct. [3] A permutation of  1, 2, , p , is a sequence 1x , 2x , , px , where each  1, 2, ,ixp and each ix is unique. (iv) Show that there exists a permutation 1v , 2v , …, pv of  1, 2, , p , such that each of the terms 1v , 12vv , 1 2 3v v v , …, and 12 ... pv v v leaves a different remainder when divided by p. [3] (v) Write down a permutation of  1, 2, , 11 as described in part (iii). [2] 5 For  \ 0, 1x , the functions f and g satisfy the equations 1f ( ) 1x x= − and 1g( ) 1x x=− . (i) Show that the composite functions fg and gf exist. [2] (ii) Show that 2f ( ) g( )xx= and 2g ( ) f ( )xx= . [2] For  \ 0, 1x , the function h satisfies the equation 1h( ) h 1xx x += − . (iii) Find h( )x . [7] 6 (a) Let x, y and z be positive integers. (i) If 1xy (mod z), show that gcd( , ) 1xz = . [2] (ii) If 1x (mod y) and 1x (mod z) with gcd( , ) 1yz = , show that 1x (mod yz). [2] (b) Let a, b and c be positive integers with 1 abc   . It is given that 1ab (mod c), 1ac (mod b), and 1bc (mod a). (i) Show that 1 1 1 1abc++ . [5] (ii) Find all the possible values of a, b and c. [3]

5 7 (a) A student-care centre has 6 classrooms, each assigned a level from Primary 1 to Primary 6. The principal needs to distribute 20 tables into the 6 classrooms. Find the number of ways of distributing the tables into the classrooms if (i) the Primary 1 classroom has at most 3 tables, [2] (ii) each of the Primary 5 and Primary 6 classrooms holds at least 3 tables. [3] One student representing each level from Primary 1 to Primary 6 are queuing up in line to collect snacks for tea break. (iii) Find the number of ways that these six students can arrange themselves in line such that no three consecutive students are in ascending order of level, from front to back. (For instance, 1 3 2 6 4 5p p p p p p is allowed but 6 3 4 5 1 2p p p p p p and 3 4 6 1 2 5p p p p p p are not allowed.) [5] (b) Given a set  1 2 3 9, , ,...,m m m m of 9 distinct integers, show that there exists three integers m , m and m , such that the difference between any two of them is divisible by 4. [3] (c) The array below shows an arrangement of the letters contained in the word VICTORY. V | V − I − V | | | V − I − C − I − V | | | | | V − I − C − T − C − I − V | | | | | | | V − I − C − T − O − T − C − I − V | | | | | | | | | V − I − C − T − O − R − O − T − C − I − V | | | | | | | | | | | V − I − C − T − O − R − Y − R − O − T − C − I − V | | | | | | | | | | | V − I − C − T − O − R − O − T − C − I − V | | | | | | | | | V − I − C − T − O − T − C − I − V | | | | | | | V − I − C − T − C − I − V | | | | |