tanjong katong girls school 2023 A math paper 1 (w answers)

TANJONG KATONG GIRLS’ SCHOOL PRELIMINARY EXAMINATION SECONDARY FOUR EXPRESS CANDIDATE NAME CLASS E INDEX NUMBER ADDITIONAL MATHEMATICS 4049/01 PAPER 1 14 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 23 printed pages and 1 blank page. For Examiner’s use

2 Tanjong Katong Girls’ School 4049/S4PRELIMS/01/2023 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/01/2023 [Turn over 3 1 A right circular cylinder has a volume of ( )6 2 3 + cm3 and a base radius of ( )13+ cm. Find, without using a calculator, the height of the cylinder, in cm, in the form ( )3ab+ , where a and b are integers. [3]

4 Tanjong Katong Girls’ School 4049/S4PRELIMS/01/2023 2 The curve 5 3y x=+ and the line 2 3 0xy− − = intersect at the points P and Q. Find the coordinates of P and of Q. [3]

Tanjong Katong Girls’ School 4049/S4PRELIMS/01/2023 [Turn over 5 3 Express 23 3 2xx−− in the form ( ) 2 a x b c++ and hence state the coordinates of the turning point of the curve 23 3 2y x x= − − . [4]

6 Tanjong Katong Girls’ School 4049/S4PRELIMS/01/2023 4 Integrate 3 84 121xx ++− with respect to x. [3]

Tanjong Katong Girls’ School 4049/S4PRELIMS/01/2023 [Turn over 7 5 Find the value of A, B, C, and D for which ( )( ) 32 22 4 7 13 2 3131 x x x Bx C D A xxxx + − − + = + + +−+− . [6]

8 Tanjong Katong Girls’ School 4049/S4PRELIMS/01/2023 6 A polynomial, P, is 32x x x k− − + , where k is a constant. (a) Find the value of k given that P leaves a remainder of 3 when divided by 2x− . [2] (b) In the case where 2k =− , the quadratic expression 2 1x ax++ is a factor of P. Find the value of the constant a. [3]

Tanjong Katong Girls’ School 4049/S4PRELIMS/01/2023 [Turn over 9 7 The diagram shows the curve cosy a bx c=+ for 20 3x  radians, where a, b and c are integers. The curve has a maximum point at ,13   and one of its minimum point at 2 ,33  − . (a) Find b. [2] (b) Explain the effect of c on the curve, and show that 1c=− . [2] (c) State the amplitude, and write down the equation of the curve. [2]

10 Tanjong Katong Girls’ School 4049/S4PRELIMS/01/2023 8 [The volume of a sphere of radius r is 34 3 r , and its surface area is 24 r .] The diagram shows a bubble tea cup of capacity 360  cm3. It consists of a cylindrical body of base radius r cm and height h cm, and a hemispheric cap. Let S cm2 be the total surface area of the cup. (a) Show that 25 720 3Sr r=+  . [4]