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1.5.10 Partial fractions (A-level only)

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Question 1

A physical experiment measures the spectral intensity III of a pulsed laser relative to its frequency shift sss. The measurements are modeled by the parametric equations

s=5+4sin⁡t s = 5 + 4 \sin t s=5+4sint I=187+cos⁡2t I = \frac{18}{7 + \cos 2t} I=7+cos2t18​

where −π2≤t≤π2-\frac{\pi}{2} \le t \le \frac{\pi}{2}−2π​≤t≤2π​.

a.

Show that the relationship between intensity and frequency shift can be expressed as

I=144(13−s)(s+3)p≤s≤q I = \frac{144}{(13 - s)(s + 3)} \quad p \le s \le q I=(13−s)(s+3)144​p≤s≤q

where ppp and qqq are constants to be found.

[5]
b.

Hence, determine a Cartesian equation for this relationship in the form

I=as+b+cs+dp≤s≤q I = \frac{a}{s + b} + \frac{c}{s + d} \quad p \le s \le q I=s+ba​+s+dc​p≤s≤q

where a,b,ca, b, ca,b,c and ddd are constants.

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1.5.10 Partial fractions (A-level only) Questions

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