2025 Prelim AM Bedok View P2 QP

[Turn over Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation ax 2 + bx + c = 0, 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 ( ) ( )11! !( )! ! n n n n rn r r n r r − − + == − 2. TRIGONOMETRY Identities sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A sin (A  B) = sin A cos B  cos A sin B cos (A  B) = cos A cos B sin A sin B tan (A  B) = tan tan 1 tan tan AB AB  sin 2A = 2 sin A cos A cos 2A = cos 2 A − sin 2 A = 2cos 2 A − 1 = 1 − 2sin 2 A tan 2A = 2 2 tan 1 tan A A− Formulae for ABC sin a A = sin b B = sin c C a 2 = b 2 + c 2 – 2bc cos A  = 1 2 bc sin A

For Examiner’s Use 2 1 The nutrient absorption rate, N, in mg/hour, of a plant depends on the diameter x, in mm, of its root hairs and is modelled by the function 2 2N( ) lnxx x =  , where 02 x . (a) Find N ( )x . [3] (b) Find the optimal root hair diameter that maximises nutrient absorption and show that the nutrient absorption is a maximum. [5]

[Turn over For Examiner’s Use 3 2 Hot coffee is heated to a temperature of 90 C . The temperature, CT  , t minutes after removal from the heat source is given by 22 e , ktTA=+ where A and k are constants. (a) Explain why A = 68. [1] (b) After 3.5 minutes, the temperature of coffee is 79 C . Find k. [3]

For Examiner’s Use 4 (c) The coffee should only be served when it is between 55 C to 70 C . Determine, with working, whether the coffee is suitable to be served 10 minutes after removal from the heat source. [2] (d) After a long time, T approaches a value. Calculate this value. [1]

[Turn over For Examiner’s Use 5 3 In the diagram, angle CED =  . EC is perpendicular to BE. The lengths of AB, BE and EC are 6 cm, 3 cm and 14 cm respectively. AB is parallel to FED and CD is parallel to AF. (a) Show that 14cos 3sin 6FD = + + . [3] A B C D E F 14 cm 3 cm 6 cm 

For Examiner’s Use 6 (b) Express FD in the form cos( )Rd −+ where 0,R 0 90   , and d is a constant. [3] (c) Find the value of  when 12FD= cm. [2] (d) Determine if it is possible for FD to be 15 cm. [2]

[Turn over For Examiner’s Use 7 4 In a biology experiment, a certain type of bacteria multiplies rapidly. The number of bacteria, N, after t hours is modelled by the formula 10 ,ktNa= where a and k are constants. The table below shows corresponding values of t and N. t 2 4 6 8 N 900 2700 8100 24 300 (a) On the grid on the next page, plot lg N against t and draw a straight line graph to illustrate the information. [4]

For Examiner’s Use 8 (b) Use the graph to estimate the value of each of the constants a and k. [2]

[Turn over For Examiner’s Use 9 5 (a) The cubic polynomial f ( )x is such that the coefficient of 3x is 2− and the roots of f ( ) 0x = are 1− , 2 and 3 2 . Find the expression of f ( )x in descending powers of x. [3] (b) Find the quotient when 32 3 2 5x x x− − + is divided by 2 1x − . [2] (c) The expression 32 72hx x kx− + − , where h and k are constants, has a factor 23 5 2.xx−− Find the value of h and k. [5]

For Examiner’s Use 10