PRSS 2021 Sec 4E5N Prelim A Maths P1 - Mark Scheme

3 PRSS_2021_Prelim_4E5N_AMaths_P1 Answer all the questions. 1 (a) Find the range of values of p for which the equation has real and distinct roots. [3] [M1 cao] Correct expression for D For real and distinct roots, [M1] State D > 0 [A1] (b) State the value(s) of p for which the curve is tangent to the x-axis. [1] For the curve to be tangent to y-axis,

4 PRSS_2021_Prelim_4E5N_AMaths_P1 , [A1] 2 Solve the following equations (i) , [4] [M1 – Change of Base] Let Therefore, [ M1 – Substitution Method ] or [M1] Values of y [A1] (ii) . [4] [M1] Break up the indices Let [M1] Correct Quadratic Equation

5 PRSS_2021_Prelim_4E5N_AMaths_P1 [M1] Substitution and values of both p found or , [A1] 3 In the expansion of , where m is a positive constant, the independent term of x is 126720. (a) Show that . [4] General Term [M1] General Term [M1cao] – Correct value of r [M1] Equate general term with [AG1] (b) Hence, find the coefficient of in the expansion of [3]

6 PRSS_2021_Prelim_4E5N_AMaths_P1 [M1] Value of r found Term with [M1] Find term with Coefficient of is 887040 [A1] 4 (a) Express in partial fraction. [3] [M1cao] Break up Substitute : [M1]Substitution Method Substitute : [A1] Both values of A and B correct

7 PRSS_2021_Prelim_4E5N_AMaths_P1 (b) Hence, evaluate . [3] [M1] Relate to [M1] and 2 seen [A1] 5 The curve , for has a stationary point at point A. Find (i) , [2] [M1] Use Product Rule [A1]

8 PRSS_2021_Prelim_4E5N_AMaths_P1 (ii) the x-coordinate of A, [2] At stationary point, , [M1] Equate or ( Not required) ( 3 s.f.) [A1] (iii) the nature of A. [3] [M1] Product Rule Or by 1st derivative test At A, [M1] Use 1st or second derivative correctly to determine its nature , Since , point A is a maximum point.

9 PRSS_2021_Prelim_4E5N_AMaths_P1 [A1] Conclusion and state 6 (a) Given that , find (i) , [1] [B1]

10 PRSS_2021_Prelim_4E5N_AMaths_P1 (ii) . [2] [M1] Use correct double angle formulae [A1] (b) The diagram below shows a right angled triangle ACD. It is given that , , and . A D C y cm 3 cm 3 cm B

11 PRSS_2021_Prelim_4E5N_AMaths_P1 Express in terms of y. [4] and [M1 cao] Since [M1] Apply Addition Formulae Let [M1] Correct Quadratic Equation Therefore, . [A1] 7 A curve has the equation . (a) Explain why the curve has no turning point. [2]

12 PRSS_2021_Prelim_4E5N_AMaths_P1 [M1 cao] Correct Since and , therefore and . [A1] Correct Explaination y has no stationary point. (b) Given that y is increasing at a rate of 0.4 units per second at the instant when , find the rate of change of x at that instant. [3] Given units per second By Chain Rule,