Question 6 - A-Level Maths

CAIE FP2 2010 June — Question 6 5 marks

Exam Board	CAIE
Module	FP2 (Further Pure Mathematics 2)
Year	2010
Session	June
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Normal Distribution
Type	Linear transformation of normal
Difficulty	Standard +0.3 This is a straightforward lognormal distribution question requiring only basic transformations and standard normal table lookups. Part (i) uses the property that median of log X equals mean (immediate), and part (ii) requires one standardization calculation. The conceptual step of recognizing the lognormal setup is standard for Further Maths students, making this slightly easier than average overall.
Spec	5.03g Cdf of transformed variables

The lifetime, \(X\) days, of a particular insect is such that \(\log_{10} X\) has a normal distribution with mean \(1.5\) and standard deviation \(0.2\). Find the median lifetime. [3] Find also P\((X \geqslant 50)\). [2]

Show mark scheme Show mark scheme source

Question 6:

Answer	Marks
6	Find relation for median M: log10 M = 1⋅5 M1 A1

Evaluate M: M = 10 1⋅5 = 31⋅6 A1

Relate P(X ≥ 50) to Normal distribution: P(X ≥ 50) = P(log X ≥ log 50)

= 1 – Φ((log 50 – 1⋅5) / 0⋅2) M1

Answer	Marks
[log 50 = 1⋅699] = 1 – Φ(0⋅995) = 0⋅160 A1	3
2	[5]

Question 6:
6 | Find relation for median M: log10 M = 1⋅5 M1 A1
Evaluate M: M = 10 1⋅5 = 31⋅6 A1
Relate P(X ≥ 50) to Normal distribution: P(X ≥ 50) = P(log X ≥ log 50)
= 1 – Φ((log 50 – 1⋅5) / 0⋅2) M1
[log 50 = 1⋅699] = 1 – Φ(0⋅995) = 0⋅160 A1 | 3
2 | [5]

Show LaTeX source

The lifetime, $X$ days, of a particular insect is such that $\log_{10} X$ has a normal distribution with mean $1.5$ and standard deviation $0.2$. Find the median lifetime. [3]

Find also P$(X \geqslant 50)$. [2]

\hfill \mbox{\textit{CAIE FP2 2010 Q6 [5]}}

This paper (11 questions)

View full paper

Q1 5 Q2 7 Q3 9 Q4 9 Q5 13 Q6 5 Q7 7 Q8 9 Q9 9 Q10 13 Q11 28