| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Sequence of transformations order |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing standard C2 transformations and techniques. Parts (i)-(iii) require basic recall of translations/stretches and sketching exponentials. Part (iv) is routine logarithm manipulation, and part (v) is a standard trapezium rule application. All parts follow textbook procedures with no problem-solving or novel insight required, making it easier than average. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02w Graph transformations: simple transformations of f(x)1.06g Equations with exponentials: solve a^x = b1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 2 (units) in the positive \(x\)-direction | M1 | Correct direction — identify translation is in \(x\)-direction (positive or negative); allow \(\binom{k}{0}\) |
| A1 | Fully correct description — must have correct magnitude and direction using precise language; A0 for 'factor 2'; allow \(\binom{2}{0}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| sf \(\frac{4}{9}\) in the \(y\)-direction | M1 | Correct direction, with sf of \(\frac{4}{9}\) or 9 — stretch in \(y\)-direction; allow just \(\frac{4}{9}\) or 9 with no mention of 'scale factor' |
| A1 | Fully correct description — must have correct scale factor and direction using precise language; A0 for in/on/along the \(y\)-axis |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct sketch in both quadrants; intersect at \(\left(0, \frac{4}{9}\right)\) | B1* | Correct sketch in both quadrants — curve must tend towards negative \(x\)-axis but not touch or cross it |
| State \(\left(0, \frac{4}{9}\right)\) | B1d* | \(x=0\), \(y=\frac{4}{9}\) as alternative; \(x=0\) must be stated explicitly; allow no brackets; allow exact decimal equiv |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\log 3^{x-2} = \log 180\) (or \(x-2 = \log_3 180\)); \((x-2)\log 3 = \log 180\) | M1* | Introduce logs and drop power — can use any base; power must be dropped; brackets must be seen around \((x-2)\) |
| \(x - 2 = 4.7268...\); \(x = 6.73\) | M1d* | Attempt to solve for \(x\) — correct order of operations; M0 for \(\log_3 180 - 2\) |
| A1 | Obtain 6.73, or better — if \(> 3\)sf allow 6.727; 0/3 for answer only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.5 \times 1.5 \times \{3^{-1} + 2 \times 3^{0.5} + 3^2\} = 9.60\) | B1 | State the 3 correct \(y\)-values, and no others — allow unsimplified; allow decimal equivs |
| M1 | Attempt use of correct trapezium rule between \(x=1\) and \(x=4\) — correct placing of \(y\)-values; must have \(h\) as 1.5 | |
| A1 | Obtain 9.60, or better (allow 9.6) — allow answers in range [9.595, 9.600] if \(> 3\)sf |
## Question 8:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| 2 (units) in the positive $x$-direction | M1 | Correct direction — identify translation is in $x$-direction (positive or negative); allow $\binom{k}{0}$ |
| | A1 | Fully correct description — must have correct magnitude and direction using precise language; A0 for 'factor 2'; allow $\binom{2}{0}$ |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| sf $\frac{4}{9}$ in the $y$-direction | M1 | Correct direction, with sf of $\frac{4}{9}$ or 9 — stretch in $y$-direction; allow just $\frac{4}{9}$ or 9 with no mention of 'scale factor' |
| | A1 | Fully correct description — must have correct scale factor and direction using precise language; A0 for in/on/along the $y$-axis |
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct sketch in both quadrants; intersect at $\left(0, \frac{4}{9}\right)$ | B1* | Correct sketch in both quadrants — curve must tend towards negative $x$-axis but not touch or cross it |
| State $\left(0, \frac{4}{9}\right)$ | B1d* | $x=0$, $y=\frac{4}{9}$ as alternative; $x=0$ must be stated explicitly; allow no brackets; allow exact decimal equiv |
### Part (iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\log 3^{x-2} = \log 180$ (or $x-2 = \log_3 180$); $(x-2)\log 3 = \log 180$ | M1* | Introduce logs and drop power — can use any base; power must be dropped; brackets must be seen around $(x-2)$ |
| $x - 2 = 4.7268...$; $x = 6.73$ | M1d* | Attempt to solve for $x$ — correct order of operations; M0 for $\log_3 180 - 2$ |
| | A1 | Obtain 6.73, or better — if $> 3$sf allow 6.727; 0/3 for answer only |
### Part (v):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.5 \times 1.5 \times \{3^{-1} + 2 \times 3^{0.5} + 3^2\} = 9.60$ | B1 | State the 3 correct $y$-values, and no others — allow unsimplified; allow decimal equivs |
| | M1 | Attempt use of correct trapezium rule between $x=1$ and $x=4$ — correct placing of $y$-values; must have $h$ as 1.5 |
| | A1 | Obtain 9.60, or better (allow 9.6) — allow answers in range [9.595, 9.600] if $> 3$sf |
---8 (i) The curve $y = 3 ^ { x }$ can be transformed to the curve $y = 3 ^ { x - 2 }$ by a translation. Give details of the translation.\\
(ii) Alternatively, the curve $y = 3 ^ { x }$ can be transformed to the curve $y = 3 ^ { x - 2 }$ by a stretch. Give details of the stretch.\\
(iii) Sketch the curve $y = 3 ^ { x - 2 }$, stating the coordinates of any points of intersection with the axes.\\
(iv) The point $P$ on the curve $y = 3 ^ { x - 2 }$ has $y$-coordinate equal to 180 . Use logarithms to find the $x$-coordinate of $P$, correct to 3 significant figures.\\
(v) Use the trapezium rule, with 2 strips each of width 1.5, to find an estimate for $\int _ { 1 } ^ { 4 } 3 ^ { x - 2 } \mathrm {~d} x$. Give your answer correct to 3 significant figures.
\hfill \mbox{\textit{OCR C2 2016 Q8 [12]}}