Two circles whose radii are equal to $4$ and $8$ intersect at right angles. The length of their common chord is

The sulphate of a metal $M$ contains $9.87 \%$ of $M$. This sulphate is isomorphous with $ZnSO_4 \cdot 7 H_2 O$. The atomic weight of $M$ is (in $.3$)

Give the correct order of initials $T$ or $F$ for the following statements. Use $T$ if the statement is true and $F$ if it is false.
$(I)$ Sulphide ions react with $Na_2[Fe(CN)_5(NO)]$ to form a purple-coloured compound $Na_4[Fe(CN)_5(NOS)]$. In the reaction,the oxidation state of iron changes.
$(II)$ $Pt(IV)$ compounds are relatively more stable than $Ni(IV)$ compounds.
$(III)$ The welding of magnesium can be done in the atmosphere of Helium.
$(IV)$ $LiAlH_4$ on hydrolysis will give $H_2$.

Acetic acid is reacted with metallic sodium to form hydrogen and $X$. When $X$ is heated with sodalime,$Y$ and sodium carbonate are formed. $Y$ is

The instantaneous angular position of a point on a rotating wheel is given by the equation $\theta(t) = 2t^3 - 6t^2$. The torque on the wheel becomes zero at $t = \dots \text{sec}$.

Six wires each of cross-sectional area $A$ and length $l$ are combined as shown in the figure. The thermal conductivities of copper and iron are $K_1$ and $K_2$ respectively. The equivalent thermal resistance between points $A$ and $C$ is