RI H3 Mathematics 2024 Test 3

RAFFLES INSTITUTION 2024 YEAR 6 LECTURE TEST 3 Higher 3 MATHEMATICS 9820 22 August 2024 2 hours Additional materials: List of Formulae (MF26) READ THESE INSTRUCTIONS FIRST 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 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 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. Write the number and part, for example 1a, you are responding to clearly. This document consists of 4 printed pages. RAFFLES INSTITUTION Mathematics Department ã RI 2024 [Turn over

2 1 Let n be a positive integer and denote by σpnq the sum of all the positive divisors of n. We say that a number n is perfect ifσpnq“ 2n. For example, the 2 smallest perfect numbers are 6 and 28 sinceσp6q“ 1` 2` 3` 6“ 12 andσp28q“ 1` 2` 4` 7` 14` 28“ 56. (a) If p and 2p´ 1 are prime numbers, show that n“ 2p´1p2p´ 1q is a perfect number. [3] (b) Write down 2 perfect numbers other than 6 and 28. [2] (c) Show that the last digit of all perfect numbers of the form n“ 2p´1p2p´ 1q is either 6 or 8. [5] 2 Let Spr, nq denote the number of ways of distributing r distinct objects into n identical boxes so that no box is empty. (a) State the value of Spr, r´ 1q. [1] (b) Explain why Spr, 2q“ 2r´1´ 1. [2] (c) Prove that Spr, nq“ nSpr´ 1, nq` Spr´ 1, n´ 1q for 1ď nď r. [3] (d) Hence use mathematical induction to show that Spr, r´ 2q“ ˆr 3 ˙ ` 3 ˆr 4 ˙ for rě 3. [5] (e) Use a combinatorial argument to explain why the result in (d) is true. [3] 3 (a) Let A “ t1, 2,..., pu and f : A Ñ A. By considering the number of functions f that are surjective, use the principle of inclusion-exclusion to show that p!“ pÿ r“0 p´1qr ˆp r ˙ pp´ rqp. [5] (b) Let p be a prime number. Hence show that pp´ 1q!”p´ 1qp`1 p´1ÿ r“1 r p´1 (modulo pq. [3] (c) Let p be a prime number and let n be a positive integer. Use mathematical induction to show that np” n (modulo p). [4] (d) Deduce thatpp´ 1q!”´ 1 (modulo p) for a prime number p. [2] c⃝RI 2024 H3 Mathematics Test 3

3 4 You may assume that all integrals in this question converges. (a) Show that 2 nÿ k“1 cosp2kxq sin x“ sinp2n` 1qx´ sin x. [3] (b) Deduce that for all positive integers n, ż π 2 0 sinp2n` 1qx sin x dx“ π 2. [3] Let f be a continuous function defined onra, bs such that it is also differentiable and its derivative f1 is also continuous onra, bs. (c) Use integration by parts to show that lim nÑ8 ż b a fpxq sinpnxq dx“ 0. [3] Let g be defined onr0, π 2s by gpxq“ $ & % 1 x´ 1 sin x if 0ă xď π 2, 0 if x“ 0. You may assume that the function g and its derivative g1 is continuous onr0, π 2s. (d) Hence evaluate ż8 0 sin x x dx. [3] c⃝RI 2024 H3 Mathematics Test 3

4 5 Lettx1, x2,..., xnu andty1, y2,..., ynu be two sequences of real numbers. We say that the sequence tx1, x2,..., xnu majorizes thety1, y2,..., ynu, if the following conditions are fulfilled: • x1ě x2ě...ě xn; • y1ě y2ě...ě yn; • x1` x2`¨¨¨` xn“ y1` y2`¨¨¨` yn; • x1` x2`¨¨¨` xkě y1` y2`¨¨¨` yk for all 1ď kď n´ 1. For example,t3, 0, 0u majorizest2, 1, 0u andt2, 1, 0u majorizest1, 1, 1u. Let f be a convex function defined over the real numbers. (a) Use a sketch to explain why if xď yă z then fpzq´ fpxq z´ x ď fpzq´ fpyq z´ y . [2] Letta1, a2,..., anu andtb1, b2,..., bnu be two sequences of real numbers such that ta1, a2,..., anu majorizes the sequencetb1, b2,..., bnu, and let ci“ fpbiq´ fpaiq bi´ ai . Define the sequences tAiu and tBiu by A0“ 0, Ak“ kÿ i“1 ai and B0“ 0, Bk“ kÿ i“1 bi. (b) (i) Show that nÿ i“1 pfpaiq´ fpbiqq“ n´1ÿ i“1 pci´ ci`1qpAi´ Biq. [3] (ii) Deduce that fpa1q` fpa2q`¨¨¨` fpanqě fpb1q` fpb2q`¨¨¨` fpbnq. [3] (c) Let a, b, c be positive real numbers. Use the result in (b) to show that 1 a` b` 1 b` c` 1 c` aď 1 2a` 1 2b` 1 2c. [3] (d) Let x1, x2,..., xnP ” ´π 6,π 6 ı . Show that cosp2x1´ x2q` cosp2x2´ x3q`¨¨¨` cosp2xn´ x1qď cos x1` cos x2`¨¨¨` cos xn. [4] c⃝RI 2024 H3 Mathematics Test 3