| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Session | Specimen |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration using inverse trig and hyperbolic functions |
| Type | Derivative of inverse hyperbolic function |
| Difficulty | Standard +0.8 This is a Further Maths question requiring knowledge of inverse hyperbolic functions. Part (i) is a standard 'show that' derivative using the chain rule and hyperbolic identities. Part (ii) requires completing the square and recognizing the inverse sinh form, which is non-trivial but follows a standard technique for this topic. The question tests multiple steps and pattern recognition beyond standard A-level, placing it moderately above average difficulty. |
| Spec | 4.07d Differentiate/integrate: hyperbolic functions4.07e Inverse hyperbolic: definitions, domains, ranges4.08h Integration: inverse trig/hyperbolic substitutions |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (i) | d (cid:11) (cid:16)1(cid:11)2x(cid:12) (cid:12) |
| Answer | Marks |
|---|---|
| 2 | e |
| Answer | Marks |
|---|---|
| [2] | c |
| Answer | Marks |
|---|---|
| 2.1 | Use formulae from formulae booklet |
| Answer | Marks |
|---|---|
| be seen | OR |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (ii) | 2(cid:16)2x(cid:14)x2 (cid:32)1(cid:14)(cid:11)x(cid:16)1(cid:12)2 |
| Answer | Marks |
|---|---|
| (cid:182) | B1 |
| Answer | Marks |
|---|---|
| [3] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | Complete square soi |
| Answer | Marks |
|---|---|
| 5(ii) | e |
Question 5:
5 | (i) | d (cid:11) (cid:16)1(cid:11)2x(cid:12) (cid:12)
sinh
dx
p
1
=
x2 (cid:14)(1)2
2 S
1 2 2
(cid:32) (cid:117) (cid:32)
x2 (cid:14)(1)2 2 4x2 (cid:14)1
2 | e
M1
E1
[2] | c
1.1a
2.1 | Use formulae from formulae booklet
AG At least one intermediate step must
be seen | OR
M1 sinhy(cid:32)2x
dy
so coshy(cid:32)2
dx
dy 2 2
E1 (cid:32) (cid:32)
dx coshy 4x2 (cid:14)1
5 | (ii) | 2(cid:16)2x(cid:14)x2 (cid:32)1(cid:14)(cid:11)x(cid:16)1(cid:12)2
(cid:180) 1
(cid:159)(cid:181) dx(cid:32)(cid:170)sinh(cid:16)1(cid:11)x(cid:16)1(cid:12)(cid:186)(cid:14)c
(cid:181) (cid:172) (cid:188)
1(cid:14)(cid:11)x(cid:16)1(cid:12)2
(cid:182) | B1
M1
A1
[3] | 3.1a
1.1
1.1 | Complete square soi
Use correct form
oe
--- 5(i) ---
5(i)
n
e
m
i
c
--- 5(ii) ---
5(ii) | e
p
S5 (i) Show that $\frac { \mathrm { d } } { \mathrm { d } x } \left( \sinh ^ { - 1 } ( 2 x ) \right) = \frac { 2 } { \sqrt { 4 x ^ { 2 } + 1 } }$.\\
(ii) Find $\int \frac { 1 } { \sqrt { 2 - 2 x + x ^ { 2 } } } \mathrm {~d} x$.
\hfill \mbox{\textit{OCR Further Pure Core 1 Q5 [5]}}