| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Show derivative equals given algebraic form |
| Difficulty | Standard +0.3 This is a straightforward quotient rule application with trigonometric functions. Part (i) requires applying the quotient rule to sin(2x)/(cos x + 1), using the chain rule for sin(2x), then algebraic manipulation with standard trig identities to reach the given form. Part (ii) is routine solving of a quadratic equation in cos x. While it involves multiple steps, all techniques are standard for P2 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use quotient rule or equivalent to find first derivative | M1 | |
| Obtain \(\frac{2\cos 2x(\cos x+1)+\sin 2x\sin x}{(\cos x+1)^2}\) or equivalent | A1 | |
| Use at least one of \(\cos 2x=2\cos^2 x-1\) and \(2x=2\sin x\cos x\) | B1 | |
| Express first derivative in terms of \(\cos x\) only | M1 | |
| Obtain \(\frac{2\cos^3 x+4\cos^2 x-2}{(\cos x+1)^2}\) or equivalent | A1 | |
| Factorise numerator or divide numerator by \((\cos x+1)\) or equivalent | M1 | |
| Confirm given answer \(\frac{2(\cos^2 x+\cos x-1)}{\cos x+1}\) correctly | A1 | [7] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use quadratic formula or equivalent to find value of \(\cos x\) | M1 | |
| Obtain \(x\)-coordinate \(0.905\) | A1 | |
| Obtain \(x\)-coordinate \(-0.905\) and no others in range | A1 | [3] |
## Question 7:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use quotient rule or equivalent to find first derivative | M1 | |
| Obtain $\frac{2\cos 2x(\cos x+1)+\sin 2x\sin x}{(\cos x+1)^2}$ or equivalent | A1 | |
| Use at least one of $\cos 2x=2\cos^2 x-1$ and $2x=2\sin x\cos x$ | B1 | |
| Express first derivative in terms of $\cos x$ only | M1 | |
| Obtain $\frac{2\cos^3 x+4\cos^2 x-2}{(\cos x+1)^2}$ or equivalent | A1 | |
| Factorise numerator or divide numerator by $(\cos x+1)$ or equivalent | M1 | |
| Confirm given answer $\frac{2(\cos^2 x+\cos x-1)}{\cos x+1}$ correctly | A1 | [7] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use quadratic formula or equivalent to find value of $\cos x$ | M1 | |
| Obtain $x$-coordinate $0.905$ | A1 | |
| Obtain $x$-coordinate $-0.905$ and no others in range | A1 | [3] |7 The equation of a curve is $y = \frac { \sin 2 x } { \cos x + 1 }$.\\
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \left( \cos ^ { 2 } x + \cos x - 1 \right) } { \cos x + 1 }$.\\
(ii) Find the $x$-coordinate of each stationary point of the curve in the interval $- \pi < x < \pi$. Give each answer correct to 3 significant figures.
\hfill \mbox{\textit{CAIE P2 2015 Q7 [10]}}