A2 June 2024 Q1

EdexcelCurrent spec5 marksNumerical Methods

(a) Given that\[y = \ln\left(3 + x^2\right)\]complete the table with the value of \(y\) corresponding to \(x = 3\), giving your answer to 4 significant figures.

\(\boldsymbol{x}\)	2	2.5	3	3.5	4	4.5	5
\(\boldsymbol{y}\)	1.946	2.225		2.725	2.944	3.146	3.332

(1)

In part (b) you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

(b) Use Simpson’s rule with all the values of \(y\) in the completed table to estimate, to 3 significant figures, the value of\[\int_2^5 \ln\left(3 + x^2\right)\,\mathrm{d}x\] (3)

(c) Using your answer to part (b) and making your method clear, estimate the value of\[\int_2^5 \ln\sqrt{\left(3 + x^2\right)}\,\mathrm{d}x\] (1)

Scheme	Marks	AO
Awrt 2.485	B1	1.1b
	(1)

Mark scheme (b)

Scheme	Marks	AO
\(h = 0.5\)	B1	1.1b
\(\dfrac{1}{3} \times \text{“}0.5\text{”}\left[1.946 + 3.332 + 2(2.485 + 2.944) + 4(2.225 + 2.725 + 3.146)\right]\) \(= \dfrac{1}{6} \times 48.52\)	M1	1.1b
8.09 cao	A1	1.1b
	(3)

Notes

B1: Correct step length used, stated or clearly implied in working.

M1: Correct structure for Simpson’s rule \(\dfrac{1}{3}\text{“}h\text{”}\left[\text{ends} + 2\text{evens} + 4\text{odds}\right]\) using their value of \(h\). If no value of \(h\) is stated nor is implied and the \(\dfrac{1}{3}h\) is not clearly seen, award bod for any multiple in front of the bracket that does not imply a clearly incorrect method – but the insides must be correct. Condone a missing closing bracket at the end, but otherwise bracketing must be correct or implied correct by the answer. Condone minor miscopies of the ordinates as long as they are in correct positions.

A1: For 8.09, correct answer only, must be to 3.s.f. Must have scored the M.
Note calculator gives 8.086 …

Mark scheme (c)

Scheme	Marks	AO
\(0.5 \times \text{“}8.09\text{”} = 4.045\) or 4.04 or 4.05 (awrt either)	B1ft	2.2a
	(1)
	(5 marks)

Notes

B1ft: Deduces the value by finding \(0.5 \times\) their answer to (b). Allow for awrt 3.s.f. answers giving tolerance with regards to prior rounding. Accept as a fraction, e.g. \(\dfrac{1213}{300}\) as long as it is half their answer to (b), or rounds correctly to it.