stgss Prelim 2021 4E AMath P2 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/AMP2/4E/Prelim/2021 1 (i) Differentiate 2 2)31ln( x+ with respect to x. [2] (ii) Hence, find  + 231 6 x x dx. [2]

3 [Turn over SGSS/AMP2/4E/Prelim/2021 2 (i) Given that 3 2 )sin( )sin( =+ − BA BA , prove that tan A = 5 tan B. [4] (ii) Hence, or otherwise, solve the equation )45 (sin 2)45 (sin 3 °+=°− AA for °≤≤° 360 0 A . [3]

4 SGSS/AMP2/4E/Prelim/2021 3 (i) Show that x – 2 is a factor of 632 23 −+− xxx . [1] (ii) Express 632 211 23 −+− − xxx x as partial fractions. [6]

5 [Turn over SGSS/AMP2/4E/Prelim/2021 4 It is given that ) ( fx is such that xxx 3cos 2sin ) ( f +=′ . Given also that 06 πf =     , show that xxx 2cos 2 5 4 3) ( f 9) ( " f −− =+ . [5]

6 SGSS/AMP2/4E/Prelim/2021 5 (a) Simplify xx 27 log 9log × without using a calculator . [3] (b) Solve elog 11) e2ln( x x +=+ . [4]

7 [Turn over SGSS/AMP2/4E/Prelim/2021 6 The diagram shows part of a straight line graph obtained by plotting xy 1against 1 . Express y in terms of x. [4] (2, 3) (4, 4) x x 0

8 SGSS/AMP2/4E/Prelim/2021 7 The equation of a circle is 016 4822 =−+−+ yxyx . (i) Find the coordinates of the centre C of the circle and find the radius. [3] (ii) The line y = 2( x + 1) cuts the circle at two distinct points A and B. Find the coordinates of A and of B. [5] (iii) Find the equation of a second circle, with same centre C, and whose area is 4 times of the original circle. [2]

9 [Turn over SGSS/AMP2/4E/Prelim/2021 8 The mass, m mg, of a radioactive substance decreases with time, t hours. Measured values of m and t are given in the table below. t (hours) 2 4 6 8 10 m (mg) 48.2 41.5 35.7 30.7 26.5 It is known that m and t are related by the equation kt emm −= 0 , where 0m and k are constants. (i) On the grid on the next page, draw a straight line graph of mln against t, using a scale of 2 cm for 0.1 unit on the ln m-axis, starting from 3.2 , and a scale of 1 cm for 1 unit on the t-axis. [3] (ii) Use your graph to estimate the value of k and of 0m . [5] (iii) Use your graph to estimate the number of hours for the mass of the substance to be reduced by 20%. [2]

10 SGSS/AMP2/4E/Prelim/2021