JEE Advanced Mix Test 66

Two trains A and B are moving with speeds 20 m/s and 30 m/s respectively in the same direction on the same straight track, with B ahead of A. The engines are at the front ends. The engine of train A blows a long whistle.

Assume that the sound of the whistle is composed of components varying in frequency from f₁ = 800 Hz to f₂ = 1,120 Hz, as shown in the figure. The spread in the frequency (highest frequency - lowest frequency) is thus 320 Hz. The speed of sound in still air is 340 m/s.

The spread of frequency (in Hz) as observed by the passengers in train B is

A small block of mass 1 kg is released from rest at the top of a rough track. The track is circular arc of radius 40 m. The block slides along the track without toppling and a frictional force acts on it in the direction opposite to the instantaneous velocity. The work done in overcoming the friction up to the point Q, as shown in the figure below, is 150 J. (Take the acceleration due to gravity, g = 10 m/s^-2)

The speed (in ms^-1) of the block when it reaches the point Q is

A ray of light travelling in the direction  is incident on a plane mirror. After reflection, it travels along the direction  What is the angle of incidence (in degrees)?

A binary star consists of two stars A (mass 2.2M_s) and B (mass 11M_S), where M_s is the mass of the Sun. They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio of the total angular momentum of the binary star to the angular momentum of star B about the centre of mass is

The total number of cyclic structural as well as stereoisomer possible for a compound with the molecular formula C₅H₁₀ is

Consider a reaction: aG + bH → products. When concentration of both the reactants G and H is doubled, the rate increases by eight times. However, when concentration of G is doubled keeping the concentration of H fixed, the rate is doubled. The overall order of the reaction is

The sum of the number of lone pairs of electrons on each central atom in the following species is

[TeBr₆]^2–, [BrF₂]⁺, SNF₃, and [XeF₃]^–

(Atomic numbers: N = 7, F = 9, S = 16, Br = 35, Te = 52, Xe = 54)

The harmonic mean of the roots of the equation (5 + √2) x² - (4 + √5) x + 8 + 2√5 = 0 is

If the system of equations x - ky - z = 0,

kx - y - z = 0,

x + y - z = 0

has a non-zero solution, then possible values of k are ±A. Calculate the value of A.

Consider the parabola y² = 8x. Let Δ₁ be the area of the triangle formed by the end points of its latus rectum and the point  on the parabola, and Δ₂ be the area of the triangle formed by drawing tangents at P and at the end points of the latus rectum.

Then Δ₁/Δ₂ is