tanjong katong girls school 2023 A math paper 2 (w answers)

TANJONG KATONG GIRLS’ SCHOOL PRELIMINARY EXAMINATION SECONDARY FOUR EXPRESS CANDIDATE NAME CLASS E INDEX NUMBER ADDITIONAL MATHEMATICS 4049/02 PAPER 2 16 August 2023 2 hour 15 minutes Candidates answer on the Question Paper READ THESE INSTRUCTIONS FIRST Write your index number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fluid. Answer all questions. If working is needed for any question it must be shown with the answer. Omission of essential working will result in loss of marks. The use of an approved scientific calculator is expected, where appropriate. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For π, use either your calculator value or 3.142, unless the question requires the answer in terms of π. The number of marks is given in brackets [ ] at the end of each question or part question. The total marks for this paper is 90. This document consists of 21 printed pages and 1 blank page. For Examiner’s use

Tanjong Katong Girls’ School 2 4049/S4PRELIMS/02/2023 1. ALGEBRA Quadratic Equation For the equation 2 0+ + =ax bx c , 2 4 2 b b acx a −  −= Binomial expansion 1 2 2( ) ... ... , 12 n n n n n r r n n n na b a a b a b a b b r − − −     + = + + + + + +           where n is a positive integer and ! ( 1)...( 1) !( )! ! n n n n n r r r n r r  − − +== − 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA=+ 22 cosec 1 cotAA=+ sin( ) sin cos cos sinA B A B A B =  cos( ) cos cos sin sinA B A B A B= tan tantan( ) 1 tan tan ABAB AB = sin 2 2sin cosA A A= 2 2 2 2cos 2 cos sin 2cos 1 1 2sinA A A A A= − = − = − 2 2 tantan 2 1 tan AA A= − Formula for ∆ABC sin sin sin a b c A B C== 2 2 2 2 cosa b c bc A= + − 1 2 sinbc A=

Tanjong Katong Girls’ School 4049/S4PRELIMS/02/2023 [Turn over 3 1 The diagram below shows the curve 24e 7xy=− and the curve 22e xy −= . (a) Using the diagram above, determine with explanation, the number of solutions for the equation 224e 2e 7xx −−= . [1] (b) Solve the equation 224e 7 2exx −−= . [4]

Tanjong Katong Girls’ School 4 4049/S4PRELIMS/02/2023 2 (a) Show that 1 is a solution of the equation 32 5 2 6x x x− = − . Hence solve the equation 32 5 2 6x x x− = − completely, expressing non-integer roots in in surd form. [4]

Tanjong Katong Girls’ School 4049/S4PRELIMS/02/2023 [Turn over 5 (b) Hence solve the equation ( ) ( ) 32 1 5 1 6 8 0x x x− − − + − = completely. Express non-integer roots in surd form. [2]

Tanjong Katong Girls’ School 6 4049/S4PRELIMS/02/2023 3 (a) Given that 4 32 xy x = − , show that ( ) 3 d 12 4 d 32 yx x x −= − . [4]

Tanjong Katong Girls’ School 4049/S4PRELIMS/02/2023 [Turn over 7 (b) Hence find the value of ( ) 3 1 0 3 d 32 x x x − − . [4]

Tanjong Katong Girls’ School 8 4049/S4PRELIMS/02/2023 4 In the diagram, BE intersects AD at F. AF = 7 cm and FD = 4 cm. ABF, ACD and DEF are right angles. BCDF is a trapezium and  is an acute angle. (a) Show that the perimeter of BCDF, P cm, is given by 4 18sin 4cosP = + + . [2] (b) Express P in the form ( )sin , where is a constant, 0 and P a R a R = + +  0 < 90   . [2]

Tanjong Katong Girls’ School 4049/S4PRELIMS/02/2023 [Turn over 9 (c) Find the maximum exact value of P and the corresponding value of . [3] (d) Find the value of  when P = 11. [2] (e) Without evaluating  , determine whether the perimeter of BCDF can have a a value of 28 cm. Justify your answer. [1]

Tanjong Katong Girls’ School 10 4049/S4PRELIMS/02/2023 5 (a) Find the set of values of x for which the curve 2 24y x x=− + + lies below another curve 2 22y x x= − − and represent this set on a number line. [4] (b) The line 2y x k=+ is a normal to the curve 2 24y x x=− + + at the point A. (i) Find the x-coordinate of A. [3]