CAT HIGH 2025 AMATH PRELIM P2 MS

1 Jane bought an electric car in January 2025. After purchase, the value of the car, $V , diminishes over time and can be modelled by 38000 ktVe −= , where k is a positive constant and t is time measured in years. The value of the car is expected to be $29000 after 3 years of driving. Jane intends to sell her car when its value drops to half of its original value. Showing clear mathematical calculations, find the year that Jane is likely to sell her electric car. [5] 3 3 29000 38000 29 38 293 ln 38 1 29ln 0.0900963 38 k k e e k k − − = = −= = −≈ 0.090096 0.090096 19000 38000 1 2 10.090096 ln 2 7.69 years t t e e t t − − = = −= ≈ The year is likely to be 2032. ______________________________________________________________________________________ 2 (i) Given that the curve 2 21y ax bx=+− lies entirely below the x-axis, determine the conditions that must be applied to the constants a and b. [2] 0a< for maximum curve (not required) ( ) ( ) ( ) 2 2 22 2 4 1 0 for no roots 4 40 0 or ba ba ba a b − −< +< + < <− (ii) If a and b are both integers, state an example of the values of a and b which satisfy the conditions found in (i). [2] 5a=− 1b= ______________________________________________________________________________________

3 (a) The variables x and y are defined such that 2 49log 3log log 3xy−= . (i) Give a reason why x and y must be positive numbers. [1] x and y must be positive for 2log x and 2log y to be defined / exist / calculable / computed. (ii) Express x in terms of y. [5] 2 2 2 2 2 22 23 22 2 2 3 2 3 23 3 log 1log 3 log 4 2 3log 1log 22 2 log 3log 1 log log 1 log 1 2 2 2 yx yx xy xy x y x y xy xy −=  −= −= −= = = = = (b) Solve the equation ( ) 15 2 5 11xx+− += , leaving non-exact value(s) of x in the form loga b . [4] ( ) 15 5 2 11 5 x x += Let 5xw= ( )( ) 2 1 5 25 11 5 11 2 0 51 20 1 or 25 5 5 or 2 1 or log 2 x w w ww ww w x − += − += − −= = = =− ______________________________________________________________________________________ A1

4 (a) (i) The polynomials ( ) 32P 10x ax x bx= +++ and ( ) 32Q5x x ax x b=+ −+ leave the same remainder when divided by 2x+ . Show that 44ab+= . [2] ( ) ( ) ( ) ( ) ( ) ( ) 32 3 2 2 2 2 10 2 2 5 2 8 2 14 8 4 10 12 3 12 44 ab a b ab a b ab ab − + − +−+= − +− −−+ − − + = −+ + + − −= − += (ii) If ( )P x′ has a factor of 32x− , find the values of a and b. [4] ( ) 2P 32x ax x b′ = ++ 2 2P0 3 2232 033 44 33 ab ab ′ =  + +=    += − 43 4 4 4 from (i) 28 4 2 ab ab b b a += − += =− =− = (b) Solve the equation 322 4 5 70xxx+ − −= , expressing non-exact solutions in the form 1 2ab± where a and b are constants to be determined. [5] ( ) 32P 2 4 57x xxx= + −− ( ) ( ) ( ) ( ) 32 P1 21 41 5170−= − + − −−−= 1x+ is a factor of ( )P x ( ) ( )