Given:
- Total resistance of the wire: \( 20 \, \Omega \)
- The wire is divided into 10 equal parts.
Step 1: Resistance of each part
The resistance of each part is:
\[ R_{\text{part}} = \frac{\text{Total resistance}}{\text{Number of parts}} = \frac{20 \, \Omega}{10} = 2 \, \Omega. \]
Step 2: Parallel combination of two parts
Each pair of two parts is connected in parallel. The equivalent resistance of a parallel combination is:
\[ R_{\text{parallel}} = \frac{R_{\text{part}}}{2} = \frac{2 \, \Omega}{2} = 1 \, \Omega. \]
Step 3: Total number of parallel combinations
Since there are 10 parts and they are paired in groups of 2, the total number of parallel combinations is:
\[ \text{Number of parallel combinations} = \frac{10}{2} = 5. \]
Step 4: Series combination of parallel pairs
The 5 parallel combinations (each of \( 1 \, \Omega \)) are connected in series. The equivalent resistance of a series combination is:
\[ R_{\text{eq}} = 5 \times R_{\text{parallel}} = 5 \times 1 \, \Omega = 5 \, \Omega. \]
Thus, the equivalent resistance of the final combination is \( R_{\text{eq}} = 5 \, \Omega \).
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32