In the binomial expansion of \( (1 + x)^{2n - 1} \), the general term is given by: \[ T_k = \binom{2n-1}{k} x^k. \] The 30th term corresponds to \( T_{30} \), and the 12th term corresponds to \( T_{12} \). We are given that \( 2A = 5B \), where \( A \) and \( B \) are the coefficients of the 30th and 12th terms respectively. Solving the equation \( 2A = 5B \), we can find the value of \( n \).
Final Answer: \( n = 21 \).
Let \( y = y(x) \) be the solution of the differential equation \[ 2\cos x \frac{dy}{dx} = \sin 2x - 4y \sin x, \quad x \in \left( 0, \frac{\pi}{2} \right). \]
If \( y\left( \frac{\pi}{3} \right) = 0 \), then \( y\left( \frac{\pi}{4} \right) + y\left( \frac{\pi}{4} \right) \) is equal to ________.
The function \( f: (-\infty, \infty) \to (-\infty, 1) \), defined by \[ f(x) = \frac{2^x - 2^{-x}}{2^x + 2^{-x}}, \] is:
If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then \(a + b + ab\) is equal to:
Given three identical bags each containing 10 balls, whose colours are as follows:
| Bag I | 3 Red | 2 Blue | 5 Green |
| Bag II | 4 Red | 3 Blue | 3 Green |
| Bag III | 5 Red | 1 Blue | 4 Green |
A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from Bag I is $ p $ and if the ball is Green, the probability that it is from Bag III is $ q $, then the value of $ \frac{1}{p} + \frac{1}{q} $ is:
If \( \theta \in \left[ -\frac{7\pi}{6}, \frac{4\pi}{3} \right] \), then the number of solutions of \[ \sqrt{3} \csc^2 \theta - 2(\sqrt{3} - 1)\csc \theta - 4 = 0 \] is equal to ______.