MFSS AMath Prelim P1 QP

Name Reg. No Class MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL MAYFLOWER SECONDARY SCHOOL 4E/5N ADDITIONAL MATHEMATICS 4049/01 Paper 1 [ 90 marks ] PRELIMINARY EXAMINATION 20 August 2024 2 hours 15 minutes Candidates answer on the question paper. READ THESE INSTRUCTIONS FIRST Do not open this booklet until you are told to do so. Write your name, index number and class 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  , use either your calculator value or 3.142, unless the question requires the answer in terms of  . The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 90. Write the brand and model of your calculator in the space provided below. For Examiner’s Use Brand/Model of Calculator Total 90 This question paper consists of 16 printed pages and 2 blank pages. Setter: Ms Shen Sirui Vetter: Mr. Narayanan

2 [Turn over Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax2 + bx + c = 0, a acbbx 2 42 −−= 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 sin2 A + cos 2 A = 1 sec 2 A = 1 + tan 2A cosec 2 A = 1 + cot 2A sin ( A  B ) = sin A cos B  cos A sin B cos ( A  B ) = cos A cos B sin A sin B tan tantan( ) 1 tan tan ABAB AB = sin 2A = 2 sin A cos A cos 2A = cos 2 A – sin 2 A = 2 cos 2 A – 1 = 1 – 2 sin 2 A 2 2 tantan 2 1 tan AA A= − Formulae for ABC C c B b A a sinsinsin == a2 = b2 +c2 – 2bc cos A Area of  1 sin2 ab C=

3 [Turn over 1 Given that 334(5 ) 20xx+− = , evaluate 10x without using a calculator. [4] 2 Solve the equations. (a) [4] (b) 510log 5 3 logy y+= [4] 22log ( 4) 2log 1xx+ = −

4 [Turn over 3 The variables x and y are related by 28 21 xy x += + . When values of (1 )xy − are plotted against y, a straight line is obtained. The straight line intersects the vertical and horizontal axes at A and B respectively. (i) Find the coordinates of A and of B. [4] (ii) State the value of tan . [1]

5 [Turn over 4 (i) Factorise completely 322 3 5 6x x x− − + . [4] (ii) Hence, solve 322 3 5 6 0y y ye e e− − + = . [3]

6 [Turn over 5 (i) Prove that   tan1)cos(sinsec sec2 2 −=+ − . [4] (ii) Hence solve the equation 2 24 2sec sec 2sec (sin cos )     − =−+ for 02  . [5]

7 [Turn over 6 The first two non-zero terms in the expansion of ( )( ) 6 11 axbx ++ in ascending powers of x are 1 and 221 4 x− . Find the value of each of the constants a and b, where ab . [7]

8 [Turn over 7 The diagram shows a circle passing through the points A, B, C and D. AC is a diameter of the circle. The line EA is a tangent to the circle and it intersects the straight line EDC at E. (i) Show that angle AED = angle DAC. [2] (ii) Show that 2AD CD DE= . [4]

9 [Turn over 8 A vessel in the shape of an inverted right pyramid has a square base of side 12 cm and a height of 30 cm. Water is leaking from the vessel at a constant rate of 5 cm3/s. (i) Show that the volume of water in the vessel, V cm3, is given by 34 75 hV = , where h is the depth of the water. [2] (ii) Find the rate of change of the depth of water when the water is 6 cm deep. [3]

10 [Turn over 9 f(x) is such that 1f '( ) sin cos 44x x x=− . Given that f (2 ) 1 = , show that 16f ''( ) f ( ) sin 4x x a x b+ = + , where a and b are constants. [6]