If $\frac{dy}{dx} = \frac{y + x \tan(\frac{y}{x})}{x}$,then $\sin(\frac{y}{x})$ is equal to

An ice ball melts at a rate proportional to the amount of ice present at that instant. Half the quantity of ice melts in $20 \text{ minutes}$. Let $x_0$ be the initial quantity of ice. If after $40 \text{ minutes}$ the amount of ice left is $k x_0$,then the value of $k$ is:

If $a = i + j + k$,$a \cdot b = 1$ and $a \times b = j - k$,then $b = \dots$

The point at which the circles $x^2+y^2-4x-4y+7=0$ and $x^2+y^2-12x-10y+45=0$ touch each other is:

The refractive index of a glass convex lens is $1.5$. The radius of curvature of each of the two surfaces of the lens is $20 ~cm$. The ratio of the power of the lens when immersed in a liquid of refractive index $1.25$ to that when placed in air is:

In the following circuit,$PN$-junction diodes $D_1, D_2$,and $D_3$ are ideal. For the following potentials of $A$ and $B$,the correct increasing order of resistance between $A$ and $B$ will be:
$(i) -10 \ V, -5 \ V$
$(ii) -5 \ V, -10 \ V$
$(iii) -4 \ V, -12 \ V$