Water at $100^{\circ} C$ cools in $15 \text{ minutes}$ to $75^{\circ} C$ in a room temperature of $25^{\circ} C$. Then the temperature of water after $30 \text{ minutes}$ is

Let $f : [0,1] \to R$ be such that $f(xy) = f(x)f(y)$ for all $x, y \in [0,1],$ and $f(0) \ne 0.$ If $y = y(x)$ satisfies the differential equation $\frac{dy}{dx} = f(x)$ with $y(0) = 1,$ then $y\left( \frac{1}{4} \right) + y\left( \frac{3}{4} \right)$ is equal to

Let $f$ be a differentiable function defined on $\left[0, \frac{\pi}{2}\right]$ such that $f(x) > 0$ and $f(x)+\int \limits_0^x f(t) \sqrt{1-\left(\log _e f(t)\right)^2} d t=e, \forall x \in\left[0, \frac{\pi}{2}\right]$. Then $\left(6 \log _{ e } f \left(\frac{\pi}{6}\right)\right)^2$ is equal to $.............$

The differential equation $x \frac{dy}{dx} = 2y$ represents ....

The number of solutions of $y' = \frac{y + 1}{x - 1}, y(1) = 2$ is

The area enclosed by the closed curve $C$ given by the differential equation $\frac{dy}{dx} + \frac{x+a}{y-2} = 0, y(1) = 0$ is $4\pi$. Let $P$ and $Q$ be the points of intersection of the curve $C$ and the $y$-axis. If normals at $P$ and $Q$ on the curve $C$ intersect the $x$-axis at points $R$ and $S$ respectively,then the length of the line segment $RS$ is