Step 1: Analyzing the scene composition.
- The examination hall should feature rows of desks and chairs.
- Students should be depicted seated and engaged in writing their exams.
- An invigilator should be present, supervising the students.
Step 2: Creating depth and perspective.
- Utilize linear perspective to add depth to the hall.
- Draw desks in rows, reducing in size as they recede into the background to maintain perspective.
Step 3: Incorporating environmental details.
- Add elements such as windows, doors, ceiling fans, and a wall clock to enhance realism.
- Illustrate students in varied postures, including writing, thinking, or reviewing their answer sheets.
Step 4: Enhancing realism.
- Apply shading techniques to add volume to the desks, chairs, and architectural elements.
- Maintain accurate proportions for students and objects to ensure a realistic composition.
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
Let $ y(x) $ be the solution of the differential equation $$ x^2 \frac{dy}{dx} + xy = x^2 + y^2, \quad x > \frac{1}{e}, $$ satisfying $ y(1) = 0 $. Then the value of $ 2 \cdot \frac{(y(e))^2}{y(e^2)} $ is ________.
The left and right compartments of a thermally isolated container of length $L$ are separated by a thermally conducting, movable piston of area $A$. The left and right compartments are filled with $\frac{3}{2}$ and 1 moles of an ideal gas, respectively. In the left compartment the piston is attached by a spring with spring constant $k$ and natural length $\frac{2L}{5}$. In thermodynamic equilibrium, the piston is at a distance $\frac{L}{2}$ from the left and right edges of the container as shown in the figure. Under the above conditions, if the pressure in the right compartment is $P = \frac{kL}{A} \alpha$, then the value of $\alpha$ is ____