RI 2023 MATH TP SOLN

RAFFLES INSTITUTION H2 Mathematics (9758) 2023 Year 6 ________________________________________________ 2023 Y6 H2 Math Timed Practice: Solutions with comments Page 1 of 17       2023 Year 6 H2 Mathematics Timed Practice: Solutions with Comments 1 (a) (i) Find 2sin dxx . [2] (ii) Hence or otherwise, show that 22 11 1sin d sin 2 cos 2 .44 8xx xx xx x c   [3] (b) (i) Sketch the graph of 2sinyx x   for 10 π2x , labelling the end points clearly. [2] (ii) Hence, find the exact area of the region bounded by the graph of 2siny xx , the x-axis and the line 1 π2x . [3] Solutions Comments (a)(i) [2]  2 1c o s 2sin d d 2 1 1c o s 2 d2 1s i n 2 22 1s i n 2 24 xxx x xx xxc xxc          (a)(ii) [3] 2 2 22 2 11 11sin d sin 2 sin 2 d24 24 11 1 1 c o s 2sin 224 2 24 2 11 11 sin 2 cos 224 48 11 1 sin 2 cos 2 (shown)44 8 x xx x x x x x x xxxx x c xx x x x c xx x x c                            “Hence” method: Integration by parts: ux   2d sind v xx    d 1d u x    1s i n 2 24 xvx

Raffles Institution H2 Mathematics 2023 Year 6 __________________________________________________________________________________________     ________________________________________________ 2023 Y6 H2 Math Timed Practice: Solutions with comments Page 2 of 17       y x 2sinyx x ππ,22   O 0, 0 π 2x Alternatively, 2 2 2 2 2 1c o s 2sin d d 2 1 cos 2 d42 1 sin 2 sin 2 d42 2 2 sin 2 1 cos 2 44 4 2 11 1 sin 2 cos 2 (shown)44 8 xxx x x x x xx x xx x xx xx x x c xx x x c                    (b)(i) [2] Students needs to zoom to the appropriate scale to see the shape of the graph. Do observe the gradient as well.

Raffles Institution H2 Mathematics 2023 Year 6 __________________________________________________________________________________________     ________________________________________________ 2023 Y6 H2 Math Timed Practic e: Solutions with comments Page 3 of 17       (b)(ii) [3]       π 22 0 π 22 0 2 2 2 2 2 Required area sin d 11 1 sin 2 cos 244 8 1 π 1 ππ 1 πsin 2 cos 242 42 2 8 2 11 100 s i n 2 0 c o s 2 044 8 π 11 116 8 8 π 1 units16 4 xx x xx x x                            Some students made careless mistakes with the signs.

Raffles Institution H2 Mathematics 2023 Year 6 __________________________________________________________________________________________     ________________________________________________ 2023 Y6 H2 Math Timed Practic e: Solutions with comments Page 4 of 17       2 (a) A point R has position vector r = 1 24 35 a a        , where a is a real number. Describe geometrically the set of all possible positions of R as a varies.