NTA JEE Mains 9th April 2024 Shift 1

If $$\vec{a}$$ and $$\vec{b}$$ makes an angle $$\cos^{-1}\left(\frac{5}{9}\right)$$ with each other, then for $$|\vec{a} + \vec{b}| = \sqrt{2}|\vec{a} - \vec{b}|$$, $$|\vec{a}| = n|\vec{b}|$$. The integer value of $$n$$ is ____

A string is wrapped around the rim of a wheel of moment of inertia $$0.40$$ kg m$$^2$$ and radius $$10$$ cm. The wheel is free to rotate about its axis. Initially the wheel is at rest. The string is now pulled by a force of $$40$$ N. The angular velocity of the wheel after $$10$$ s is $$x$$ rad/s, where $$x$$ is _______

Two persons pull a wire towards themselves. Each person exerts a force of $$200$$ N on the wire. Young's modulus of the material of wire is $$1 \times 10^{11}$$ N m$$^{-2}$$. Original length of the wire is $$2$$ m and the area of cross section is $$2$$ cm$$^2$$. The wire will extend in length by ______ $$\mu$$m.

The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of $$4$$ m, $$2$$ ms$$^{-1}$$ and $$16$$ ms$$^{-2}$$ at a certain instant. The amplitude of the motion is $$\sqrt{x}$$ m, where $$x$$ is ________

At the centre of a half ring of radius $$R = 10$$ cm and linear charge density $$4$$ nCm$$^{-1}$$, the potential is $$x\pi$$ V. The value of $$x$$ is _______

The current flowing through the $$1\Omega$$ resistor is $$\frac{n}{10}$$ A. The value of $$n$$ is ________

A square loop of edge length $$2$$ m carrying current of $$2$$ A is placed with its edges parallel to the $$x$$-$$y$$ axis. A magnetic field is passing through the $$x - y$$ plane and expressed as $$\vec{B} = B_0(1 + 4x)\hat{k}$$, where $$B_0 = 5$$ T. The net magnetic force experienced by the loop is _______ N.

When a coil is connected across a $$20$$ V dc supply, it draws a current of $$5$$ A. When it is connected across $$20$$ V, $$50$$ Hz ac supply, it draws a current of $$4$$ A. The self inductance of the coil is ______ mH. (Take $$\pi = 3$$)

In a Young's double slit experiment, the intensity at a point is $$\left(\frac{1}{4}\right)^{th}$$ of the maximum intensity, the minimum distance of the point from the central maximum is _______ $$\mu$$m. (Given : $$\lambda = 600$$ nm, $$d = 1.0$$ mm, $$D = 1.0$$ m)

A star has $$100\%$$ helium composition. It starts to convert three $$^4$$He into one $$^{12}$$C via triple alpha process as $$^4$$He $$+ ^4$$He $$+ ^4$$He $$\rightarrow ^{12}$$C $$+ Q$$. The mass of the star is $$2.0 \times 10^{32}$$ kg and it generates energy at the rate of $$5.808 \times 10^{30}$$ W. The rate of converting these $$^4$$He to $$^{12}$$C is $$n \times 10^{42}$$ s$$^{-1}$$, where n is ________ [Take, mass of $$^4$$He $$= 4.0026$$ u, mass of $$^{12}$$C $$= 12$$ u]