| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/1 (Pre-U Further Mathematics Paper 1) |
| Session | Specimen |
| Marks | 6 |
| Topic | Integration by Parts |
| Type | Reduction formula or recurrence |
| Difficulty | Challenging +1.2 This is a standard reduction formula question requiring integration by parts to establish the recurrence relation, followed by a straightforward induction proof. While it involves multiple techniques (integration by parts, induction, and inequality manipulation), each step follows a well-established pattern that Further Maths students practice regularly. The induction in part (ii) is relatively routine once the reduction formula is established. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps4.08b Standard Maclaurin series: e^x, sin, cos, ln(1+x), (1+x)^n |
**(i)** $I_n = \left[-t^n e^{-t}\right]_0^1 + n\int_0^1 t^{n-1}e^{-t}\, dt$ M1A1
$I_n = nI_{n-1} - e^{-1}$ (AG) A1 **[3]**
**(ii)** $H_n: I_n < n!$
$I_1 = 1 - 2e^{-1} < 1! \Rightarrow H_1$ is true B1
$H_k: I_k < k!$ for some $k$ B1
$I_{k+1} = (k+1)I_k - e^{-1} \Rightarrow I_{k+1} < (k+1)! - e^{-1}$ M1A1
$\Rightarrow I_{k+1} < (k+1)!$ (since $e^{-1} > 0$) A1
Hence $H_k \Rightarrow H_{k+1}$, so by PMI $H_n$ is true $\forall$ positive integers $n$ A1
(Final A1 requires all previous marks and full statement of conclusion) **[6]**5 Let
$$I _ { n } = \int _ { 0 } ^ { 1 } t ^ { n } \mathrm { e } ^ { - t } \mathrm {~d} t$$
where $n \geqslant 0$.\\
(i) Show that, for all $n \geqslant 1$,
$$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
(ii) Hence prove by induction that, for all positive integers $n$,
$$I _ { n } < n ! .$$
\hfill \mbox{\textit{Pre-U Pre-U 9795/1 Q5 [6]}}