NTA JEE Main 30th January 2023 Shift 1

A horse rider covers half the distance with $$5$$ m s$$^{-1}$$ speed. The remaining part of the distance was travelled with speed $$10$$ m s$$^{-1}$$ for half the time and with speed $$15$$ m s$$^{-1}$$ for other half of the time. The mean speed of the rider averaged over the whole time of motion is $$\frac{x}{7}$$ m s$$^{-1}$$. The value of $$x$$ is ______.

A thin uniform rod of length $$2$$ m, cross sectional area $$A$$ and density $$d$$ is rotated about an axis passing through the centre and perpendicular to its length with angular velocity $$\omega$$. If value of $$\omega$$ in terms of its rotational kinetic energy $$E$$ is $$\sqrt{\frac{\alpha E}{Ad}}$$, then the value of $$\alpha$$ is ______.

The general displacement of a simple harmonic oscillator is $$x = A \sin \omega t$$. Let $$T$$ be its time period. The slope of its potential energy $$(U)$$ - time $$(t)$$ curve will be maximum when $$t = \frac{T}{\beta}$$. The value of $$\beta$$ is ______.

A capacitor of capacitance $$900$$ $$\mu$$F is charged by a $$100$$ V battery. The capacitor is disconnected from the battery and connected to another uncharged identical capacitor such that one plate of uncharged capacitor connected to positive plate and another plate of uncharged capacitor connected to negative plate of the charged capacitor. The loss of energy in this process is measured as $$x \times 10^{-2}$$ J. The value of $$x$$ is ______.

In the following circuit, the magnitude of current $$I_1$$, is ______ A.

As per the given figure, if $$\frac{dI}{dt} = -1$$ A s$$^{-1}$$, then the value of $$V_{AB}$$ at this instant will be ______ V.

In an experiment for estimating the value of focal length of converging mirror, image of an object placed at $$40$$ cm from the pole of the mirror is formed at distance $$120$$ cm from the pole of the mirror. These distances are measured with a modified scale in which there are 20 small divisions in $$1$$ cm. The value of error in measurement of focal length of the mirror is $$\frac{1}{K}$$ cm. The value of $$K$$ is ______.

In Young's double slit experiment, two slits $$S_1$$ and $$S_2$$ are $$d$$ distance apart and the separation from slits to screen is $$D$$ (as shown in figure). Now if two transparent slabs of equal thickness $$0.1$$ mm but refractive index $$1.51$$ and $$1.55$$ are introduced in the path of beam $$\lambda = 4000$$ $$\text{\AA}$$ from $$S_1$$ and $$S_2$$ respectively. The central bright fringe spot will shift by ______ number of fringes.

A point source of light is placed at the centre of curvature of a hemispherical surface. The source emits a power of $$24$$ W. The radius of curvature of hemisphere is $$10$$ cm and the inner surface is completely reflecting. The force on the hemisphere due to the light falling on it is ______ $$\times 10^{-8}$$ N.

In a screw gauge, there are 100 divisions on the circular scale and the main scale moves by $$0.5$$ mm on a complete rotation of the circular scale. The zero of circular scale lies 6 divisions below the line of graduation when two studs are brought in contact with each other. When a wire is placed between the studs, 4 linear scale divisions are clearly visible while $$46^{th}$$ division the circular scale coincide with the reference line. The diameter of the wire is ______ $$\times 10^{-2}$$ mm.