Check whether the matrix
is invertible or not.
If the mean and variance of a binomial distribution are \( 18 \) and \( 12 \) respectively, then the value of \( n \) is __________.
The integrating factor of the linear differential equation \[ x \frac{dy}{dx} + 2y = x^2 \log x \] is __________.
If \( \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} x^3 \sin^4 x \, dx = k \), then \( k \) is ____________.
The slope of the tangent to the curve \( x = \sin\theta \) and \( y = \cos 2\theta \) at \( \theta = \frac{\pi}{6} \) is ___________.
The perpendicular distance of the plane \( r \cdot (3\hat{i} + 4\hat{j} + 12\hat{k}) = 78 \) from the origin is __________.
If \( \alpha, \beta, \gamma \) are direction angles of a line and \( \alpha = 60^\circ, \beta = 45^\circ \), then \( \gamma \) is _________.
The principal solutions of the equation \( \cos\theta = \frac{1}{2} \) are _________.
The dual of statement \( t \lor (p \lor q) \) is _________.
Interpret the given diagrams A and B. Enlist the changes occurring during inspiration and expiration.
