Mathematics · Oct / Nov 2023 · Paper 21 · Question 6

YMA01

As printed in the paper

The question as text

6. (0, 0) (15, 0) x (3, 1.52) (6, 2.74) (12, 3.08) (9, 3.12) y Figure 1 A river is being studied. At one particular place, the river is 15 m wide. The depth, y metres, of the river is measured at a point x metres from one side of the river. Figure 1 shows a plot of the cross‑section of the river and the coordinate values (x, y)

Read automatically from the paper; the printed page above is exact.

Show the answer and mark scheme

6(a)[3]
Identifies h = 3
3 0
(
)
Area =
+
0
+
2 1.52
+
2.74
+
3.12
+
3.08
2
= 31.38 m2
(b) Calculates ''31.38'' 1.5 60
×
×
= awrt 2800 m3
Underestimate with valid reason, e.g. area of trapezia are smaller than cross
6(b)[2]
Identifies h = 3
3 0
(
)
Area =
+
0
+
2 1.52
+
2.74
+
3.12
+
3.08
2
= 31.38 m2
(b) Calculates ''31.38'' 1.5 60
×
×
= awrt 2800 m3
Underestimate with valid reason, e.g. area of trapezia are smaller than cross
sectional area
(a)
3
(
)
B1: Identifies h = 3. This may be stated or implied by the sight of
....
+
2 ... ... ...
+
+
2
M1: Correct method of finding area using the trapezium rule. The form of the rule is given so look for the form
3 0
(
)
+
0
+
2 1.52
+
2.74
+
3.12
+
3.08
but you can condone slips on the digits of the heights, e.g 3.21 for
2
3.12. You may not see the zeros or the trailing which is fine. The bracketing may be hard to decipher
but the correct answer implies the method. If they state another value of h e.g. 15
6 =
2.5
then look for
2.5 0
(
)
+
0
+
2 1.52
+
2.74
+
3.12
+
3.08
condoning slips. If h is not stated, then this mark can be implied by
2
... 0
(
)
+
0
+
2 1.52
+
2.74
+
3.12
+
3.08
condoning slips.
2
Condone an attempt using two triangles and three trapezia. Use the same criteria for this award. The area
of any triangle and the area of any trapezium must be found from a correct formula.
A1: Accept 31.38 or awrt 31.4 (m2) following correct expression for area.
Units are not necessary. The exact fractional answer is 1569
50 is also acceptable
As the area is below the axis, a minority of candidates are taking h or the y values to be negative. This is
fine for the first two marks and all three marks provided the modulus is taken.
(b)
M1: Attempts their (a) ∈1.5∈60 or equivalent such as their (a) ∈90
A1: AWRT 2800 m3 Units are not necessary
(c)
B1: States underestimate AND gives a valid reason.
The reason, in almost all cases, should explicitly compare the area of the trapezia with the shaded area.
All reasons must make some reference to ‘trapezia’ or ‘area of a trapezium’
The Figure could be used to explain their reasoning so look carefully at what the candidate states
Condone ‘the trapezia are smaller than the cross section’
‘the bottom lines of the trapezia are above the curve’
‘the curve is above the lines of the trapezia’ (remember the y axis is pointing down)
Do not accept any reference to convex or concave without further clarification.
Merely stating '' the area we get from the trapezium rule is less than the shaded area'' is too vague. There
needs to be some explanation of why it is an underestimate, not just the fact that it is.
2
6(c)[1]
sectional area
(a)
3
(
)
B1: Identifies h = 3. This may be stated or implied by the sight of
....
+
2 ... ... ...
+
+
2
M1: Correct method of finding area using the trapezium rule. The form of the rule is given so look for the form
3 0
(
)
+
0
+
2 1.52
+
2.74
+
3.12
+
3.08
but you can condone slips on the digits of the heights, e.g 3.21 for
2
3.12. You may not see the zeros or the trailing which is fine. The bracketing may be hard to decipher
but the correct answer implies the method. If they state another value of h e.g. 15
6 =
2.5
then look for
2.5 0
(
)
+
0
+
2 1.52
+
2.74
+
3.12
+
3.08
condoning slips. If h is not stated, then this mark can be implied by
2
... 0
(
)
+
0
+
2 1.52
+
2.74
+
3.12
+
3.08
condoning slips.
2
Condone an attempt using two triangles and three trapezia. Use the same criteria for this award. The area
of any triangle and the area of any trapezium must be found from a correct formula.
A1: Accept 31.38 or awrt 31.4 (m2) following correct expression for area.
Units are not necessary. The exact fractional answer is 1569
50 is also acceptable
As the area is below the axis, a minority of candidates are taking h or the y values to be negative. This is
fine for the first two marks and all three marks provided the modulus is taken.
(b)
M1: Attempts their (a) ∈1.5∈60 or equivalent such as their (a) ∈90
A1: AWRT 2800 m3 Units are not necessary
(c)
B1: States underestimate AND gives a valid reason.
The reason, in almost all cases, should explicitly compare the area of the trapezia with the shaded area.
All reasons must make some reference to ‘trapezia’ or ‘area of a trapezium’
The Figure could be used to explain their reasoning so look carefully at what the candidate states
Condone ‘the trapezia are smaller than the cross section’
‘the bottom lines of the trapezia are above the curve’
‘the curve is above the lines of the trapezia’ (remember the y axis is pointing down)
Do not accept any reference to convex or concave without further clarification.
Merely stating '' the area we get from the trapezium rule is less than the shaded area'' is too vague. There
needs to be some explanation of why it is an underestimate, not just the fact that it is.
2

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