If the points with position vectors $60 \hat{i}+3 \hat{j}$, $40 \hat{i}-8 \hat{j}$, and $a \hat{i}-52 \hat{j}$ are collinear, then $a$ is equal to

$\sum_{k=1}^{\infty} \frac{1}{k!}\left(\sum_{n=1}^k 2^{n-1}\right)$ is equal to

Which of the following series of transition metal ions all have a $3d^2$ electronic configuration?

Which of the following ligands is expected to be bidentate?

$\int \frac{x^{49} \tan ^{-1}\left(x^{50}\right)}{1+x^{100}} d x = k\left(\tan ^{-1}\left(x^{50}\right)\right)^2 + c$,then $k$ is equal to

Three rods of same dimensions have thermal conductivities $3K$,$2K$,and $K$. They are arranged as shown in the figure. In the steady state,the temperature of the junction $P$ is