Date | May 2014 | Marks available | 7 | Reference code | 14M.1.hl.TZ0.5 |
Level | HL only | Paper | 1 | Time zone | TZ0 |
Command term | Determine | Question number | 5 | Adapted from | N/A |
Question
(a) Assuming the Maclaurin series for ex, determine the first three non-zero terms in the Maclaurin expansion of ex−e−x2.
(b) The random variable X has a Poisson distribution with mean μ. Show that P(X≡1(mod where a, b and c are constants whose values are to be found.
Markscheme
(a) {{\text{e}}^x} = 1 + x + \frac{{{x^2}}}{{2!}} + \frac{{{x^3}}}{{3!}} + \frac{{{x^4}}}{{4!}} + \frac{{{x^5}}}{{5!}} + \ldots
{{\text{e}}^{ - x}} = 1 - x + \frac{{{x^2}}}{{2!}} - \frac{{{x^3}}}{{3!}} + \frac{{{x^4}}}{{4!}} - \frac{{{x^5}}}{{5!}} + \ldots A1
\frac{{{{\text{e}}^x} - {{\text{e}}^{ - x}}}}{2} = x + \frac{{{x^3}}}{{3!}} + \frac{{{x^5}}}{{5!}} + \ldots (M1)A1
Note: Accept any valid (otherwise) method.
[3 marks]
(b) {\text{P}}\left( {X \equiv 1(\bmod 2)} \right) = {\text{P}}(X = 1,{\text{ }}3,{\text{ }}5,{\text{ }} \ldots ) (M1)
= {{\text{e}}^{ - \mu }}\left( {\mu + \frac{{{\mu ^3}}}{{3!}} + \frac{{{\mu ^5}}}{{5!}} + \ldots } \right) A1
= \frac{{{{\text{e}}^{ - \mu }}({{\text{e}}^\mu } - {{\text{e}}^{ - \mu }})}}{2} A1
= \frac{1}{2} - \frac{1}{2}{{\text{e}}^{ - 2\mu }} A1
\left( {a = \frac{1}{2},{\text{ }}b = - \frac{1}{2},{\text{ }}c = - 2} \right)
[4 marks]