UPSC NDA & NA Official Solved Papers

A bicycle racing track is a level circular road of radius $10\text{ m}$. Three cyclists P, Q and R with identical cycles are cycling on the track with speeds $2\text{ m/s}$, $3\text{ m/s}$ and $5\text{ m/s}$ respectively. If the coefficient of friction between the tyres and the road is $0.1$, then which cyclist will slip? (Take $g = 10\text{ m/s}^2$)

The bob of a simple pendulum is displaced from its mean position and then released from rest $0.20\text{ m}$ above its mean position. What is the speed of the bob as it passes through the mean position? (Take $g = 10\text{ m/s}^2$)

A tennis ball of mass $350\text{ g}$ is thrown vertically upwards at a speed of $4\text{ m/s}$. If $M$ is the maximum gravitational potential energy of the ball and $H$ is the maximum height it reaches, then which one of the following is correct? (Take $g = 10\text{ m/s}^2$)

Which of the following is correct regarding the total internal reflection at the interface between two different optical media?

Consider the following statements:

1. The weight of an object is the same everywhere on the surface of the Earth, but its inertial mass could be different.
2. The inertial mass and gravitational mass of an object are proportional and the proportionality constant is the same everywhere.
3. The inertial mass and weight of an object are the same at different places on Earth.

Which of the statements given above is/are correct?

Consider the following:

1. Gravitational field due to a spherical body
2. Electrostatic field due to a point charge
3. Strong nuclear force field due to a proton
4. Magnetic field due to a bar magnet

How many of the above follow the inverse square law?

While pouring water slowly from a jug, a general observation is that a water layer usually flows along the side of the jug. Which property of water among the following is responsible for this phenomenon?

A fish is at a depth $d$ in water of refractive index $4/3$. If a transparent slab of refractive index $3/2$ and thickness $d/2$ is placed on the surface of water, then which one of the following is the apparent depth of the fish as observed by an observer in air?

A particle of mass $m$ moves along x-direction under the force $F(x) = F_0\left(1 - \frac{x}{L}\right)$ for $0 \le x \le L$. If the particle starts from rest at $x = 0\text{ (zero)}$, then what is the speed ($v$) of the particle at $x = L$?

A narrow tunnel passes through the centre of a planet of radius $R$ and uniform mass density $\rho$. A particle is released into the tunnel. What is the angular frequency of oscillation of the particle ($\omega$) equal to? (Symbols carry their usual meaning)

On a rainy day, rain drops fall vertically with speed $v$. An observer is moving towards east with speed $v$, while another observer moves towards north with the same speed $v$. What is the angle between apparent rain directions observed by them?

What is the minimum horizontal speed given to the bob of a simple pendulum of length $50\text{ cm}$ so that it describes a complete circle? (Take $g = 10\text{ m/s}^2$)

Two conducting wires made of the same material have lengths $L$ and $1.2L$. If the volume of the longer wire is half that of the shorter wire, then what is the electric resistance of the longer wire in comparison with the shorter wire?

A biconvex lens is made of a material of refractive index $2$ such that the ratio of the radii of curvature of its two faces is $1 : 2$. A point source in air is placed on the principal axis of this lens at a distance $u$ from the lens. Which one of the following is correct if the image is formed exactly at the same distance as that of the point source? (where $R$ is the radius of curvature of the first face)

In a carrom game, a striker of mass $15\text{ g}$ hits a coin of mass $5\text{ g}$ head on and the coin moves with a speed of $0.36\text{ m/s}$. If the time of contact between the striker and the coin is $3\text{ milliseconds}$, then what is the average force applied by the striker on the coin?

A hollow prism is filled with water of refractive index $1.34$ and white light is incident on the prism. The white light is assumed to have wavelengths ranging from $4020\ \text{\AA}$ to $6700\ \text{\AA}$. What is the corresponding wavelength range in water?

A heating filament and an ideal resistor of fixed value are connected in series with a DC voltage source. After some time, the filament has reached its operating point so that the current in the circuit is steady. Which one of the following is correct?

Two astronauts X and Y float in deep space far from external forces. Initially, they are at rest relative to one another. X throws a tool towards Y. Consider the two astronauts and the tool as one system. Which one of the following statements is correct?

A uniform rope hangs symmetrically over a smooth pulley. One end of the rope is given a slight mass, making that side slightly heavier. Which one of the following is correct?

An insulating container contains $250\text{ g}$ of water at $20^\circ\text{C}$. Steam at $100^\circ\text{C}$ is passed through it, heating it up to $25^\circ\text{C}$. During this process, part of the steam also gets condensed, raising the mass of water to $252\text{ g}$. If the specific heat of water is $1\text{ Cal g}^{-1\,\circ}\text{C}^{-1}$, then what is the value of the latent heat of steam?

Three parallel glass slabs of refractive indices 1.5, $\sqrt{3}$ and 1.5 respectively have widths 2 mm, 1 mm and 4 mm. The combination is in air. Light is incident at the first slab at an angle of incidence $\pi/3$. What is the angle with respect to the normal at which the light will emerge from the final surface?

Consider three wires X, Y and Z of the same length such that the ratio of their conductivities is $1 : 2 : 3$. Further, the cross-sectional area of wire Z is twice that of Y and the resistance of X is equal to the resistance of Z. What is the ratio of the resistance of Y to that of X?

Two point charges of 4 C and 16 C are kept at a distance of 0.18 m. A third point charge of 1 C is kept between them such that the potential energy of the system is minimum. What is the magnitude of the electric field at the 1 C charge?

Consider the following statements in respect of sound waves:

I. The speed of sound waves decreases with an increase in temperature if the atmospheric pressure remains the same.
II. The speed of sound waves increases with increasing humidity if the atmospheric pressure remains the same.
III. The speed of sound waves is not affected by a change in atmospheric pressure, provided the temperature is kept constant.

Which of the statements given above are correct?

Consider the following statements:

Statement-I : Marine animals remain alive in winter and move near the bottom of lakes despite frozen water on the surface.
Statement-II: The density of water is maximum at $4^{\circ}\text{C}$ and water at the bottom of lakes remains warm in winter.

Which one of the following is correct in respect of the above statements?