$H^{+}$,$D^{+}$,and $T^{+}$ differ in all of the following $EXCEPT$ one. Identify it.

When a bucket is half full,the weight of the bucket and the water is $10 \ kg$. When the bucket is two-thirds full,the total weight is $11 \ kg$. What is the total weight (in $kg$),when the bucket is completely full?

If $\frac{\cos x}{\cos (x-2y)} = \lambda$,then $\tan (x-y) \tan y$ is equal to

If $\frac{3x+2}{(x+1)(2x^2+3)} = \frac{A}{x+1} + \frac{Bx+C}{2x^2+3}$,then $A+C-B$ is equal to :

$A$ bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. The refractive index of the first lens is $1.5$ and that of the second lens is $1.2$. Both the curved surfaces have the same radius of curvature $R = 14\; cm$. For this bi-convex lens,if the object distance is $40\; cm$,the image distance will be: (in $; cm$)

$A$ metal wire $PQ$ slides on parallel metallic rails having separation $0.25 \ m,$ each having negligible resistance. There is a $2 \ \Omega$ resistor and $10 \ V$ battery as shown in the figure. There is a uniform magnetic field directed into the plane of the paper of magnitude $0.5 \ T.$ $A$ force of $0.5 \ N$ to the left is required to keep the wire $PQ$ moving with constant speed to the right. With what speed is the wire $PQ$ moving? (Neglect self-inductance of the loop).