stgss Prelim 2021 4E AMath P1 Question

Mathematical Formulae 1. ALGEBRA Quadratic Equation For the equation 2 0ax bx c + + = , 2 4 2 b b ac x a − ± − = Binomial expansion ( ) 1 2 2 .... .... 1 2 n n n n n r r n n n n a 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 sin 2 A + cos 2 A = 1 sec 2 A = 1 + tan 2 A cosec 2 A = 1 + cot 2 A sin ( ) sin cos cos sin A B A B A B ± = ± cos ( ) cos cos sin sin A B A B A B ± = ∓ tan tan tan ( ) 1 tan tan A B A B A B ±± = ∓ sin 2 A = 2 sin A cos A cos 2 A = cos 2 A − sin 2 A = 2 cos 2 A − 1 = 1 − 2 sin 2 A 2 2 tan tan 2 1 tan AA A= − Formulae for ∆ ABC sin sin sin a b c A B C = = 2 2 2 2 cos a b c bc A = + − ∆ = Abc sin 2 1

2 SGSS/AMP1/4E/Prelim/2021 1 Given that sin A p = , where A is obtuse, find (i) cos A, [2] (ii) cot A. [1]

3 [Turn over SGSS/AMP1/4E/Prelim/2021 2 Find the range of values of k for which x2 + 12 x + 9 is always greater than 4 x + k. [4] 3 Given that 23e 2 1 x y x= + , find the value of k for which d d 2 1 y kxy x x = + . [4]

4 SGSS/AMP1/4E/Prelim/2021 4 The radius of a circle increases at a rate of 1s cm 3− . Find, in terms of π , the rate of increase of the area when (i) the radius is 5 cm, [3] (ii) the area is 24π cm . [3]

5 [Turn over SGSS/AMP1/4E/Prelim/2021 5 The function f is defined by f ( ) 1 2cos 2 x x = + for 0 πx≤ ≤ . (i) State the amplitude and the period of f(x). [2] (ii) State the minimum and maximum values of f( x). [2] (iii) Sketch the graph of f ( ) y x = for 0 πx≤ ≤ . [2]

6 SGSS/AMP1/4E/Prelim/2021 6 A certain species of fish was introduced into a pond and its number was tracked daily. It was observed that t days later, its population, P, was given by kt AP e5000 += , where A and k are constants. Initially, there were 3000 fishes in the pond. 10 days later, 3800 fishes were left. (i) Show that 2000 A = − . [1] (ii) Find the value of k, giving your answer correct to 3 significant figures. [3] (iii) Explain why the number of fishes can never reach 5000. [1]

7 [Turn over SGSS/AMP1/4E/Prelim/2021 7 The diagram shows a quadrilateral ABCD whose vertices lie on the circumference of the circle. The point E lies on the extended line CB such that AE is a tangent to the circle. CE and DA are parallel lines. (i) Explain why angle BAE = angle CAD . [2] (ii) Show that triangles BAE and DAC are similar. [2] (iii) Given that AB = BE , explain why the line AC bisects the angle BCD . [3] A B C D E

8 SGSS/AMP1/4E/Prelim/2021 8 The first two non-zero terms in the expansion of (1 + bx )(1 + ax )6 are 1 and 2 4 21 x− . Given that a < b, find the value of a and of b. [7]

9 [Turn over SGSS/AMP1/4E/Prelim/2021 9 (a) Solve the simultaneous equations. ( ) 5 15125 1) 8 )( 4 ( 3 =÷ = xy yx [5] (b) A cylinder has a radius of ( )2 1 − cm and a volume of ( ) 312 4 2 π cm + . Find, without using a calculator, the exac t value of its height, h, in the form 2a b + , where a and b are integers. [4]

10 SGSS/AMP1/4E/Prelim/2021 10 Given that 3) 42 ( −= xxy , (i) find the coordinates of the stationary points of th e curve, [6] (ii ) determine the nature of each of these points. [ 4]