2025 Prelim AM Bedok View P1 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 1 Solve the simultaneous equations [4]

For Examiner’s Use 2 4,yx−= 24 6 6 4.x xy y− − = 2 Solve the equation 9 sin 2 secxx=− for π 0.x−   [4]

[Turn over For Examiner’s Use 3 3 (a) Explain why the principal value of cos – 1 2 2 − cannot be π .4− [1]

For Examiner’s Use 4 (b) Prove that cot tan sec cosec .A A A A+= [4]

[Turn over For Examiner’s Use 5 4 Express ( ) 3 2 24 2 x xx + + in partial fractions. [5]

For Examiner’s Use 6 5 The function f is given by 2f ( ) . 3 xax x += + (a) Find f '( ).x [2] It is given that f increases for 1bx . (b) Find the values of a and b. [4]

[Turn over For Examiner’s Use 7 6 Find the set of values of the constant k for which the curve ( ) 268y k x x k= − − + does not intersect the x-axis and has a minimum point. [6]

For Examiner’s Use 8 7 (a) Find ( )d 1 e .d xxx − − [2] (b) Hence find e d .xxx−  [4]

[Turn over For Examiner’s Use 9 8 The radius of a circle, r cm, decreases at a rate of ( ) 2 2 1t + cm per minute. (a) Given that the initial radius is 4 cm, find an expression for r in terms of t. [3] (b) Find the rate of change of the area of the circle when the radius is 2.6 cm. [3]

For Examiner’s Use 10 9 (a) Find the value of the constant k such that the line 6y kx=+ is a tangent to the curve 22 3.x xy−= [3] (b) Solve ( )4 2 2 2 3 2 1,xx− + = − giving your answer in the form 2,ab + where a and b are rational numbers. [4]