People are given a maximum of six hours to complete a puzzle. The time spent on the puzzle, in hours, can be modelled using the continuous random variable which has probability density function

The graph of the probability density function is shown below. The graph has a local maximum.
 

1. Show that  .  (2 marks)
   

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2. Show that the mode of is two hours.  (2 marks)
   

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3. Show that  .  (2 marks)
   

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4. The Intelligence Quotient (IQ) scores of people are normally distributed with a mean of 100 and standard deviation of 15.
5. It has been observed that the puzzle is generally completed more quickly by people with a high IQ.
6. It is known that 80% of people with an IQ greater than 130 can complete the puzzle in less than two hours.
7. A person chosen at random can complete the puzzle in less than two hours.
8. What is the probability that this person has an IQ greater than 130? Give your answer correct to three decimal places.  (2 marks)
   

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