NTA JEE Main 31st August 2021 Shift 2

Statement I: If three forces $$\vec{F}_1$$, $$\vec{F}_2$$ and $$\vec{F}_3$$ are represented by three sides of a triangle and $$\vec{F}_1 + \vec{F}_2 = -\vec{F}_3$$, then these three forces are concurrent forces and satisfy the condition for equilibrium.
Statement II: A triangle made up of three forces $$\vec{F}_1$$, $$\vec{F}_2$$ and $$\vec{F}_3$$ as its sides were taken in the same order, satisfies the condition for translatory equilibrium.
In the light of the above statements, choose the most appropriate answer from the options given below:

Statement-I: Two forces $$\vec{P} + \vec{Q}$$ and $$\vec{P} - \vec{Q}$$ where $$\vec{P} \perp \vec{Q}$$, when act at an angle $$\theta_1$$ each other, the magnitude of their resultant is $$\sqrt{3P^2 + Q^2}$$, when they act at an angle $$\theta_2$$, the magnitude of their resultant becomes $$\sqrt{2P^2 + Q^2}$$. This is possible only when $$\theta_1 < \theta_2$$.
Statement-II: In the situation given above. $$\theta_1 = 60°$$ and $$\theta_2 = 90°$$
In the light of the above statement, choose the most appropriate answer from the options given below :

If velocity $$V$$, time $$T$$ and force $$F$$ are chosen as the base quantities, the dimensions of the mass will be:

A block moving horizontally on a smooth surface with a speed of 40 m s$$^{-1}$$ splits into two parts with masses in the ratio of 1 : 2. If the smaller part moves at 60 m s$$^{-1}$$ in the same direction, then the fractional change in kinetic energy is:

A system consists of two identical spheres each of mass 1.5 kg and radius 50 cm at the ends of a light rod. The distance between the centres of the two spheres is 5 m. What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its midpoint?

If $$R_E$$ be the radius of Earth, then the ratio between the acceleration due to gravity at a depth $$r$$ below and a height $$r$$ above the earth surface is: (Given: $$r \lt R_E$$)

Four identical hollow cylindrical columns of mild steel support a big structure of mass $$50 \times 10^3$$ kg. The inner and outer radii of each column are 50 cm and 100 cm respectively. Assuming uniform local distribution, calculate the compression strain of each column.
[use $$Y = 2.0 \times 10^{11}$$ Pa, $$g = 9.8$$ m s$$^{-2}$$]

Two thin metallic spherical shells of radii $$r_1$$ and $$r_2$$ ($$r_1 < r_2$$) are placed with their centres coinciding. A material of thermal conductivity $$K$$ is filled in the space between the shells. The inner shell is maintained at temperature $$\theta_1$$ and the outer shell at temperature $$\theta_2$$ ($$\theta_1 < \theta_2$$). The rate at which heat flows radially through the material is:

A mixture of hydrogen and oxygen has volume 500 cm$$^3$$, temperature 300 K, pressure 400 kPa and mass 0.76 g. The ratio of masses of oxygen to hydrogen will be:

A bob of mass $$m$$ suspended by a thread of length $$\ell$$ undergoes simple harmonic oscillations with time period $$T$$. If the bob is immersed in a liquid that has density $$\frac{1}{4}$$ times that of the bob and the length of the thread is increased by $$\frac{1}{3}$$rd of the original length, then the time period of the simple harmonic oscillations will be: