AISS Prelim AMath Paper 1 Marking Scheme

3 AISS PRELIM/4E/4049/01/2025 [Turn over 1 It is given that P, Q and R are the angles of a triangle. (a) Show that cos P = – cos (Q + R). [2] (b) Given that Q = 45 and R = 60, find cos P in the form 1 4 ( a – b ), where a and b are integers. [3] cos P = cos [180   (Q + R)] [M1 – for replacing] = cos 180  cos (Q + R) + sin 180 sin (Q + R) = –cos ( Q + R) + 0 [A1 – apply addition formula & = –cos (Q + R) [evaluate to arrive at result] cos P = –cos (45  + 60) = –cos 45  cos 60 + sin 45 sin 60 [M1 – correct use of formula expansion] = – 2 2 ( 1 2 ) + 2 2 ( 3 2 ) [M1 – correct special angles trigo ratios] = 1 4 ( 6 – 2 ) [A1]

4 AISS PRELIM/4E/4049/01/2025 2 Baking powder is poured onto a flat surface at a constant rate of 312 c m s  and formed a right circular cone. The radius of the cone is always 1 18 of its height. Find the rate of change of the radius of the cone after 3 seconds of pouring. [5]   2 2 3 11 2 1 1 3 1 183 6 d 18 [B1d 6 6 [M1 - finding c a orresponding ] Vol. of cone, After 3 sec 1[ A 1 ] dd d [M1 - T o connect o e 2 nd d s, 6 rate of change]dd d d d8 d 11 he rate f h 9 c 1 d rr r r r V r r r VV r tr t r t V t r r r V r r                    1 [A1 o.e.]nge required is = cm9 /s.

5 AISS PRELIM/4E/4049/01/2025 [Turn over 3 (a) Determine the set of values of m for which the equation 26242 2  mxmxx has real roots. [4] (b) Hence state what can be deduced about the curve 2)1(2  xy and the line .26  xy Justify your statement. [2] 22( 4 6 ) 2 2 0 2 40 2(4 6 ) 4(2)(2 2) 0 216 48 36 8(2 2) 0 236 64 0 (9 16) 0 160 9 xm x m ba c mm mm m mm mm mo r m                [M1 - correct Discriminant] [M1 – simplification in factors] [A2, minus 1 mark if inequality sign is wrong due to earlier wrong D sign] 224 2 6 2xx x  By comparing with (a), m = 1 [B1 – correct m value] When m = 1, it is not within the set of values of m for which there will be real roots, hence the curve will not meet the line/ the curve will not cut the line. [B1]

6 AISS PRELIM/4E/4049/01/2025 4 (a) Show that d ln cos tand x xx  . [2] (b) Differentiate tanx x with respect to x. [2] (c) Using the results from part (a) and (b), find 2sec dx xx and hence show that 24 0 1sec d ln 2 42xx x   . [4]  d1 dln cos cosdc o s d sin = tancos x xxx x x xx    [M1 – show working] [A1 – show fraction] 2d tan sec