NTA JEE Mains 1st Feb 2024 Shift 1

A particle is moving in one dimension (along $$x$$ axis) under the action of a variable force. Its initial position was $$16$$ m right of origin. The variation of its position $$x$$ with time $$t$$ is given as $$x = -3t^3 + 18t^2 + 16t$$, where $$x$$ is in m and $$t$$ is in s. The velocity of the particle when its acceleration becomes zero is _________ m s$$^{-1}$$.

The identical spheres each of mass $$2M$$ are placed at the corners of a right angled triangle with mutually perpendicular sides equal to $$4$$ m each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is $$\frac{4\sqrt{2}}{x}$$, where the value of $$x$$ is ________.

A plane is in level flight at constant speed and each of its two wings has an area of $$40$$ m$$^2$$. If the speed of the air is $$180$$ km h$$^{-1}$$ over the lower wing surface and $$252$$ km h$$^{-1}$$ over the upper wing surface, the mass of the plane is ________kg. (Take air density to be $$1$$ kg m$$^{-3}$$ and $$g = 10$$ m s$$^{-2}$$)

A tuning fork resonates with a sonometer wire of length $$1$$ m stretched with a tension of $$6$$ N. When the tension in the wire is changed to $$54$$ N, the same tuning fork produces $$12$$ beats per second with it. The frequency of the tuning fork is _______ Hz.

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle $$\theta$$ with each other. When suspended in water the angle remains the same. If density of the material of the sphere is $$1.5$$ g/cc, the dielectric constant of water will be ______. (Take density of water $$= 1$$ g/cc)

The current in a conductor is expressed as $$I = 3t^2 + 4t^3$$, where $$I$$ is in Ampere and $$t$$ is in second. The amount of electric charge that flows through a section of the conductor during $$t = 1$$ s to $$t = 2$$ s is ____________ C.

A regular polygon of $$6$$ sides is formed by bending a wire of length $$4\pi$$ meter. If an electric current of $$4\pi\sqrt{3}$$ A is flowing through the sides of the polygon, the magnetic field at the centre of the polygon would be $$x \times 10^{-7}$$ T. The value of $$x$$ is ______.

A rectangular loop of sides $$12$$ cm and $$5$$ cm, with its sides parallel to the $$x$$-axis and $$y$$-axis respectively moves with a velocity of $$5$$ cm s$$^{-1}$$ in the positive $$x$$ axis direction, in a space containing a variable magnetic field in the positive $$z$$ direction. The field has a gradient of $$10^{-3}$$ T cm$$^{-1}$$ along the negative $$x$$ direction and it is decreasing with time at the rate of $$10^{-3}$$ T s$$^{-1}$$. If the resistance of the loop is $$6$$ m$$\Omega$$, the power dissipated by the loop as heat is ______ $$\times 10^{-9}$$ W.

The distance between object and its $$3$$ times magnified virtual image as produced by a convex lens is $$20$$ cm. The focal length of the lens used is ________ cm.

The radius of a nucleus of mass number $$64$$ is $$4.8$$ fermi. Then the mass number of another nucleus having radius of $$4$$ fermi is $$\frac{1000}{x}$$, where $$x$$ is _________.