If the line segment joining the points \( (1,0) \) and \( (0,1) \) subtends an angle of \( 45^\circ \) at a variable point \( P \), then the equation of the locus of \( P \) is:
If \( (a, b) \) be the orthocenter of the triangle whose vertices are \( (1, 2) \), \( (2, 3) \), and \( (3, 1) \), and \( I_1 = \int_a^b x \sin(4x - x^2) dx \), \( I_2 = \int_a^b \sin(4x - x^2) dx \), then \( 36 \frac{I_1}{I_2} \) is equal to:
Let $ [.] $ denote the greatest integer function. If $$ \int_1^e \frac{1}{x e^x} dx = \alpha - \log 2, \quad \text{then} \quad \alpha^2 \text{ is equal to:} $$
If the area of the region $$ \{(x, y): |4 - x^2| \leq y \leq x^2, y \geq 0\} $$ is $ \frac{80\sqrt{2}}{\alpha - \beta} $, $ \alpha, \beta \in \mathbb{N} $, then $ \alpha + \beta $ is equal to:
Three distinct numbers are selected randomly from the set $ \{1, 2, 3, ..., 40\} $. If the probability that the selected numbers are in an increasing G.P. is $ \frac{m}{n} $, where $ \gcd(m, n) = 1 $, then $ m + n $ is equal to: