Question 4 - A-Level Maths

OCR C3 2007 June — Question 4 7 marks

Exam Board	OCR
Module	C3 (Core Mathematics 3)
Year	2007
Session	June
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Numerical integration
Type	Simpson's rule application
Difficulty	Moderate -0.8 This is a straightforward two-part question requiring routine application of standard techniques: (i) integration using the reverse chain rule with a simple linear substitution, and (ii) mechanical application of Simpson's rule with only two strips. Both parts are textbook exercises with no problem-solving or insight required, making this easier than average.
Spec	1.08d Evaluate definite integrals: between limits 1.09f Trapezium rule: numerical integration

4 The integral I is defined by $$I = \int _ { 0 } ^ { 13 } ( 2 x + 1 ) ^ { \frac { 1 } { 3 } } d x$$

Use integration to find the exact value of I .
Use Simpson's rule with two strips to find an approximate value for I. Give your answer correct to 3 significant figures.

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Answer	Marks	Guidance
(i) Obtain integral of form $k(2x + 1)^4$	M1	or equiv using substitution; any constant $k$
Obtain correct $\frac{1}{5}(2x + 1)^4$	A1	or equiv
Substitute limits in expression of form $(2x + 1)^6$ and subtract the correct way round	M1	using adjusted limits if subn used
Obtain $30$	A1	4
(ii) Attempt evaluation of $k(y_0 + 4y_1 + y_2)$	M1	any constant $k$
Identify $k$ as $\frac{1}{3} \times 6.5$	A1
Obtain $29.6$	A1	3 or greater accuracy (29.554566…)
[SR: (using Simpson's rule with 4 strips)
Obtain $\frac{1}{3} \times 3.25(1 + 4 \times \sqrt{7.5} + 2 \times \sqrt{14} + 4 \times \sqrt{20.5} + 3)$ and hence $29.9$	B1	or greater accuracy (29.897…)]

(i) Obtain integral of form $k(2x + 1)^4$ | M1 | or equiv using substitution; any constant $k$
Obtain correct $\frac{1}{5}(2x + 1)^4$ | A1 | or equiv
Substitute limits in expression of form $(2x + 1)^6$ and subtract the correct way round | M1 | using adjusted limits if subn used
Obtain $30$ | A1 | 4

(ii) Attempt evaluation of $k(y_0 + 4y_1 + y_2)$ | M1 | any constant $k$
Identify $k$ as $\frac{1}{3} \times 6.5$ | A1 |
Obtain $29.6$ | A1 | 3 or greater accuracy (29.554566…)
[SR: (using Simpson's rule with 4 strips) | |
Obtain $\frac{1}{3} \times 3.25(1 + 4 \times \sqrt{7.5} + 2 \times \sqrt{14} + 4 \times \sqrt{20.5} + 3)$ and hence $29.9$ | B1 | or greater accuracy (29.897…)]

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4 The integral I is defined by

$$I = \int _ { 0 } ^ { 13 } ( 2 x + 1 ) ^ { \frac { 1 } { 3 } } d x$$

(i) Use integration to find the exact value of I .\\
(ii) Use Simpson's rule with two strips to find an approximate value for I. Give your answer correct to 3 significant figures.

\hfill \mbox{\textit{OCR C3 2007 Q4 [7]}}

This paper (9 questions)

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Q1 5 Q2 5 Q3 7 Q4 7 Q5 7 Q6 9 Q7 9 Q8 11 Q9 12

Answer	Marks	Guidance
(i) Obtain integral of form \(k(2x + 1)^4\)	M1	or equiv using substitution; any constant \(k\)
Obtain correct \(\frac{1}{5}(2x + 1)^4\)	A1	or equiv
Substitute limits in expression of form \((2x + 1)^6\) and subtract the correct way round	M1	using adjusted limits if subn used
Obtain \(30\)	A1	4
(ii) Attempt evaluation of \(k(y_0 + 4y_1 + y_2)\)	M1	any constant \(k\)
Identify \(k\) as \(\frac{1}{3} \times 6.5\)	A1
Obtain \(29.6\)	A1	3 or greater accuracy (29.554566…)
[SR: (using Simpson's rule with 4 strips)
Obtain \(\frac{1}{3} \times 3.25(1 + 4 \times \sqrt{7.5} + 2 \times \sqrt{14} + 4 \times \sqrt{20.5} + 3)\) and hence \(29.9\)	B1	or greater accuracy (29.897…)]