| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration using inverse trig and hyperbolic functions |
| Type | Derivative of inverse trig function |
| Difficulty | Moderate -0.5 Part (i) is a standard bookwork proof requiring implicit differentiation of tan(y)=x, which is routine for FP2 students. Part (ii) involves straightforward differentiation of the result from (i) and algebraic verification. While this is Further Maths content, both parts follow standard techniques with no problem-solving insight required, making it easier than average overall. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07l Derivative of ln(x): and related functions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Get \(\sec^2 y \frac{dy}{dx} = 1\) or equivalent | M1 | Clearly use \(1 + \tan^2 y = \sec^2 y\) |
| Clearly arrive at A.G. | A1 | |
| (ii) Reasonable attempt to diff. to \(\frac{-2x}{(1+x^2)^2}\) | M1 | Use of chain/quotient rule |
| Substitute their expressions into D.E. | M1 | Or attempt to derive diff. equ'n. |
| Clearly arrive at A.G. | A1 | Attempt diff. of \((1+x^2)\frac{dy}{dx} = 1\) |
| SC | Clearly arrive at A.G. | B1 |
**(i)** Get $\sec^2 y \frac{dy}{dx} = 1$ or equivalent | M1 | Clearly use $1 + \tan^2 y = \sec^2 y$
Clearly arrive at A.G. | A1 |
**(ii)** Reasonable attempt to diff. to $\frac{-2x}{(1+x^2)^2}$ | M1 | Use of chain/quotient rule
Substitute their expressions into D.E. | M1 | Or attempt to derive diff. equ'n.
Clearly arrive at A.G. | A1 | Attempt diff. of $(1+x^2)\frac{dy}{dx} = 1$
| SC | Clearly arrive at A.G. | B12 (i) Given that $y = \tan ^ { - 1 } x$, prove that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }$.\\
(ii) Verify that $y = \tan ^ { - 1 } x$ satisfies the equation
$$\left( 1 + x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 0$$
\hfill \mbox{\textit{OCR FP2 2006 Q2 [6]}}