Using a unit step size, the value of integral $$\int\limits_1^2 {x\,\ln \,xdx\,\,\,} $$ by trapezoidal rule is ___________.

A bullet spins as the shot is fired from a gun. For this purpose, two helical slots as shown in the figure are cut in the barrel. Projections A and B on the bullet engage in each of the slots.
Helical slots are such that one turn of helix is completed over a distance of 0.5m. If velocity of bullet when it exits the barrel is 20m/s, its spinning speed (in rad/s) is ________

A weight of 500N is supported by two metallic ropes as shown in the figure. The values of tensions T₁ and T₂ are respectively.

A Prandtl tube (Pitot-static tube with $$C=1$$) is used to measure the velocity of water. The differential manometer reading is $$10$$ mm of liquid column with a relative density of $$10.$$ Assuming $$g = 9.8\,\,m/{s^2},$$ the velocity of water (in $$m/s$$) is ___________