Mathematics · May / June 2022 · Paper 41 · Question 5

( ) − × 8 2 180 oe 8 2 × [3]

M1 for ( ) 8 or ( ) M2 − × ×− × 8 2 180 or 360 2 8 4 90 8 8 174 or 173.8.... [4]

M3 for 1 2 ××OM oe 6 1 ( ) 2 × 2 × or OA sin45 oe or 1 2 ×× × 6 OA sin 67.5 oe where OA and OM are as in the M2 OM =× 3 tan67.5

or M2 for oe or 6 ×sin 67.5 3 = or for OA oe cos 67.5 sin 45 OM = tan67.5

or M1 for oe 3 or for 3 =cos 67.5 oe OA sin 45 sin 67.5 = or for oe 6 OA

193 or 193.0 to 193.1 [3]

2 3 π ×

M2 for oe cos 67.5

or M1 for 3 sin 45 sin 67.5 =cos = 67.5 or r 6 r

1.27 or 1.272 to 1.273 [2]

M1 for 1 × π × × 0.45 2 4 2 1 [ ] × π × 2 × or 0.45 4 2

742 or 743 [6]

M5 for a method leading to the volume of water 0.15 inv cos 0.45 4 {2 × × × π × 0.45 2 e.g. 360 1 0.15 − × 2 × 0.45 sin 2 inv cos } oe 2 0.45 OR 0.15 inv cos 0.45 [ ] × × π × 2 M2 2 0.45 oe 360 0.15 − 90 inv cos 0.45 [ ] × × π × 2 or 2 0.45 oe 360 θ × π × 0.45 2

or M1 for use of oe 360 1 0.15 × 2 ×

M2 for 0.45 sin 2 inv cos oe 2 0.45 1 0.15 [ ] × × × × or 0.15 0.45 sin inv cos 2 oe 2 0.45 [1]

2 × 2 0.45 × sinθ oe

or M1 for use of ]1 or [ 0.15 0.45 sinβ × × × × 2 2 oe If 0 scored, 0.15 0.15 0.45 or invsin 0.45

SC1 for invcos 2 − 2 0.45 0.15 or soi