JEE Advanced 2016 · Paper 1

A uniform wooden stick of mass 1.6 kg and length l rests in an inclined manner on a smooth, vertical wall of height h (< l) such that a small portion of the stick extends beyond the wall. The reaction force of the wall on the stick is perpendicular to the stick. The stick makes an angle of 30° with the wall and the bottom of the stick is on a rough floor. The reaction of the wall on the stick is equal in magnitude to the reaction of the floor on the stick. The ratio h/l and the frictional force f at the bottom of the stick are (g = 10 m s⁻²)

1. Ah/l = √3/16, f = 16√3/3 N
2. Bh/l = 3/16, f = 16√3/3 N
3. Ch/l = 3√3/16, f = 8√3/3 N
4. Dh/l = 3√3/16, f = 16√3/3 N

Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principal quantum number n, where n ≫ 1. Which of the following statement(s) is(are) true?

1. ARelative change in the radii of two consecutive orbitals does not depend on Z
2. BRelative change in the radii of two consecutive orbitals varies as 1/n
3. CRelative change in the energy of two consecutive orbitals varies as 1/n³
4. DRelative change in the angular momenta of two consecutive orbitals varies as 1/n

An incandescent bulb has a thin filament of tungsten that is heated to high temperature by passing an electric current. The hot filament emits black-body radiation. The filament is observed to break up at random locations after a sufficiently long time of operation due to non-uniform evaporation of tungsten from the filament. If the bulb is powered at constant voltage, which of the following statement(s) is(are) true?

1. AThe temperature distribution over the filament is uniform
2. BThe resistance over small sections of the filament decreases with time
3. CThe filament emits more light at higher band of frequencies before it breaks up
4. DThe filament consumes less electrical power towards the end of the life of the bulb

A plano-convex lens is made of a material of refractive index n. When a small object is placed 30 cm away in front of the curved surface of the lens, an image of double the size of the object is produced. Due to reflection from the convex surface of the lens, another faint image is observed at a distance of 10 cm away from the lens. Which of the following statement(s) is(are) true?

1. AThe refractive index of the lens is 2.5
2. BThe radius of curvature of the convex surface is 45 cm
3. CThe faint image is erect and real
4. DThe focal length of the lens is 20 cm

A length-scale (l) depends on the permittivity (ε) of a dielectric material, Boltzmann constant (kB), the absolute temperature (T), the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles. Which of the following expression(s) for l is(are) dimensionally correct?

1. Al = √(nq²/(εkBT))
2. Bl = √(εkBT/(nq²))
3. Cl = √(q²/(εn²ᐟ³kBT))
4. Dl = √(q²/(εn¹ᐟ³kBT))

The position vector r⃗ of a particle of mass m is given by the following equation r⃗(t) = αt³ î + βt² ĵ, where α = 10/3 m s⁻³, β = 5 m s⁻² and m = 0.1 kg. At t = 1 s, which of the following statement(s) is(are) true about the particle?

1. AThe velocity v⃗ is given by v⃗ = (10î + 10ĵ) m s⁻¹
2. BThe angular momentum L⃗ with respect to the origin is given by L⃗ = −(5/3)k̂ N m s
3. CThe force F⃗ is given by F⃗ = (î + 2ĵ) N
4. DThe torque τ⃗ with respect to the origin is given by τ⃗ = −(20/3)k̂ N m