Question 7 - A-Level Maths

AQA C4 2006 June — Question 7 6 marks

Exam Board	AQA
Module	C4 (Core Mathematics 4)
Year	2006
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Differential equations
Type	Separable variables - standard (polynomial/exponential x-side)
Difficulty	Moderate -0.8 This is a straightforward separable variables question requiring only standard technique: separate variables, integrate both sides (giving -1/y = 3x² + c), then apply the initial condition to find c and rearrange for y. It's routine C4 material with no conceptual challenges, making it easier than average.
Spec	1.08k Separable differential equations: dy/dx = f(x)g(y)

7 Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x y ^ { 2 }$$ given that $y = 1$ when $x = 2$. Give your answer in the form $y = \mathrm { f } ( x )$.

Show mark scheme Show mark scheme source

Answer	Marks	Guidance
$\int\frac{dy}{y^2} = \int 6x\,dx$	M1
$-\frac{1}{y} = 3x^2 (+C)$	A1A1
$x = 2$ $y = 1$ $C = -13$	M1, A1
$y = \frac{1}{13-3x^2}$	A1	6 marks

Question 8(a)(i)

Answer	Marks	Guidance
$(5000-x)$ seen in a product	B1

Question 8(a)(ii)

Answer	Marks	Guidance
$\frac{dx}{dt} = kx(5000-x)$	B1	2 marks

Question 8(b)(i)

Answer	Marks	Guidance
$200 = k \times 1000 \times (5000 - 1000)$	M1
$k = 0.00005$	A1	2 marks

Question 8(b)(ii)

Answer	Marks	Guidance
$t = 4\ln\left(\frac{4 \times 2500}{5000-2500}\right) = 5.5$ (hours)	M1, A1	2 marks

Question 8(b)(iii)

Answer	Marks	Guidance
$e^{\frac{30}{4}}$	B1
$e^{7.5} = \frac{4x}{5000-x}$	M1
$5000 \times e^{7.5} = x(4 + e^{7.5})$	m1
$x = 4988.96... \Rightarrow 4989$ rabbits infected	A1	4 marks
TOTAL		75 marks

$\int\frac{dy}{y^2} = \int 6x\,dx$ | M1 | | Attempt to separate Either $dx$ or $dy$ in right place
$-\frac{1}{y} = 3x^2 (+C)$ | A1A1 | | | $\frac{1}{y}$; $3x^2$
$x = 2$ $y = 1$ $C = -13$ | M1, A1 | | Use $(2,1)$ to find a constant. CAO
$y = \frac{1}{13-3x^2}$ | A1 | 6 marks | CAO OE

## Question 8(a)(i)
$(5000-x)$ seen in a product | B1 | | Could be implied, eg $5000a - xa$

## Question 8(a)(ii)
$\frac{dx}{dt} = kx(5000-x)$ | B1 | 2 marks | $\frac{dx}{dt} = 200, x = 1000$ in their diff. equation Condone $ts$ and $t=0$ for M1 CAO OE

## Question 8(b)(i)
$200 = k \times 1000 \times (5000 - 1000)$ | M1 | |
$k = 0.00005$ | A1 | 2 marks |

## Question 8(b)(ii)
$t = 4\ln\left(\frac{4 \times 2500}{5000-2500}\right) = 5.5$ (hours) | M1, A1 | 2 marks | $x \to 2500$ (or 4 ln 4) CAO

## Question 8(b)(iii)
$e^{\frac{30}{4}}$ | B1 | |
$e^{7.5} = \frac{4x}{5000-x}$ | M1 | | OE
$5000 \times e^{7.5} = x(4 + e^{7.5})$ | m1 | | Solvable for $x$
$x = 4988.96... \Rightarrow 4989$ rabbits infected | A1 | 4 marks | Or 4988 or 4990; integer value only

**TOTAL** | | 75 marks |

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7 Solve the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x y ^ { 2 }$$

given that $y = 1$ when $x = 2$. Give your answer in the form $y = \mathrm { f } ( x )$.

\hfill \mbox{\textit{AQA C4 2006 Q7 [6]}}

This paper (8 questions)

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Q1 8 Q2 8 Q3 8 Q4 9 Q5 14 Q6 12 Q7 6 Q8 10

Answer	Marks	Guidance
\(\int\frac{dy}{y^2} = \int 6x\,dx\)	M1
\(-\frac{1}{y} = 3x^2 (+C)\)	A1A1
\(x = 2\) \(y = 1\) \(C = -13\)	M1, A1
\(y = \frac{1}{13-3x^2}\)	A1	6 marks

Answer	Marks	Guidance
\(200 = k \times 1000 \times (5000 - 1000)\)	M1
\(k = 0.00005\)	A1	2 marks

Answer	Marks	Guidance
\(e^{\frac{30}{4}}\)	B1
\(e^{7.5} = \frac{4x}{5000-x}\)	M1
\(5000 \times e^{7.5} = x(4 + e^{7.5})\)	m1
\(x = 4988.96... \Rightarrow 4989\) rabbits infected	A1	4 marks
TOTAL		75 marks