A2 June 2025 Q1

EdexcelCurrent spec6 marksNumerical Methods

(a) Given that\[y = \mathrm{e}^{\operatorname{cosec}^2 x}\]complete the table below with the value of \(y\) corresponding to \(x = 2\), giving your answer to 2 decimal places.

\(x\)	1.5	1.75	2	2.25	2.5
\(y\)	2.73	2.81		5.22	16.31

(1)

(b) Use Simpson’s rule, with all the values of \(y\) in the completed table, to estimate, to one decimal place, the value of\[\int_{1.5}^{2.5} \mathrm{e}^{\operatorname{cosec}^2 x}\,\mathrm{d}x\] (3)

(c) Using your answer to part (b) and making your method clear, estimate the value of\[\int_{1.5}^{2.5} \mathrm{e}^{\cot^2 x}\,\mathrm{d}x\] (2)

Scheme	Marks	AO
3.35 cao	B1	1.1b
	(1)

Mark scheme (b)

Scheme	Marks	AO
Step length = 0.25	B1	1.1b
\(\dfrac{1}{3} \times \text{“}0.25\text{”}\left[2.73 + 16.31 + 2(\text{“}3.35\text{”}) + 4(2.81 + 5.22)\right]\)	M1	1.1b
\(= 4.8\)	A1	1.1b
	(3)

Notes

B1: Correct step length, may be seen in the application of Simpson’s rule formula. Could be seen as \(1.75 - 1.5\) or \(\dfrac{2.5 - 1.5}{4}\)

M1: Correct application of Simpson’s rule using their step length. If no step length stated and uses \(\dfrac{1}{3}\left[\text{ends} + 2(\text{evens}) + 4(\text{odd})\right]\) this is M0

A1: Awrt 4.8 must be using 3.35 from (a)
Note calculator answer is 4.67 is no marks
Note Answer only with no working is no marks,
Note: If they write \(\dfrac{1}{3} \times h\left[\text{ends} + 2(\text{evens}) + 4(\text{odd})\right]\) which leads to awrt 4.8 this gets B1M1A1
If their answer is incorrect this would be B0 (if no \(h\) stated) M0A0

Mark scheme (c)

Scheme	Marks	AO
\(\displaystyle\int_{1.5}^{2.5} \mathrm{e}^{\cot^2 x}\,\mathrm{d}x = \int_{1.5}^{2.5} \mathrm{e}^{\operatorname{cosec}^2 x - 1}\,\mathrm{d}x = \frac{1}{\mathrm{e}}\int_{1.5}^{2.5} \mathrm{e}^{\operatorname{cosec}^2 x}\,\mathrm{d}x = \frac{1}{\mathrm{e}} \times \text{“}4.8\text{”}\)	M1	3.1a
\(=\) awrt 1.8	A1ft	1.1b
	(2)
	(6 marks)

Notes

M1: Uses a correct identity to express \(\displaystyle\int \mathrm{e}^{\cot^2 x}\,\mathrm{d}x\) in terms of \(\displaystyle\int \mathrm{e}^{\operatorname{cosec}^2 x}\,\mathrm{d}x\) and then uses correct index work to allow \(\displaystyle\int_{1.5}^{2.5} \mathrm{e}^{\cot^2 x}\,\mathrm{d}x\) to be evaluated using part (b) \(\dfrac{1}{\mathrm{e}} \times \text{“}4.8\text{”}\)

A1ft: Awrt 1.8 or follow through \(\dfrac{1}{\mathrm{e}} \times \text{their (b)}\) which must be evaluated and correct for their (b) rounded to 1 d.p.
If uses Simpson’s rule again this is M0 A0