Two aeroplanes $I$ and $II$ bomb a target in succession. The probabilities of $I$ and $II$ scoring a hit are $0.3$ and $0.2$,respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane is

If the solution curve of the differential equation $\frac{dy}{dx} = \frac{x+y-2}{x-y}$ passing through the point $(2,1)$ is $\tan^{-1}\left(\frac{y-1}{x-1}\right) - \frac{1}{\beta} \log_e\left(\alpha + \left(\frac{y-1}{x-1}\right)^2\right) = \log_e|x-1|$,then $5\beta + \alpha$ is equal to

The solution of $\frac{dy}{dx} = \frac{x+y}{x-y}$ is

Express $\frac{dt}{dx} = \frac{t}{x + t e^{-2x/t}}$ in the form of $\frac{dx}{dt} = \phi\left(\frac{x}{t}\right)$.

Let $y = y(x)$ be the solution of the differential equation $x \sin(\frac{y}{x}) dy = (y \sin(\frac{y}{x}) - x) dx$,$y(1) = \frac{\pi}{2}$ and let $\alpha = \cos(\frac{e^{12}}{e^{12}})$. Then the number of integral values of $p$,for which the equation $x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$ represents a circle of radius $r \leq 6$,is . . . . . . .

The substitution required to reduce the differential equation $t^2 dx + (x^2 - tx + t^2) dt = 0$ to a differential equation which can be solved by the variables separable method is