| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2019 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find derivative after algebraic simplification (fractional/mixed powers) |
| Difficulty | Moderate -0.8 This is a straightforward differentiation question requiring students to rewrite terms as powers of x and apply the power rule. While it has multiple terms and requires algebraic manipulation (converting fractions and roots to index form), it's a standard C1/C2 exercise with no problem-solving or conceptual challenge—purely procedural application of basic differentiation rules. |
| Spec | 1.07i Differentiate x^n: for rational n and sums |
| HAVI SIHI NI JINM ION OC | VIIV SIHI NI JINAM ION OA | VIUV SIHI NI JIIIM ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\frac{2x^4-8}{5\sqrt{x}} = \frac{2x^4}{5\sqrt{x}} - \frac{8}{5\sqrt{x}} = \frac{2}{5}x^{\frac{7}{2}} - \frac{8}{5}x^{-\frac{1}{2}}\) | M1 | Attempts to split into two terms where one is of the form \(px^{\frac{7}{2}}\) or \(qx^{-\frac{1}{2}}\); may write as two separate fractions or multiply out \((2x^4-8)\times\frac{1}{5}x^{-\frac{1}{2}}\) |
| \(\frac{dy}{dx} = \ldots x^1 + \ldots x^{-2} + \ldots x^{\frac{5}{2}} + \ldots x^{-\frac{3}{2}}\) | M1 | Reduces a correct power by 1 on any term \(x^n \to x^{n-1}\); indices must have been processed |
| \(\frac{dy}{dx} = 10x - \frac{1}{2}x^{-2} + \frac{7}{5}x^{\frac{5}{2}} + \frac{4}{5}x^{-\frac{3}{2}}\) | A1A1A1A1 | A1 any one term correct; A1 any two terms correct; A1 any three terms correct; A1 all four terms correct and fully simplified on one line. Accept \(-\frac{1}{2x^2}\), \(+\frac{7\sqrt{x^5}}{5}\), \(+\frac{4}{5x^{\frac{3}{2}}}\), \(0.8x^{-\frac{3}{2}}\) but NOT e.g. \(10x^1\). Beware \(\frac{1}{2x}\to 2x^{-1}\to -2x^{-2}\to -\frac{1}{2x^2}\) is M1A0 |
## Question 4:
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{2x^4-8}{5\sqrt{x}} = \frac{2x^4}{5\sqrt{x}} - \frac{8}{5\sqrt{x}} = \frac{2}{5}x^{\frac{7}{2}} - \frac{8}{5}x^{-\frac{1}{2}}$ | M1 | Attempts to split into two terms where one is of the form $px^{\frac{7}{2}}$ or $qx^{-\frac{1}{2}}$; may write as two separate fractions or multiply out $(2x^4-8)\times\frac{1}{5}x^{-\frac{1}{2}}$ |
| $\frac{dy}{dx} = \ldots x^1 + \ldots x^{-2} + \ldots x^{\frac{5}{2}} + \ldots x^{-\frac{3}{2}}$ | M1 | Reduces a correct power by 1 on any term $x^n \to x^{n-1}$; indices must have been processed |
| $\frac{dy}{dx} = 10x - \frac{1}{2}x^{-2} + \frac{7}{5}x^{\frac{5}{2}} + \frac{4}{5}x^{-\frac{3}{2}}$ | A1A1A1A1 | A1 any one term correct; A1 any two terms correct; A1 any three terms correct; A1 all four terms correct and fully simplified on one line. Accept $-\frac{1}{2x^2}$, $+\frac{7\sqrt{x^5}}{5}$, $+\frac{4}{5x^{\frac{3}{2}}}$, $0.8x^{-\frac{3}{2}}$ but NOT e.g. $10x^1$. Beware $\frac{1}{2x}\to 2x^{-1}\to -2x^{-2}\to -\frac{1}{2x^2}$ is M1A0 |
---4. Given that
$$y = 5 x ^ { 2 } + \frac { 1 } { 2 x } + \frac { 2 x ^ { 4 } - 8 } { 5 \sqrt { x } } \quad x > 0$$
find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving each term in its simplest form.\\
(6)
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HAVI SIHI NI JINM ION OC & VIIV SIHI NI JINAM ION OA & VIUV SIHI NI JIIIM ION OC \\
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\hfill \mbox{\textit{Edexcel C12 2019 Q4 [6]}}