Question 8 - A-Level Maths

Edexcel C1 2013 January — Question 8 6 marks

Exam Board	Edexcel
Module	C1 (Core Mathematics 1)
Year	2013
Session	January
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Standard Integrals and Reverse Chain Rule
Type	Find curve equation from derivative (straightforward integration + point)
Difficulty	Moderate -0.8 This is a straightforward integration question requiring standard power rule techniques and finding a constant using initial conditions. The algebraic manipulation to simplify the fraction before integrating is routine, and all steps follow standard C1 procedures with no problem-solving insight needed.
Spec	1.08a Fundamental theorem of calculus: integration as reverse of differentiation 1.08b Integrate x^n: where n != -1 and sums

8. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 3 } + \frac { 4 x - 5 } { 2 x ^ { 3 } } , \quad x \neq 0$$ Given that $y = 7$ at $x = 1$, find $y$ in terms of $x$, giving each term in its simplest form.

Show mark scheme Show mark scheme source

Question 8:

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
$\frac{dy}{dx} = -x^3 + "2"x^{-2} - "\frac{5}{2}"x^{-3}$	M1	Expresses as three term polynomial with powers 3, -2 and -3
$y = -\frac{1}{4}x^4 + \frac{"2"x^{-1}}{(-1)} - "\frac{5}{2}"\frac{x^{-2}}{(-2)} (+c)$	M1	Attempt to integrate at least one term, $x^n \to x^{n+1}$
Any two integration terms correct (unsimplified)	A1ft	Any two follow through terms correct
$y = -\frac{1}{4}x^4 + \frac{2x^{-1}}{(-1)} - \frac{5}{2}\frac{x^{-2}}{(-2)} (+c)$	A1	Correct three terms, coefficients may be unsimplified
Given $y=7$ at $x=1$: $7 = -\frac{1}{4} - 2 + \frac{5}{4} + c \Rightarrow c=$	M1	Need constant; uses $y=7$ and $x=1$
$y = -\frac{1}{4}x^4 - 2x^{-1} + \frac{5}{4}x^{-2} + 8$, $c=8$	A1	All four terms simplified, $c=8$

Total: 6 marks

## Question 8:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = -x^3 + "2"x^{-2} - "\frac{5}{2}"x^{-3}$ | M1 | Expresses as three term polynomial with powers 3, -2 and -3 |
| $y = -\frac{1}{4}x^4 + \frac{"2"x^{-1}}{(-1)} - "\frac{5}{2}"\frac{x^{-2}}{(-2)} (+c)$ | M1 | Attempt to integrate at least one term, $x^n \to x^{n+1}$ |
| Any two integration terms correct (unsimplified) | A1ft | Any two follow through terms correct |
| $y = -\frac{1}{4}x^4 + \frac{2x^{-1}}{(-1)} - \frac{5}{2}\frac{x^{-2}}{(-2)} (+c)$ | A1 | Correct three terms, coefficients may be unsimplified |
| Given $y=7$ at $x=1$: $7 = -\frac{1}{4} - 2 + \frac{5}{4} + c \Rightarrow c=$ | M1 | Need constant; uses $y=7$ and $x=1$ |
| $y = -\frac{1}{4}x^4 - 2x^{-1} + \frac{5}{4}x^{-2} + 8$, $c=8$ | A1 | All four terms simplified, $c=8$ |

**Total: 6 marks**

---

Show LaTeX source

8.

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = - x ^ { 3 } + \frac { 4 x - 5 } { 2 x ^ { 3 } } , \quad x \neq 0$$

Given that $y = 7$ at $x = 1$, find $y$ in terms of $x$, giving each term in its simplest form.\\

\hfill \mbox{\textit{Edexcel C1 2013 Q8 [6]}}

This paper (11 questions)

View full paper

Q1 3 Q2 2 Q3 6 Q4 5 Q5 7 Q6 12 Q7 8 Q8 6 Q9 7 Q10 7 Q11 12