C2 June 2013 (R) Q2

EdexcelOld spec5 marksIntegration

2.\[y = \frac{x}{\sqrt{(1 + x)}}\]

(a) Complete the table below with the value of \(y\) corresponding to \(x = 1.3\), giving your answer to 4 decimal places.

\(x\)	1	1.1	1.2	1.3	1.4	1.5
\(y\)	0.7071	0.7591	0.8090		0.9037	0.9487

(1)

(b) Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an approximate value for\[\int_1^{1.5} \frac{x}{\sqrt{(1 + x)}}\,\mathrm{d}x\]giving your answer to 3 decimal places.
You must show clearly each stage of your working. (4)

Scheme	Marks
\(\{x = 1.3\}\ \ y = 0.8572\) (only)	B1 cao
	(1)

Mark scheme (b)

Scheme	Marks
\(\dfrac{1}{2} \times 0.1\ldots\ldots..\)	B1
\(\{0.7071 + 0.9487 + 2(0.7591 + 0.8090 + \text{"}0.8572\text{"} + 0.9037)\}\)	M1
\(\ldots\{0.7071 + 0.9487 + 2(0.7591 + 0.8090 + \text{"}0.8572\text{"} + 0.9037)\}\)	A1ft
\(\{0.05(8.3138)\} = 0.41569 =\) awrt 0.416	A1
	(4)
	Total 5

Notes

B1 for using \(\tfrac{1}{2} \times 0.1\) or 0.05 or equivalent.

M1 It needs the first bracket to contain first \(y\) value plus last \(y\) value and the second bracket to be multiplied by 2 and to be the summation of the remaining \(y\) values in the table with no additional values. If the only mistake is a copying error or is to omit one value from 2nd bracket this may be regarded as a slip and the M mark can be allowed ( An extra repeated term forfeits the M mark however). M0 if values used in brackets are \(x\) values instead of \(y\) values

A1ft for the correct bracket \(\{\ldots\ldots\}\) following through candidate’s \(y\) value found in part (a).

NB: Separate trapezia may be used : B1 for 0.05, M1 for 1/2 \(h(a + b)\) used 4 or 5 times (and A1ft if it is all correct ) Then A1 as before. (Equivalent correct formulae may be used)

Special case: Bracketing mistake
\(0.05 \times (0.7071 + 0.9487) + 2(0.7591 + 0.8090 + \text{"}0.8572\text{"} + 0.9037)\) scores B1 M1 A0 A0 (usually for 6.74079) unless the final answer implies that the calculation has been done correctly (then full marks can be given).