NTA JEE Main 27th June 2022 Shift 2

A mass of $$10$$ kg is suspended vertically by a rope of length $$5$$ m from the roof. A force of $$30$$ N is applied at the middle point of rope in horizontal direction. The angle made by upper half of the rope with vertical is $$\alpha = \tan^{-1}(x \times 10^{-1})$$. The value of $$x$$ is ______.
(Given, $$g = 10$$ m s$$^{-2}$$)

A rolling wheel of $$12$$ kg is on an inclined plane at position $$P$$ and connected to a mass of $$3$$ kg through a string of fixed length and pulley as shown in figure. Consider PR as friction free surface. The velocity of centre of mass of the wheel when it reaches at the bottom $$Q$$ of the inclined plane $$PQ$$ will be $$\frac{1}{2}\sqrt{xgh}$$ m s$$^{-1}$$. The value of $$x$$ (rounded off to the nearest integer) is ______.

A diatomic gas $$(\gamma = 1.4)$$ does $$400$$ J of work when it is expanded isobarically. The heat given to the gas in the process is ______ J.

A particle executes simple harmonic motion. Its amplitude is $$8$$ cm and time period is $$6$$ s. The time it will take to travel from its position of maximum displacement to the point corresponding to half of its amplitude, is ______ s.

A parallel plate capacitor is made up of stair like structure with a plate area $$A$$ of each stair and that is connected with a wire of length $$b$$, as shown in the figure. The capacitance of the arrangement is $$\frac{x}{15}\frac{\varepsilon_0 A}{b}$$. The value of $$x$$ is ______.

The current density in a cylindrical wire of radius $$r = 4.0$$ mm is $$1.0 \times 10^6$$ A$$^2$$ m$$^2$$. The current through the outer portion of the wire between radial distances $$\frac{r}{2}$$ and $$r$$ is $$x\pi$$ A; where $$x$$ is ______

In the given circuit 'a' is an arbitrary constant. The value of $$m$$ for which the equivalent circuit resistance is minimum, will be $$\sqrt{\frac{x}{2}}$$. The value of $$x$$ is ______.

A metallic rod of length $$20$$ cm is placed in North-South direction and is moved at a constant speed of $$20$$ m s$$^{-1}$$ towards East. The horizontal component of the Earth's magnetic field at that place is $$4 \times 10^{-3}$$ T and the angle of dip is $$45°$$. The emf induced in the rod is ______ mV.

A deuteron and a proton moving with equal kinetic energy enter into a uniform magnetic field at right angle to the field. If $$r_d$$ and $$r_p$$ are the radii of their circular paths respectively, then the ratio $$\frac{r_d}{r_p}$$ will be $$\sqrt{x} : 1$$ where $$x$$ is ______

The cut-off voltage of the diodes (shown in figure) in forward bias is $$0.6$$ V. The current through the resister of $$40\Omega$$ is ______ mA.