For a reaction taking place in three steps at the same temperature, the overall rate constant \( K = \frac{K_1 K_2}{K_3} \). If \( E_{a1} \), \( E_{a2} \), and \( E_{a3} \) are 40, 50, and 60 kJ/mol respectively, the overall \( E_a \) is ____ kJ/mol.
For \( 0 < a < 1 \), the value of the integral \[ \int_{0}^{\frac{\pi}{2}} \frac{dx}{1 - 2a \cos x + a^2} \] is:
Four distinct points \( (2k, 3k), (1, 0), (0, 1) \) and \( (0, 0) \) lie on a circle for \( k \) equal to:
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: