What is the maximum value of the force $F$ such that the block shown in the arrangement does not move (in $N$)? (Given: $m = \sqrt{3} \ kg$,$\mu = \frac{1}{2\sqrt{3}}$,$\theta = 60^\circ$)

Two particles having position vectors $r_1 = (3 \hat{i} + 5 \hat{j}) \ m$ and $r_2 = (-5 \hat{i} - 3 \hat{j}) \ m$ are moving with velocities $v_1 = (4 \hat{i} + 3 \hat{j}) \ m/s$ and $v_2 = (a \hat{i} + 7 \hat{j}) \ m/s$. If they collide after $2 \ s$,then the value of $a$ is:

$A$ curve passes through the point $\left(1, \frac{\pi}{6}\right)$. Let the slope of the curve at each point $(x, y)$ be $\frac{y}{x}+\sec \left(\frac{y}{x}\right)$,where $x>0$. Then,the equation of the curve is:

If $PM$ is the perpendicular from $P(2, 3)$ onto the line $x+y=3$,then the coordinates of $M$ are

Find the perimeter of the triangle shown in the figure (in $cm$).

If $x^3 + y^3 - 3axy = 0$,then $\frac{dy}{dx}$ equals