Question 5:
5 | (a) | (i) | Quadratic curve is symmetrical about x = 2,
and therefore so is L | E1 | 2.4 | Implied symmetry of lamina
about x = 2 | x = 2 is a line of symmetry
of the lamina is enough
[1]
(ii) | DR
Area = ∫ 4 (−𝑥2 +4𝑥+2)d𝑥
0 | M1 | 1.1a | Condone missing limits / dx
2
Area = 18.7 or 18
3 | A1 | 1.1 | 56
Or
3
1 
 (x2 4x2)2dx
 
2 
∫𝑥4 −8𝑥3 +12𝑥2 +16𝑥 +4 dx | M1
A1 | 3.1b
1.1 | Consider integral of y2
Correctly expanding y2 to get x4
and at least 2 other correct terms. | Condone omission of ρ
Condone omission of
56
𝜌𝑦̅ =
3
1
[ 𝑥5 −2𝑥4 +4𝑥3 +8𝑥2 +4𝑥]
5 | A1FT | 1.1 | Follow through on a five term
quartic with leading term x4.
56 232
y =
3 5 | M1 | 1.1 | Use limits and equate to
56 56
y or y
3 3
87 17
[𝑦̅ =] 2.49 or or 2
35 35 | A1 | 1.1 | 2.4857…
[7]
(b) | 2.49
tan𝑂𝐴𝐺 =
2 | M1 | 3.1b
Angle is 51.2° | A1 | 1.1 | 51.1799… | Allow final answers in
range [51.179,51.23]
[2]
(c) | The x coordinate will stay the same (𝑥̅ = 2) | B1 | 1.1 | No reason needed for this one
The y coordinate will be less (𝑦̅ < 2.49)
because the part of M not included in N is
nearer to the x axis (or less tall). | B1 | 1.1 | “Extra mass is below y = 2.49” | Any answer which applies
that there is more mass /
area below 𝑦̅ < 2.49 than
before.
[2]