NTA JEE Mains 27th Jan 2024 Shift 1

A particle starts from origin at $$t = 0$$ with a velocity $$5\hat{i} \text{ m s}^{-1}$$ and moves in $$x - y$$ plane under action of a force which produces a constant acceleration of $$(3\hat{i} + 2\hat{j}) \text{ m s}^{-2}$$. If the $$x$$-coordinate of the particle at that instant is $$84$$ m, then the speed of the particle at this time is $$\sqrt{\alpha} \text{ m s}^{-1}$$. The value of $$\alpha$$ is _______.

Four particles, each of mass $$1$$ kg are placed at four corners of a square of side $$2$$ m. The moment of inertia of the system about an axis perpendicular to its plane and passing through one of its vertex is ______ kg m$$^2$$.

If average depth of an ocean is $$4000$$ m and the bulk modulus of water is $$2 \times 10^9 \text{ N m}^{-2}$$, then fractional compression $$\frac{\Delta V}{V}$$ of water at the bottom of ocean is $$\alpha \times 10^{-2}$$. The value of $$\alpha$$ is _______, (Given, $$g = 10 \text{ m s}^{-2}, \rho = 1000 \text{ kg m}^{-3}$$)

A particle executes simple harmonic motion with an amplitude of $$4$$ cm. At the mean position, velocity of the particle is $$10 \text{ cm s}^{-1}$$. The distance of the particle from the mean position when its speed becomes $$5 \text{ cm s}^{-1}$$ is $$\sqrt{\alpha}$$ cm, where $$\alpha =$$ ______.

A thin metallic wire having cross sectional area of $$10^{-4} \text{ m}^2$$ is used to make a ring of radius $$30$$ cm. A positive charge of $$2\pi$$ C is uniformly distributed over the ring, while another positive charge of $$30$$ pC is kept at the centre of the ring. The tension in the ring is _______ N; provided that the ring does not get deformed (neglect the influence of gravity). (Given, $$\frac{1}{4\pi\epsilon_0} = 9 \times 10^9$$ SI units)

The charge accumulated on the capacitor connected in the following circuit is ______ $$\mu$$C. (Given $$C = 150 \; \mu$$F)

Two long, straight wires carry equal currents in opposite directions as shown in figure. The separation between the wires is $$5.0$$ cm. The magnitude of the magnetic field at a point P midway between the wires is ______ $$\mu$$T. (Given: $$\mu_0 = 4\pi \times 10^{-7} \text{ T m A}^{-1}$$)

Two coils have mutual inductance $$0.002$$ H. The current changes in the first coil according to the relation $$i = i_0 \sin \omega t$$, where $$i_0 = 5$$ A and $$\omega = 50\pi \text{ rad s}^{-1}$$. The maximum value of emf in the second coil is $$\frac{\pi}{\alpha}$$ V. The value of $$\alpha$$ is

Two immiscible liquids of refractive indices $$\frac{8}{5}$$ and $$\frac{3}{2}$$ respectively are put in a beaker as shown in the figure. The height of each column is $$6$$ cm. A coin is placed at the bottom of the beaker. For near normal vision, the apparent depth of the coin is $$\frac{\alpha}{4}$$ cm. The value of $$\alpha$$ is _______.

In a nuclear fission process, a high mass nuclide $$(A \approx 236)$$ with binding energy $$7.6$$ MeV/Nucleon dissociated into two middle mass nuclides $$(A \approx 118)$$, having binding energy of $$8.6$$ MeV/Nucleon. The energy released in the process would be _______ MeV.