MFSS AMath Prelim P2 QP

1 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 4049/02 PAPER 2 [90 marks] PRELIMINARY EXAMINATION 23 August 2024 2 hours 15 minutes Candidates answer in the Question Paper No additional material required INSTRUCTIONS TO CANDIDATES Do not open this booklet until you are told to do so. Write your name, register number and class on all the work you hand in. Write in dark blue or black pen. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. Answer ALL questions. You are reminded of the need for clear presentation in your answers. Write the brand and model of your calculator in the space provided below. INFORMATION FOR CANDIDATES You are expected to use an electronic calculator to evaluate explicit numerical expressions. If the degree of accuracy is not specified in the question, and if the answer is not exact, the answer should be given to three significant figures. Answers in degrees should be given 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. This question paper consists of 17 printed pages, including 1 blank page. Setter: Mr Tan Boon Yong Vetter: Mr Nara Brand / Model of Calculator For Examiner’s Use

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 nn 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 rn r n r r r − − + == − . 2. TRIGONOMETRY Identities 22sin cos 1AA+= 22sec 1 tanAA=+ 22cosec 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= − Formulae for ABC C c B b A a sinsinsin == a2 = b2 +c2 – 2bc cos A Area of  = ab2 1 sinC

3 [Turn Over 1 (a) Find the remainder when 323 4 2x x x− + + is divided by 2 1x + . [2] (b) The remainder when 3x ax+ , where a is a constant, is divided by x + 2 is the same as the remainder when it is divided by x – 1. Find the value of a. [3] 2 The area and width of a rectangle is ( )19 3 5− cm2 and ( )2 2 5+ cm respectively. Express the length of the rectangle in the form of ( )5ab + cm where a and b are rational numbers. [5]

4 [Turn Over 3 (i) Find the term independent of x and the 3 1 x term in the binomial expansion of 9 2 2     − xx . [4] (ii) Hence, find the term independent of x in the expansion of ( ) 9 32 23 xx x −−  . [2]

5 [Turn Over 4 Given that 5 3sin −=A and 17 8cos −=B , where angle A and angle B are in the same quadrant, find the exact values of (i) tan( )AB+ [3] (ii) 2cos B [3]

6 [Turn Over 5 (a) A curve has the equation y = 43 32 + − x x , where 3 4−x . The normal to the point P on the curve where x > 0, is parallel to the line 217 25 +−= xy . Find the coordinates of P. [4] (b) Show the function 1033 4 23 −+−= xxxy is always increasing for all real values of x. [3]

7 [Turn Over 6 The line 2 12yx+= intersects the curve 22 6 4y x x= − − at two distinct points A and B. (i) Find the coordinates of A and of B. [3] (ii) Hence, find the equation of the perpendicular bisector of AB. [4]

8 [Turn Over 7 (a) Find the range of values of x that satisfy the inequality ( )5 3 2 5x x x−  − . [3] (b) The diagram shows a horizontal bridge of 100 metres supported by 2 pillars at the side, with a cable being suspended from point A and B. Each pillar is vertical and is 40 metres tall. The lowest point C of the cable is 10 meters above the bridge. The origin, O, is vertically below point A, where the foot of the pillar meets the bridge. A quadratic function can be used to model the cable. Find the quadratic equation. [3] Bridge 40 m 40 m 100 m 10 m O A B C Cable

9 [Turn Over 8 The function f ( )x , a polynomial of degree four, has stationary points at A(1, 5) and B(4, 0). f ( )x is an increasing function when x > 4. f ( )x is not an increasing function when x < 4. (i) State the nature of the stationary points A and B. Explain your answer. [4] (ii) Given that ( ) ( ) 2 f ( ) 1 4' x a x x= − − , where a is a non-zero constant, find an expression for f ( )x . [5]

10 [Turn Over 9 A cone of height 12 cm and base radius 5 cm is placed over a cylinder of radius r cm and height h cm. The cone is in contact with the cylinder along the cylinder’s upper rim. The diagram shows a vertical cross-section of the cone and the cylinder. (i) Express h in terms of r. [2] (ii) Hence show that the volume, V cm3, of the cylinder is given by 23 1212 5V r r =− . [1] 12 cm 5 cm r cm h cm