$\sum_{k=1}^5 \frac{1^3+2^3+\ldots+k^3}{1+3+5+\ldots+(2k-1)}$ is equal to (in $.5$)

The correct output for the given logic circuit and inputs is:

Observe the following reactions at $T(K)$:
$I. A \rightarrow \text{products}$
$II. 5Br^{-}_{(aq)} + Br{O_{3}}^{-}_{(aq)} + 6H^{+}_{(aq)} \rightarrow 3Br_{2(aq)} + 3H_{2}O_{(l)}$
Both the reactions are started at $10.00 \text{ am}$. The rates of these reactions at $10.10 \text{ am}$ are same. The value of $-\frac{\Delta[Br^{-}]}{\Delta t}$ at $10.10 \text{ am}$ is $2 \times 10^{-4} \text{ mol L}^{-1} \text{ min}^{-1}$. The concentration of $A$ at $10.10 \text{ am}$ is $10^{-1} \text{ mol L}^{-1}$. What is the first order rate constant (in $\text{min}^{-1}$) of reaction $I$?

$A$ radioactive sample consists of two distinct species having an equal number of atoms $N_0$ initially. The mean-life of one species is $\tau$ and of the other is $5\tau$. The decay products in both cases are stable. The total number of radioactive nuclei at $t = 5\tau$ is:

$A$ particle of charge '$q$' and mass '$m$' moves in a circular orbit of radius '$r$' with angular speed '$\omega$'. The ratio of the magnitude of its magnetic moment to that of its angular momentum depends on

In a diffraction experiment from a single slit,the angular width of the central maxima does $NOT$ depend upon