(Bedok South) AM4 P1 (2025) Students

1 BEDOK SOUTH SECONDARY SCHOOL PRELIMINARY EXAMINATION 2025 4E5N CANDIDATE NAME CLASS REGISTER NUMBER ADDITIONAL MATHEMATICS Paper 1 4049 / 01 2 hours 15 minutes 90 Answer all the questions. 1 Prove that sin 2 cos 2 tan tanx x x x−= . [4] 2 Show that the solution of 3 4 2 1 3( 1)2 5 5 16x x x x+ + + =  is 4 52lg . [4] 3 Giving your answer in the form 2 3 cd+ , solve, without using a calculator, 18 3 32xx =+ . [5] 4 A square of area (11 120)+ cm2 has a length of () ab+ cm, where ab . Without using a calculator, find the values of a and b. [5] 5 The line 3 4 13xy+= intersects the curve 6 10 31 xy x −= − at points P and Q. Calculate the exact length of the line segment PQ. [5] 6 The coefficient of 3x in the expansion of ( )( ) 8 110 2 2kx x+− is zero. Find the value of the constant k. [5] 7 Solve the equation 5cos sin 2 cotcos 2 5sin xx xxx + =+ for 0 180x   . [6] 8 (a) Show that the derivative of 4( 3)xx − with respect to x is ( )( ) 3 5 3 3xx−− . [2] (b) Hence, given that ( ) 4     3 5     3 xxy x −= − and y is decreasing at a constant rate of 70 units/s, calculate the rate of change of x when 1x= . [4]

2 9 (a) Express 25 12 3y x x= − − in the form 2()y a x b c= + + and hence show that y can never be greater than 20. [3] (b) Explain why there are no values of k for which the curve 2( 1) 2( 2) 3y k x k x k= − + + + + is always positive. [4] 10 A rectangular field has sides (3 5)x− m and ( 10)x− m. Its area is at most 200 m2. (a) Find the range of values of x that satisfies the above sides and area conditions. [4] (b) Justify whether a fence of 98 m is enough to enclose the field. [3] 11 (a) State the range of values of x for which the equation below is valid. 2 392log (4 ) log ( 4) 2xx− − − = [1] (b) Express 2 392log (4 ) log ( 4) 2xx− − − = as a quadratic equation 2 0x bx c+ + = and explain why there is only one real solution. [6] 12 A quadratic curve is given by 2 2 9 6y hx x h= − − + , where h is a constant. (a) Show that the equation 0y= has real roots for all values of h except 0h= . [3] (b) State the value of h in the case where 0y= has two real and equal roots. [1] (c) Given that the line 2 12y x h= − − meets the curve 2 2 9 6y hx x h= − − + , find the range of values of h. [5] 13 It is given that 22f ( ) xx x e += . (a) Show that the range of values of x for which f ( )x is a decreasing function is 20 x−   . [4] (b) The gradient with the least value is in the range 20 x−   . Find the value of this gradient, giving your answer in exact form. [5] 14 It is given that 32f ( ) 3 2 16x x x= + + . The remainder when f ( )x is divided by 3xa− , where a is a constant, is the same as the remainder when