RI 2022 H3 Test 2 (Questions and Solutions)

RI 2022 RAFFLES INSTITUTION 2022 Year 6 H3 9820 Test 2 Time allocated: 1 hour 50 minutes Total Marks: 60 Instructions: Write your name and CT group on all the work you hand in. Answer all questions. 1 (a) Expand and simplify ( )( ) 1 2 2 1 ...n n n n na b a a b a b ab b− − −− + + + + + . [2] (b) The prime number 3 has the property that it is one less than a perfect square. Determine all prime numbers with this property, justifying your answer. [2] (c) Find all prime numbers that are one more than a perfect cube, justifying your answer. [3] (d) Is 2021 202132 − a prime number? Explain your reasoning carefully. [2] (e) Is there a positive integer k for which 32 2 2 1k k k+ + + is a perfect cube? Explain your reasoning carefully. [3] 2 (a) Suppose that a, b and c are positive real numbers such that the polynomial 32f ( ) 3 3x x ax bx c= − + − has three positive real roots , and    . (i) Express a, b and c in terms of , and    . [3] (ii) Show that 3bc . [2] (iii) Use the graph of f ( )yx= to e xplain why the polynomial f '( )x has 2 positive roots. [2] (iv) Hence by considering f '( )x , show that ab . [2] (b) Let A, B and C be the angles of a triangle. (i) Show that tan tan tan tan tan tan 12 2 2 2 2 2 A B B C C A+ + = . [3] (ii) Hence, using the results established in (a) and (b)(i), show that tan tan tan 32 2 2 A B C+ +  and 3tan tan tan2 2 2 9 A B C  . [3]

RI 2022 3 A sequence 12, ,...xx of real numbers is defined by 2 1 2nnxx+ =− for 1n and 1xa= . (a) Show that if 2a then ( ) 12 4 2 .n nxa − + − [5] (b) Show also that nx → as n→ if and only if 2a  . [5] 4 Throughout this question, no marks will be awarded for any use of the exponential or the logarithmic function. For positive real numbers x, define 1 1F( ) d x xt t= . (a) Show that F is a strictly increasing function. [1] (b) Show in any order, that for all positive real numbers ,ab , (i) F( ) F( ) F( )ab a b=+ , (ii) F F( ) F( )a abb  =− . [4] (c) If there exists a real number L such that lim F( ) x xL → = , state lim F(2 ) x x → . Hence deduce that lim F( ) x x → =+ . [3] (d) Show that 1F F( ) xx  =− for all positive real numbers x and hence find 0 lim F( ) x x +→ , explaining your reasoning clearly. [2] (e) Show also that F(2) 1 F(3) . [3] 5 Let 1, 2,3,...,2 1 .Sn=− Remove at least 1n− numbers from S using the following rules: • If the number sS is removed and 2sS , then 2s must be removed, • If the numbers ,s t S are removed and s t S+ , then st+ must be removed. After all the possible numbers are removed from S, let T denote the sum of the remaining numbers.