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If the position vectors of $A, B$ and $C$ are respectively $2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}$ and $3 \hat{i}-4 \hat{j}-4 \hat{k}$, then $\cos ^2 A$ is equal to

Which one of the following statements in regard to the excretion by the human kidneys is correct?

The point $(3,2)$ undergoes the following three transformations in the order given:
$(i)$ Reflection about the line $y=x$.
(ii) Translation by the distance $1$ unit in the positive direction of $x$-axis.
(iii) Rotation by an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction.
Then,the final position of the point is:

The solubility product of silver bromide $(AgBr)$ is $5.0 \times 10^{-13}$. Calculate the mass of potassium bromide $(KBr)$ (molar mass = $120 \, g \, mol^{-1}$) that must be added to $1 \, L$ of a $0.05 \, M$ silver nitrate $(AgNO_3)$ solution to initiate the precipitation of $AgBr$.

If $\vec{a}, \vec{b}$ and $\vec{c}$ are vectors with magnitudes $2, 3$ and $4$ respectively,then the best upper bound of $|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2$ among the given values is

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