| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Tangent parallel to given line |
| Difficulty | Easy -1.2 This is a routine C1 differentiation question requiring algebraic manipulation to express in index form, then applying standard power rule differentiation. Part (c) involves basic tangent gradient comparison. All steps are mechanical with no problem-solving insight needed, making it easier than average. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations |
| 6. continued | Leave blank |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{(2x+1)(x+4)}{\sqrt{x}} = \frac{2x^2+9x+4}{\sqrt{x}} = 2x^{\frac{3}{2}}+9x^{\frac{1}{2}}+4x^{-\frac{1}{2}}\) | M1 A2(1,0) | \([P=2, Q=9, R=4]\) — 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(f'(x) = 3x^{\frac{1}{2}}+\frac{9}{2}x^{-\frac{1}{2}}-2x^{-\frac{3}{2}}\) | M1 A1 ft A1 | A1 ft for one term with fractional power — 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Gradient of tangent \(= f'(1) = 3 + \frac{9}{2} - 2 = \frac{11}{2}\) | M1 A1 ft | |
| Gradient of line \(= \frac{11}{2}\), equal gradients \(\therefore\) parallel | A1 | 3 marks |
## Question 6:
### Part (a):
$\frac{(2x+1)(x+4)}{\sqrt{x}} = \frac{2x^2+9x+4}{\sqrt{x}} = 2x^{\frac{3}{2}}+9x^{\frac{1}{2}}+4x^{-\frac{1}{2}}$ | M1 A2(1,0) | $[P=2, Q=9, R=4]$ — **3 marks**
### Part (b):
$f'(x) = 3x^{\frac{1}{2}}+\frac{9}{2}x^{-\frac{1}{2}}-2x^{-\frac{3}{2}}$ | M1 A1 ft A1 | A1 ft for one term with fractional power — **3 marks**
### Part (c):
Gradient of tangent $= f'(1) = 3 + \frac{9}{2} - 2 = \frac{11}{2}$ | M1 A1 ft |
Gradient of line $= \frac{11}{2}$, equal gradients $\therefore$ parallel | A1 | **3 marks**
**Total: 9 marks**
---6.
$$f ( x ) = \frac { ( 2 x + 1 ) ( x + 4 ) } { \sqrt { x } } , \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { f } ( x )$ can be written in the form $P x ^ { \frac { 3 } { 2 } } + Q x ^ { \frac { 1 } { 2 } } + R x ^ { - \frac { 1 } { 2 } }$, stating the values of the constants $P , Q$ and $R$.
\item Find f ${ } ^ { \prime } ( x )$.
\item Show that the tangent to the curve with equation $y = \mathrm { f } ( x )$ at the point where $x = 1$ is parallel to the line with equation $2 y = 11 x + 3$.\\
(3)\\
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\hfill \mbox{\textit{Edexcel C1 Q6 [9]}}