OCR PURE — Question 8 7 marks

Exam BoardOCR
ModulePURE
Marks7
PaperDownload PDF ↗
TopicIndefinite & Definite Integrals
TypeIntegration with given constant
DifficultyModerate -0.3 This is a straightforward definite integration problem requiring evaluation of a standard integral (power of x) and solving a simple equation. While it requires showing working and involves solving for a limit, the techniques are routine A-level methods with no conceptual difficulty beyond basic integration rules.
Spec1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits
8 In this question you must show detailed reasoning. Given that \(\int _ { 4 } ^ { a } \left( \frac { 4 } { \sqrt { x } } + 3 \right) \mathrm { d } x = 7\), find the value of \(a\).
Question 8:
AnswerMarks Guidance
AnswerMarks Guidance
Either term integrated correctlyM1* 2.1
\(8x^{\frac{1}{2}} + 3x\)A1 1.1
\(\left(8a^{\frac{1}{2}} + 3a\right) - (16+12) = 7\)M1dep* 1.1 - Correct use of correct limits and equating to 7; allow one substitution error
\(3a + 8a^{\frac{1}{2}} - 35 = 0\)M1 1.1 - Forming a 3TQ in \(a^{\frac{1}{2}}\); any three-term form
\(\left(3a^{\frac{1}{2}} - 7\right)\left(a^{\frac{1}{2}} + 5\right) = 0\)M1 3.1a - Dependent on all previous M marks; correct method for solving for \(a^{\frac{1}{2}}\). Or: \(8a^{\frac{1}{2}} = 35-3a\), \(9a^2 - 274a + 1225 = 0\), \((9a-49)(a-25)=0\)
\(a^{\frac{1}{2}} \neq -5\) as \(a^{\frac{1}{2}}\) can't be negativeA1 2.3 - Explicit rejection of \(-5\); no specific justification required. Or explicit rejection of \(a=25\)
\(a^{\frac{1}{2}} = \frac{7}{3} \Rightarrow a = \frac{49}{9}\)A1 2.2a - Correct value only
Total: [7]
# Question 8:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Either term integrated correctly | M1* | 2.1 |
| $8x^{\frac{1}{2}} + 3x$ | A1 | 1.1 |
| $\left(8a^{\frac{1}{2}} + 3a\right) - (16+12) = 7$ | M1dep* | 1.1 - Correct use of correct limits and equating to 7; allow one substitution error |
| $3a + 8a^{\frac{1}{2}} - 35 = 0$ | M1 | 1.1 - Forming a 3TQ in $a^{\frac{1}{2}}$; any three-term form |
| $\left(3a^{\frac{1}{2}} - 7\right)\left(a^{\frac{1}{2}} + 5\right) = 0$ | M1 | 3.1a - Dependent on all previous M marks; correct method for solving for $a^{\frac{1}{2}}$. Or: $8a^{\frac{1}{2}} = 35-3a$, $9a^2 - 274a + 1225 = 0$, $(9a-49)(a-25)=0$ |
| $a^{\frac{1}{2}} \neq -5$ as $a^{\frac{1}{2}}$ can't be negative | A1 | 2.3 - Explicit rejection of $-5$; no specific justification required. Or explicit rejection of $a=25$ |
| $a^{\frac{1}{2}} = \frac{7}{3} \Rightarrow a = \frac{49}{9}$ | A1 | 2.2a - Correct value only |

**Total: [7]**

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8 In this question you must show detailed reasoning.
Given that $\int _ { 4 } ^ { a } \left( \frac { 4 } { \sqrt { x } } + 3 \right) \mathrm { d } x = 7$, find the value of $a$.

\hfill \mbox{\textit{OCR PURE  Q8 [7]}}
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