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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 exex2.

(b)     The random variable X has a Poisson distribution with mean μ. Show that P(X1(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]

Examiners report

[N/A]

Syllabus sections

Topic 5 - Calculus » 5.6 » Maclaurin series for {{\text{e}}^x} , \sin x , \cos x , \ln (1 + x) , {(1 + x)^p} , P \in \mathbb{Q} .

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