(A) Police officers are in pursuit of a stolen vehicle. Officer Baker is directly behind the stolen car. Officer Lopez is behind Baker; Officer O'Malley is behind Lopez. Officer Reinhart is ahead of the stolen car and coming from the opposite direction. Officer Reinhart makes a U-turn and joins the pursuit. He pulls in behind Officer Lopez. Officer Baker pulls up on the driver's side of the stolen vehicle and Officer Lopez pulls up on the other side. Which officer is directly behind the vehicle? (B) Define a Histogram. How is it constructed? What information may be deduced from it? (C) Look at this series: F2, __, D8, C16, B32. What number should fill the blank? (D) In a certain code language LANDMINE is written as PYRBQGRC. How will HOMEMADE be written in that code language?

(A) Police pursuit

Start with the order from front to back: stolen car, Baker, Lopez, O'Malley.

Reinhart makes a U-turn and pulls in behind Lopez, in front of O'Malley. The order becomes: stolen car, Baker, Lopez, Reinhart, O'Malley.

Baker then moves up on the driver's side, and Lopez on the other side. Both are now alongside the stolen car, so the next vehicle in line is directly behind it.

Officer Reinhart is directly behind the stolen vehicle.

(B) Histogram

Definition: a histogram is a graph of the frequency distribution of continuous or grouped numerical data. It uses adjacent bars, whose area shows the frequency of each class interval. Unlike a bar chart, its bars touch each other, since the classes are continuous.

Construction

(i) find the range of the data, and divide it into suitable class intervals; (ii) count the frequency of values in each class and prepare a frequency table; (iii) mark the classes on the horizontal axis and the frequency on the vertical axis, with suitable scales; (iv) draw a bar over each class, with its height showing the frequency and no gaps between bars; (v) add a title and axis labels. If classes are of unequal width, heights must be adjusted so that area remains proportional to frequency.

Information deduced

Shape: whether the distribution is symmetric or skewed, and whether it has one peak or several. Centre: roughly where the data cluster, and the modal class, shown by the tallest bar. Spread: how widely the data vary. Unusual values: gaps and outliers. Comparisons: how frequently values occur in one interval compared with another.

(C) Series

The letters move back by one each time: F, E, D, C, B. The numbers double each time: 2, 4, 8, 16, 32.

The missing term is E4, so the number that fills the blank is 4.

(D) Coding

Finding the pattern: comparing LANDMINE with PYRBQGRC letter by letter gives:

L→P (+4), A→Y (−2), N→R (+4), D→B (−2), M→Q (+4), I→G (−2), N→R (+4), E→C (−2).

The rule therefore alternates +4 and −2, with A→Y wrapping around the alphabet.

Applying the pattern to HOMEMADE

H→L (+4), O→M (−2), M→Q (+4), E→C (−2), M→Q (+4), A→Y (−2), D→H (+4), E→C (−2).

HOMEMADE is written as LMQCQYHC.