NTA JEE Mains 24th jan 2023 Shift 1

Vectors $$a\hat{i} + b\hat{j} + \hat{k}$$ and $$2\hat{i} - 3\hat{j} + 4\hat{k}$$ are perpendicular to each other when $$3a + 2b = 7$$, the ratio of $$a$$ to $$b$$ is $$\frac{x}{2}$$. The value of $$x$$ is _____.

A spherical body of mass 2 kg starting from rest acquires a kinetic energy of 10000 J at the end of $$5^{th}$$ second. The force acted on the body is _____ N.

Solid sphere $$A$$ is rotating about an axis $$PQ$$. If the radius of the sphere is 5 cm, then its radius of gyration about $$PQ$$ will be $$\sqrt{x}$$ cm. The value of $$x$$ is _____.

A hole is drilled in a metal sheet. At 27 °C, the diameter of hole is 5 cm. When the sheet is heated to 177 °C, the change in the diameter of hole is $$d \times 10^{-3}$$ cm. The value of $$d$$ will be _____, if coefficient of linear expansion of the metal is $$1.6 \times 10^{-5} /°C$$

A block of mass 2 kg is attached with two identical springs of spring constant 20 $$\text{N m}^{-1}$$ each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure). When the mass is displaced from its equilibrium position, it executes a simple harmonic motion. The time period of oscillation is $$\frac{\pi}{\sqrt{X}}$$ in SI unit. The value of $$X$$ is _____.

A stream of a positively charged particles having $$\frac{q}{m} = 2 \times 10^{11} \text{ C kg}^{-1}$$ and velocity $$\vec{v}_0 = 3 \times 10^7 \hat{i} \text{ m s}^{-1}$$ is deflected by an electric field $$1.8\hat{j} \text{ kV m}^{-1}$$. The electric field exists in a region of 10 cm along $$x$$ direction. Due to the electric field, the deflection of the charge particles in the $$y$$ direction is _____ mm.

A hollow cylindrical conductor has length of 3.14 m, while its inner and outer diameters are 4 mm and 8 mm respectively. The resistance of the conductor is $$n \times 10^{-3} \Omega$$. If the resistivity of the material is $$2.4 \times 10^{-8} \Omega$$ m. The value of n is _____.

In the circuit shown in the figure, the ratio of the quality factor and the band width is _____ s.

As shown in the figure, a combination of a thin plano-concave lens and a thin plano-convex lens is used to image an object placed at infinity. The radius of curvature of both the lenses is 30 cm and refraction index of the material for both the lenses is 1.75. Both the lenses are placed at distance of 40 cm from each other. Due to the combination, the image of the object is formed at distance $$x =$$ ______ cm, from concave lens.

Assume that protons and neutrons have equal masses. Mass of a nucleon is $$1.6 \times 10^{-27}$$ kg and radius of nucleus is $$1.5 \times 10^{-15} A^{1/3}$$ m. The approximate ratio of the nuclear density and water density is $$n \times 10^{13}$$. The value of n is _____.