Standard +0.3 This is a straightforward differentiation and equation-solving problem. Students must differentiate (using power rule, not really chain rule despite the topic label), set the derivative equal to 2, and solve a quadratic equation. While it requires multiple steps and 'detailed reasoning', each step is standard A-level technique with no novel insight required, making it slightly easier than average.
13 In this question you must show detailed reasoning.
The equation of a curve is \(y = 3 x + \frac { 7 } { x } - \frac { 3 } { x ^ { 2 } }\).
Determine the coordinates of the points on the curve where the curve is parallel to the line \(y = 2 x\). [0pt]
[9]
END OF QUESTION PAPER
M1: at least two of three terms correct. A1: all correct
their \(\frac{dy}{dx} = 2\)
M1
\(x^3 - 7x + 6 = 0\)
M1
their \(\frac{dy}{dx} = 2\) rearranged to give cubic expression equal to zero
Use of Factor theorem with factor of their 6
M1
Long division oe to obtain \((x-1)(x^2+x-6)\) or \((x-2)(x^2+2x-3)\) or \((x+3)(x^2-3x+2)\)
M1
Allow sign error in quotient; quotient and factor may appear separately in algebraic division
\((x-1)(x+3)(x-2)\)
A1
Evaluation of \(f(\text{their } 1)\), \(f(\text{their } 2)\) and \(f(\text{their } -3)\) seen
M1
\((1, 7)\), \(\left(2, \frac{35}{4}\right)\) and \(\left(-3, -\frac{35}{3}\right)\)
A1
[9]
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## Question 13:
| Answer | Mark | Guidance |
|--------|------|----------|
| $\left[\frac{dy}{dx}\right] = 3 - \frac{7}{x^2} + \frac{6}{x^3}$ | M1, A1 | M1: at least two of three terms correct. A1: all correct |
| their $\frac{dy}{dx} = 2$ | M1 | |
| $x^3 - 7x + 6 = 0$ | M1 | their $\frac{dy}{dx} = 2$ rearranged to give cubic expression equal to zero |
| Use of Factor theorem with factor of their 6 | M1 | |
| Long division **oe** to obtain $(x-1)(x^2+x-6)$ or $(x-2)(x^2+2x-3)$ or $(x+3)(x^2-3x+2)$ | M1 | Allow sign error in quotient; quotient and factor may appear separately in algebraic division |
| $(x-1)(x+3)(x-2)$ | A1 | |
| Evaluation of $f(\text{their } 1)$, $f(\text{their } 2)$ and $f(\text{their } -3)$ seen | M1 | |
| $(1, 7)$, $\left(2, \frac{35}{4}\right)$ and $\left(-3, -\frac{35}{3}\right)$ | A1 | |
| **[9]** | | |
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- Phone/fax/email information
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**There is no mark scheme content visible on this page.**
To extract mark scheme content, please share the actual mark scheme pages (typically containing question numbers, answers, mark allocations such as M1, A1, B1, etc.).
13 In this question you must show detailed reasoning.\\
The equation of a curve is $y = 3 x + \frac { 7 } { x } - \frac { 3 } { x ^ { 2 } }$.\\
Determine the coordinates of the points on the curve where the curve is parallel to the line $y = 2 x$.\\[0pt]
[9]
END OF QUESTION PAPER
\hfill \mbox{\textit{OCR MEI AS Paper 2 2021 Q13 [9]}}