CAIE A-Level Further Math A2 4.1.3 Pdf and Cdf Questions

Practise deriving probability density and cumulative distribution functions, medians, percentiles and interquartile ranges with Further Mathematics Paper 2 questions and mark schemes.

Exam points

Apply de Moivre's theorem to powers and roots of complex numbers.

[Maximum number: 5]

Figure for Question CAIE A-Level Further Math A2 4.1.3 Pdf and Cdf Questions question 1 — CAIE A-Level Further Math A2

As shown in the diagram, the continuous random variable X has probability density function f given by

f(x)= \begin{cases}m x & 0 \leqslant x \leqslant 2 \\ \frac{k}{x^{2}}+c & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise }\end{cases}

where m, k and c are constants.

Find the exact numerical value of the interquartile range of X.

LQ: $\int_{0}^{L} \frac{1}{6} x \mathrm{~d} x=\frac{1}{4}, \quad L=\sqrt{3}$
UQ: $\frac{1}{3}+\int_{2}^{U}\left(\frac{k}{x^{2}}+c\right) \mathrm{d} x=\frac{3}{4}$
Or find and use CDF. May be in terms of m.

$\begin{aligned} -\frac{k}{U}+U c+\frac{k}{2}-2 c=\frac{5}{12} U^{2}-U-12=0 \end{aligned}$
Attempt at integral and correct use of correct limits.
Simplify to quadratic equation, may be in terms of k and
c.

U=4
$\mathrm{IQR}=4-\sqrt{3}$

A1 FT

FT their UQ - their LQ, exact values only.

5

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CAIE A-Level Further Math A2 4.1.3 Pdf and Cdf Questions question 1