Question 4 - A-Level Maths

OCR C4 2012 June — Question 4 6 marks

Exam Board	OCR
Module	C4 (Core Mathematics 4)
Year	2012
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Differential equations
Type	Separable variables - standard (polynomial/exponential x-side)
Difficulty	Moderate -0.3 This is a straightforward separable variables question requiring standard integration techniques (exponential and tangent). The separation is immediate, both integrals are routine A-level results, and applying the initial condition to find the constant is mechanical. Slightly easier than average due to its directness and lack of algebraic manipulation.
Spec	1.08k Separable differential equations: dy/dx = f(x)g(y)

4 Solve the differential equation $$\mathrm { e } ^ { 2 y } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \tan x = 0 ,$$ given that $x = 0$ when $y = 0$. Give your answer in the form $y = \mathrm { f } ( x )$.

Show mark scheme Show mark scheme source

Question 4:

Answer	Marks	Guidance
Answer	Marks	Guidance
$+/-\int e^{2y}\,dy$ and $+/-\int\tan x\,dx$ seen	M1	may be implied later
$\int e^{2y}\,dy = \frac{1}{2}e^{2y}$	B1
\(\int\tan x\,dx = \ln	\sec x	\) or \(-\ln
Subst $x=0$, $y=0$ into equation containing $f(x)$, $g(y)$ and $c$	M1	S.R. Using def integrals: M1 $\int_0^x = \int_0^y$ followed by A2 or A0
$c = \frac{1}{2}$ WWW (or possibly $-\frac{1}{2}$ if $c$ on LHS)	A1
\(y = \frac{1}{2}\ln(1-2\ln	\sec x	)\) or \(\frac{1}{2}\ln(1+2\ln
Total: [6]

# Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $+/-\int e^{2y}\,dy$ and $+/-\int\tan x\,dx$ seen | M1 | may be implied later |
| $\int e^{2y}\,dy = \frac{1}{2}e^{2y}$ | B1 | |
| $\int\tan x\,dx = \ln|\sec x|$ or $-\ln|\cos x|$ | B1 | Accept $\ln\sec x$ or $-\ln\cos x$ |
| Subst $x=0$, $y=0$ into equation containing $f(x)$, $g(y)$ and $c$ | M1 | S.R. Using def integrals: M1 $\int_0^x = \int_0^y$ followed by A2 or A0 |
| $c = \frac{1}{2}$ **WWW** (or possibly $-\frac{1}{2}$ if $c$ on LHS) | A1 | |
| $y = \frac{1}{2}\ln(1-2\ln|\sec x|)$ or $\frac{1}{2}\ln(1+2\ln|\cos x|)$ oe WWW | A1 | Accept omission of modulus |
| **Total: [6]** | | |

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4 Solve the differential equation

$$\mathrm { e } ^ { 2 y } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \tan x = 0 ,$$

given that $x = 0$ when $y = 0$. Give your answer in the form $y = \mathrm { f } ( x )$.

\hfill \mbox{\textit{OCR C4 2012 Q4 [6]}}

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Q1 6 Q2 5 Q3 7 Q4 6 Q5 5 Q6 7 Q7 7 Q8 10 Q9 9 Q10 10