2022 SCGS AMath Prelims P1 w Ans

SINGAPORE CHINESE GIRLS’ SCHOOL PRELIMINARY EXAMINATION 2022 SECONDARY FOUR O-LEVEL PROGRAMME CANDIDATE NAME Solution CLASS 4 REGISTER NUMBER CENTRE NUMBER INDEX NUMBER ADDITIONAL MATHEMATICS 4049/01 Paper 1 Wednesday 30 August 2022 2 hours 15 minutes Candidates answer on the Question Paper. No Additional Materials are required.

2 1. A quadratic curve is given by the equation 2f ( ) 6 13 3xx x= −− . By expressing f ( x) in the form 2()ax h k−+ , explain why the curve has no real roots. [4] 2f ( ) 6 13 3xx x= −− 2 1332 3xx= − −+ ( ) 2 133 11 3x= − − −+ ( ) 2 1031 3x= − −+ ( ) 2 3 1 10x= − −− Since the curve has a maximum point of (1, –10) which is below the x-axis, the curve doesn’t cut the x-axis and thus it has no real roots. Alternative Method Since ( ) 2 1 10x −≥ , ( ) 2 310x− −≤ ( ) 2 3 1 10 10x− − − ≤− f ( ) 10x∴ ≤− , f( ) 0x ≠ for all values of x Hence, the curve has no real roots. 2. (a) Without the use of a calculator, find the value of 6n , given that 1 3 322 9 n nn n + + −= . [3] (b) The value of an antique is estimated to increase by k % per year. Given that its value at the beginning of 2017 was $20 000. The value V, after t years, can be modelled by (1.11)tVA= . (i) State the value of A and of k. [2] (ii) Calculate the year when its estimated value will first reach $190 000. [2] (a) 1 3 322 9 n nn n + + −= ( ) 3 322 1 3 n n−= 36 7 n = (b) (i) (ii) A = 20 000, k = 11 190000 20000(1.11) t= 1.11 9.5t = lg 9.5 lg1.11t = 21.6t = The year 2038

3 [Turn over 3. The term containing the highest power of x in the polynomial f (x) is 43x . Two of the roots of the equation f (x) = 0 are x = –1 and x = k, where k in an integer. Given that 2 26xx −+ is a quadratic factor of f( x) and f( x) leaves a remainder of – 36 when divided by x, (a) show that k = 2. [3] Hence, (b) determine the number of real roots of the equation f (x) = 0. [3] (a) (b) 2f ( ) 3( 2 6)( 1)( )x x x x xk= −+ + − f (0) 36= − 3(6)( 1)( ) 36k−= − k = 2 (shown) 2 2 60xx − += Discriminant = ( ) ( ) 2 2 46 0−− < ∴ no real roots When f (x) = 0, there are 2 real roots. 4. Given that o 1cos 20 k= , where k > 0, find an expression, in terms of k , for the following trigonometric ratios. (a) osec( 20 )− , [1] (b) osin160 , [2] (c) otan110 . [1] (a) osec( 20 ) k−= (b) osin160 osin 20= 2 1k k −= (c) ootan110 tan 70= − 2 1 1k = − − k 1

4 5. The concentration of a drug, C( t), in a patient’s bloodstream, t hours after an injection of the drug, can be modelled by 3 16() 75 5 ttCt = −+ . Find the length of time in which the drug concentration in the bloodstream is increasing after an injection [4] 2 16'( ) 25 5 tCt = −+ '(