$A$ uniform cube of side $a$ and mass $m$ rests on a rough horizontal table. $A$ horizontal force $F$ is applied normal to one of the faces at a point that is directly above the centre of the face,at a height $\frac{3a}{4}$ above the base. The minimum value of $F$ for which the cube begins to tilt about the edge is (Assume that the cube does not slide):

In the Fraunhofer diffraction experiment,$L$ is the distance between the screen and the obstacle,$b$ is the size of the obstacle,and $\lambda$ is the wavelength of the incident light. The general condition for the applicability of Fraunhofer diffraction is:

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 $H_2$.

$\begin{aligned} & A(\alpha, \beta)=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & e^\beta\end{array}\right] \\ & \Rightarrow[A(\alpha, \beta)]^{-1}=\end{aligned}$

Two coins $A$ and $B$ are kept in an urn. When coin $A$ is flipped,the probability of getting a Head is $1/4$,while for coin $B$ it is $3/4$. One coin is randomly chosen from this bag and tossed twice,and it falls heads on both occasions. The probability that it is coin $A$ is:

If $|a|=1, |b|=2$ and the angle between $a$ and $b$ is $120^{\circ}$,then ${(a+3b) \times (3a-b)}^2$ is equal to