If $y$ is a function of $x$ and $\log(x+y) = 2xy$,then the value of $y'(0)$ is

Three rods made of the same material and having the same cross-section are joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at $0^{\circ}C$ and $90^{\circ}C$ respectively. The temperature of the junction of the three rods will be ...... $^{\circ}C$.

The magnetic needle of a vibration magnetometer makes $12$ oscillations per minute in the horizontal component of the earth's magnetic field. When an external short bar magnet is placed at some distance along the axis of the needle in the same line,it makes $15$ oscillations per minute. If the poles of the bar magnet are interchanged,the number of oscillations it makes per minute is

Let $f(x)$ be defined for all $x > 0$ and be continuous. Let $f(x)$ satisfy $f(\frac{x}{y}) = f(x) - f(y)$ for all $x, y > 0$ and $f(e) = 1$. Then:

In which of the following organisms is a three-chambered heart found?

If the function $f$ defined on $\left( \frac{\pi }{6}, \frac{\pi }{3} \right)$ by $f(x) = \begin{cases} \frac{\sqrt{2} \cos x - 1}{\cot x - 1}, & x \neq \frac{\pi}{4} \\ k, & x = \frac{\pi}{4} \end{cases}$ is continuous,then $k$ is equal to