9 A skydiver drops from a helicopter. Before she opens her parachute, her speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after time \(t\) seconds is modelled by the differential equation $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 10 \mathrm { e } ^ { - \frac { 1 } { 2 } t }$$ When \(t = 0 , v = 0\).

1. Find \(v\) in terms of \(t\).
2. According to this model, what is the speed of the skydiver in the long term? She opens her parachute when her speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Her speed \(t\) seconds after this is \(w \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and is modelled by the differential equation $$\frac { \mathrm { d } w } { \mathrm {~d} t } = - \frac { 1 } { 2 } ( w - 4 ) ( w + 5 )$$
3. Express \(\frac { 1 } { ( w - 4 ) ( w + 5 ) }\) in partial fractions.
4. Using this result, show that \(\frac { w - 4 } { w + 5 } = 0.4 \mathrm { e } ^ { - 4.5 t }\).
5. According to this model, what is the speed of the skydiver in the long term? RECOGNISING ACHIEVEMENT \section*{ADVANCED GCE} \section*{4754/01B} \section*{MATHEMATICS (MEI)} Applications of Advanced Mathematics (C4) Paper B: Comprehension
   WEDNESDAY 21 MAY 2008
   Afternoon
   Time: Up to 1 hour
   Additional materials: Rough paper
   MEI Examination Formulae and Tables (MF 2) \section*{Candidate Forename}
   \includegraphics[max width=\textwidth, alt={}]{8ad99e2a-4cef-40b3-af8d-673b97536227-05_125_547_986_516}
   This document consists of \(\mathbf { 6 }\) printed pages, \(\mathbf { 2 }\) blank pages and an insert. 1 Complete these Latin square puzzles.

6. 2	1	3
   3

7. 
   \includegraphics[max width=\textwidth, alt={}, center]{8ad99e2a-4cef-40b3-af8d-673b97536227-06_391_419_836_854} 2 In line 51, the text says that the Latin square

   1	2	3	4
   3	1	4	2
   2	4	1	3
   4	3	2	1

   could not be the solution to a Sudoku puzzle.
   Explain this briefly.
   3 On lines 114 and 115 the text says "It turns out that there are 16 different ways of filling in the remaining cells while keeping to the Sudoku rules. One of these ways is shown in Fig.10." Complete the grid below with a solution different from that given in Fig. 10.

   1	2	3	4

   4 Lines 154 and 155 of the article read "There are three other embedded Latin squares in Fig. 14; one of them is illustrated in Fig. 16." Indicate one of the other two embedded Latin squares on this copy of Fig. 14.

   4	2	3	1
   		2	4
   		4	2
   2	4	1	3

   5 The number of \(9 \times 9\) Sudokus is given in line 121 .
   Without doing any calculations, explain why you would expect 9! to be a factor of this number.
   6 In the table below, \(M\) represents the maximum number of givens for which a Sudoku puzzle may have no unique solution (Investigation 3 in the article). \(s\) is the side length of the Sudoku grid and \(b\) is the side length of its blocks.

   Block side length, \(b\)	Sudoku, \(s \times s\)	\(M\)
   1	\(1 \times 1\)	-
   2	\(4 \times 4\)	12
   3	\(9 \times 9\)
   4	\(16 \times 16\)
   5

8. Complete the table.
9. Give a formula for \(M\) in terms of \(b\).
   7 A man is setting a Sudoku puzzle and starts with this solution.

   1	2	3	4	5	6	7	8	9
   4	5	6	8	9	7	3	1	2
   7	8	9	3	1	2	5	6	4
   2	3	1	5	6	4	8	9	7
   5	6	4	9	7	8	1	2	3
   8	9	7	1	2	3	6	4	5
   3	1	2	6	4	5	9	7	8
   6	4	5	7	8	9	2	3	1
   9	7	8	2	3	1	4	5	6

   He then removes some of the numbers to give the puzzles in parts (i) and (ii). In each case explain briefly, and without trying to solve the puzzle, why it does not have a unique solution.
   [0pt] [2,2]

10. 1	2		4		6			9
    4			8	9			1
    	8						6
    2		1			4			7
    	6	4		7	8	1	2
    8	9			2			4
    	1		6	4		9	7
    6	4		7		9			1
    9		8	2		1	4		6

11. 1	2	3	4	5	6	7	8	9
    4	5	6	8	9	7	3	1	2
    7	8	9				5	6	4
    2	3	1	5	6	4	8	9	7
    5	6	4	9	7	8	1	2	3
    8	9	7				6	4	5
    3	1	2	6	4	5	9	7	8
    6	4	5	7	8	9	2	3	1
    9	7	8				4	5	6

12. \(\_\_\_\_\)
13. \(\_\_\_\_\)