Differentiate between 'w' and 'a' file modes in Python.
\[ \begin{array}{|c|c|} \hline \textbf{File Mode} & \textbf{Description} \\ \hline 'w' & Opens the file for writing. If the file exists, its content is erased. \\ 'a' & Opens the file for appending. Data is added to the end of the file. \\ \hline \end{array} \]
\[ \begin{array}{|c|c|c|c|} \hline \textbf{S\_id} & \textbf{S\_name} & \textbf{Address} & \textbf{S\_type} \\ \hline S001 & Sandhya & Rohini & Day Boarder \\ S002 & Vedanshi & Rohtak & Day Scholar \\ S003 & Vibhu & Raj Nagar & NULL \\ S004 & Atharva & Rampur & Day Boarder \\ \hline \end{array} \]
\[ \begin{array}{|c|c|c|} \hline \textbf{S\_id} & \textbf{Bus\_no} & \textbf{Stop\_name} \\ \hline S002 & TSS10 & Sarai Kale Khan \\ S004 & TSS12 & Sainik Vihar \\ S005 & TSS10 & Kamla Nagar \\ \hline \end{array} \]
Write a function, c_words(), in Python that separately counts and displays the number of uppercase and lowercase alphabets in a text file, Words.txt
Ms. Veda created a table named Sports in a MySQL database, containing columns Game_id, P_Age, and G_name.
After creating the table, she realized that the attribute Category has to be added.
Help her to write a command to add the Category column.
Thereafter, write the command to insert the following record in the table: Game_id: G42 P\_Age: Above 18 G_name: Chess Category: Senior
Write a user-defined function in Python named Puzzle(W, N) which takes the argument W as an English word and N as an integer and returns the string where every Nth alphabet of the word W is replaced with an underscore ("_").
Example: If W contains the word "TELEVISION" and N is 3, then the function should return the string "TE_EV_SI_N". Likewise, for the word "TELEVISION" if N is 4, the function should return "TEL_VIS_ON".
A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is \(3\) m as shown. Based on the given information answer the following: 
(i) Express \(y\) as a function of \(x\) from the given equation of ellipse.
(ii) Integrate the function obtained in (i) with respect to \(x\).
(iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(iii)(b) Write the coordinates of the points \(P\) and \(Q\) where the outer edge of the track cuts \(x\)-axis and \(y\)-axis in first quadrant and find the area of triangle formed by points \(P,O,Q\).