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A turn of radius $20\text{m}$ is banked for vehicles going at a speed of $36\text{km/h}$. If the coefficient of friction between the road and the tyres is $0.4$, what are the possible speeds of a vehicle so that it neither slips down nor skids up ?

A vector has component along the $x$-axis equal to $25\text{unit}$ and along the $y$-axis equal to $60\text{unit}$. The magnitude of the vector is

The angular width of the central maximum in the Fraunhofer diffraction pattern of a slit is measured. The slit is illuminated by light of wavelength $6000\dot{A}$. When the slit is illuminated by light of another wavelength$\left(\lambda \right)$, the angular width decreases by $30\%$. Calculate the wavelength $\left(\lambda \right)$ of this light in $\dot{A}$. The same decrease in the angular-width of central maximum is obtained when the original apparatus is immersed in a liquid. Find the refractive index $\mu$ of the liquid.

Two parallel long wires carry currents $i_1$ and $i_2$ with $i_1>i_2$. when the currents are in the same direction, the magnetic field at a point midway between the wires is $10\mu \text{T}$. If the direction of $i_2$ is reversed, the field becomes $30\mu \text{T}$. The ratio $\frac{i_1}{i_2}$ is

Find the derivative of $x^2$ using first principle.

Two blocks of mass $m_1=4\text{kg}$ and $m_2=2\text{kg}$, connected by a weightless rod on a plane having inclination of ${37}^{\circ }$ is shown in figure. If the coefficient of dynamic friction is $\mu =0.25$, then the common acceleration of the two blocks and the tension $\left(T\right)$ in the rod are

Two coherent sources have intensity ratio of $100:1$, and are used for obtaining the phenomenon of interference. Then the ratio of maximum and minimum intensity will be -

In the figure shown, the two particles are constrained to roll on a flat horizontal surface. They are fired simultaneously. What should be the initial speed of particle $\text{A}$ such that the two particles collide?

The position x of a particle varies with time (t) as

x = at² - bt³

The acceleration of the particle will be zero at time

The derivative of $\ln x$ with respect to $\cot x$ is

A composite string is made up by joining two strings of different mass per unit length $r$ and $4r$. The composite string is under the same tension. A transverse wave pulse is sent along the lighter string towards the joint. The ratio of wave number of incident wave to the transmitted wave is

A body of mass $10\text{kg}$ lies on a rough inclined plane of inclination $\theta ={\sin }^{-1}\frac{3}{5}$ with the horizontal. When a force of $30\text{N}$ is applied on the block parallel to and upward the plane, the total reaction by the plane on the block is nearly along (take $g=10m{s}^{-2}$)

A ball is projected upwards from the top of a tower with a velocity $50\text{m/s}$ making an angle ${30}^{\circ }$ with the horizontal. The height of the tower is $70\text{meter}$. After how much time from the instant of throwing, the ball will reach the ground?

Angle of the prism is $A$ and one of its refracting surfaces is silvered. If light rays falling at an angle of incidence $2A$ on the first surface returns back through the same path after suffering reflection at the silvered surface, the refractive index $\mu$, of the prism is:

A conducting wire $\text{PQ}$ of mass $1.8\text{kg}$ and length $2\text{m}$ can freely slide on a pair of parallel, smooth rails placed in a magnetic field of magnitude $1\text{T}$. The rails are connected by a capacitor of capacitance $0.5\text{F}$. The electric resistance of the rails and wire is zero. A constant force $F$ acts on the wire as shown. Find the value of $F$ if the acceleration of the wire is $5{\text{m/s}}^2$.

A rocket which is ejecting gas with a speed of $100\text{m/s}$ and the rate of ejection of the gas from the rocket is $800\text{kg/s}$. If the mass of the rocket is $4000\text{kg}$, find out the initial acceleration of the rocket. Take $(g=10{\text{m/s}}^2)$

A particle comes round a circle of radius 1 m once. The time taken by it is 10 sec. The average velocity of motion is

A particle of mass m moving with velocity v strikes a stationary particle of mass 2m and sticks to it. The speed of the system will be

Two circular discs $A$ and $B$ are of equal masses and thickness but made of metals with densities $d_A$ and $d_B(d_A>d_B)$ . If their moments of inertia about an axis passing through their centres and perpendicular to the circular faces are $I_A$ and $I_B$ then

Two blocks $m_1=5\text{gm}$ and $m_2=10\text{gm}$ are hung vertically over a light frictionless pulley as shown here. What is the acceleration of the masses when they are left free?

(where $g$ is acceleration due to gravity)

A light string passing over a smooth light pulley connects two blocks of masses m${}_1$ and m${}_2$ (vertically). If the acceleration of the system is g/8, then the ratio of the masses is

A body X with a momentum p collides with another identical stationary body Y one dimensionally. During the collision, Y gives an impulse J to body X. Then coefficient of restitution is

A battery having emf $E_1=100\text{V}$ and internal resistance $r=0.5\Omega$ is connected to a storage battery of emf $E_2=90\text{V}$ and external resistance $R$ as shown in figure. For what value of $R$ no current will pass through storage battery $E_2$?

A beaker contains water up to a height $h_1$ and oil above water up to another height $h_2.$ Find the apparent shift in the position of the bottom of the beaker when viewed from above. Refractive index of water is ${\mu }_1$ and that of oil is ${\mu }_2.$

Twelve wires, each having equal resistance of 2 Ω, are joined to form a cube. What is the equivalent resistance between diagonally opposite points?

A uniform cylindrical rod, of length $L$ and radius $r$, is made from a material, whose Young’s modulus of Elasticity equals $Y$. When this rod is heated by temperature $T$ and simultaneously subjected to a net longitudinal compressional force $F$, its length remains unchanged. The coefficient of volume expansion, of the material of the rod, is (nearly) equal to :