| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Standard +0.3 This is a multi-part question requiring Simpson's rule application (routine C3 skill) followed by logarithm law manipulations to deduce related integral values. Part (i) is standard numerical integration; parts (ii) and (iii) require recognizing ln(a²)=2ln(a) and ln(ab)=ln(a)+ln(b), which are straightforward applications of A-level log laws with minimal problem-solving demand. Slightly above average due to the deduction aspect, but the algebraic insights are direct. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt evaluation involving \(y\) values | M1 | With coefficients 1, 4 and 2 each occurring at least once; allow for wrong \(y\)-values; solution must include sufficient evidence of method |
| Obtain \(k(\ln 3 + 4\ln 7 + 2\ln 19 + 4\ln 39 + \ln 67)\) | A1 | Any constant \(k\); or decimal equivs; correct use of brackets required unless subsequent working shows their 'presence' |
| Identify value of \(k\) as \(\frac{2}{3}\) | A1 | As factor for their complete expression |
| Obtain 22.4 | A1 | Allow any value rounding to 22.4; answer only is 0/4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(9+6x^2+x^4 = (3+x^2)^2\) | B1 | Or, if proceeding numerically, demonstrate in at least three cases that \(\ln 9 = \ln 3^2\), \(\ln 49 = \ln 7^2\), \(\ln 361 = \ln 19^2\), … |
| Show relevant property \(\ln(3+x^2)^2 = 2\ln(3+x^2)\) and conclude with value \(2A\) | B1 | AG; necessary detail needed; if proceeding numerically, needs all five cases with relevant property. Note: using Simpson's rule again here is B0B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Recognise \(\ln(3e+ex^2)\) as \(1+\ln(3+x^2)\) | B1 | |
| Indicate in some way that \(\int_0^8 1\, dx\) is 8 and conclude with value \(A+8\) | B1 | AG; necessary detail needed. Note: using Simpson's rule again here is B0B0 |
## Question 6:
### Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt evaluation involving $y$ values | M1 | With coefficients 1, 4 and 2 each occurring at least once; allow for wrong $y$-values; solution must include sufficient evidence of method |
| Obtain $k(\ln 3 + 4\ln 7 + 2\ln 19 + 4\ln 39 + \ln 67)$ | A1 | Any constant $k$; or decimal equivs; correct use of brackets required unless subsequent working shows their 'presence' |
| Identify value of $k$ as $\frac{2}{3}$ | A1 | As factor for their complete expression |
| Obtain 22.4 | A1 | Allow any value rounding to 22.4; answer only is 0/4 |
### Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $9+6x^2+x^4 = (3+x^2)^2$ | B1 | Or, if proceeding numerically, demonstrate in at least three cases that $\ln 9 = \ln 3^2$, $\ln 49 = \ln 7^2$, $\ln 361 = \ln 19^2$, … |
| Show relevant property $\ln(3+x^2)^2 = 2\ln(3+x^2)$ and conclude with value $2A$ | B1 | AG; necessary detail needed; if proceeding numerically, needs all five cases with relevant property. Note: using Simpson's rule again here is B0B0 |
### Part (iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Recognise $\ln(3e+ex^2)$ as $1+\ln(3+x^2)$ | B1 | |
| Indicate in some way that $\int_0^8 1\, dx$ is 8 and conclude with value $A+8$ | B1 | AG; necessary detail needed. Note: using Simpson's rule again here is B0B0 |
---6 The value of $\int _ { 0 } ^ { 8 } \ln \left( 3 + x ^ { 2 } \right) \mathrm { d } x$ obtained by using Simpson's rule with four strips is denoted by $A$.\\
(i) Find the value of $A$ correct to 3 significant figures.\\
(ii) Explain why an approximate value of $\int _ { 0 } ^ { 8 } \ln \left( 9 + 6 x ^ { 2 } + x ^ { 4 } \right) \mathrm { d } x$ is $2 A$.\\
(iii) Explain why an approximate value of $\int _ { 0 } ^ { 8 } \ln \left( 3 \mathrm { e } + \mathrm { e } x ^ { 2 } \right) \mathrm { d } x$ is $A + 8$.
\hfill \mbox{\textit{OCR C3 2013 Q6 [8]}}