Given the equation: \(x^2 - 2^y = 2023\)
Step 1. By trial, we find that \( x = 45 \) and \( y = 1 \) satisfy the equation, as:
\(45^2 - 2^1 = 2025 - 2 = 2023\)
Step 2. Thus, the only solution in \( C \) is \( (x, y) = (45, 1) \).
Step 3. Calculate \( \sum_{(x, y) \in C} (x + y) \):
\(\sum_{(x, y) \in C} (x + y) = 45 + 1 = 46\)
The Correct Answer is: 46
Let \( S = \{p_1, p_2, \dots, p_{10}\} \) be the set of the first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y) \), where \( x \in S \), \( y \in A \), and \( x \) divides \( y \), is _________.
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