If $\overrightarrow{a}=\hat{i}-\hat{j}-\hat{k}$ and $\overrightarrow{b}=\lambda \hat{i}-3 \hat{j}+\hat{k}$ and the orthogonal projection of $\overrightarrow{b}$ on $\overrightarrow{a}$ is $\frac{4}{3}(\hat{i}-\hat{j}-\hat{k})$, then $\lambda$ is equal to

Choose the correct option from the following:
$I$. Dehydration and dormancy of mature seed are crucial for seed storage.
$II$. Seed of $Lupinus \; arcticus$ is the oldest one which germinated after $2,000 \; years$.
$III$. Orchid seed is one of the largest seeds in the plant kingdom.
$IV$. Seeds of parasitic plants $Orobanche$ and $Striga$ are tiny seeds.

If $x + by + c = 0$ is a normal to the parabola $y^2 = 12x$,then the complete set of all values of $c$ is

The bob of a simple pendulum is hanging vertically down from a fixed identical bob by means of a string of length $l$. If both bobs are charged with a charge $q$ each,what is the time period of the pendulum? (Ignore the radii of the bobs.)

The point at which the circles $x^2+y^2-4x-4y+7=0$ and $x^2+y^2-12x-10y+45=0$ touch each other is: