JEE Advanced (Matrix Match & Integer Answer): Waves

Match the Following

DIRECTIONS (Q. No. 1-2) : Each question contains statements given in two columns, which have to be matched. The statements in Column-I are labelled A, B, C and D, while the statements in ColumnII are labelled p, q, r and s. Any given statement in Column-I can have correct matching with ONE OR MORE statement(s) in Column-II. The appropriate bubbles corresponding to the answers to these questions have to be darkened as illustrated in the following example :

If the correct matches are A-p, s and t; B-q and r; C-p and q; and D-s then the correct darkening of bubbles will look like the given.

Q.1. Each of the properties of sound listed in the column A primarily depends on one of the quantities in column B. Write down the matching pairs from the two columns.        (1980)

Ans. A → q; B → p; C → r

Solution. 

Q.2. Column I shows four systems, each of the same length L, for producing standing waves. The lowest possible natural frequency of a system is called its fundamental frequency, whose wavelength is denoted as λ_f. Match each system with statements given in Column II describing the nature and wavelength of the standing waves.   (2011)

Ans. A → p, t; B → p, s; C → q, s; D → q, r

Solution. (A) Pipe closed at one end 

Waves produced are longitudinal (sound waves)

(p, t) are correct matching
(B) Pipe open at both ends 

waves produced are longitudinal (sound waves)

(p, s) are correct matching. 

(c) Stretched wire clamped at both ends 

Waves produced are transverse in nature. (waves on string)

(q, s) are correct matching. (D) Stretched wave clamped at both ends & mid point Waves produced are transverse in nature (waves on string)

(q, r) are correct matching.

Integer Value Correct Type

Q.1. A  20  cm long string, having a mass of 1.0 g, is fixed at both the ends. The tension in the string is 0.5 N. The string is set into vibrations using an external vibrator of frequency 100 Hz. Find the separation (in cm) between the successive nodes on the string.        (2009)

Ans. 5

Solution.

The wavelength of the wave established

∴ The distance between two successive nodes

Q.2. A stationary source is emitting sound at a fixed frequency f₀, which is reflected by two cars approaching the source.

The difference between the frequencies of sound reflected from the cars is 1.2% of f₀. What is the difference in the speeds of the cars (in km per hour) to the nearest integer ?

The cars are moving at constant speeds much smaller than the speed of sound which is 330 ms^-1.         (2010)

Ans. 7

Solution.

Let v be the speed of sound and v_c and f₀ the speed and frequency of car.

The frequency of sound reflected by the car is

Differentiating the above equation w.r.t. v_c ,we get

∴ dv_c = 0.198 m/s ≈ 7 km/h

Q.3. Wh en two progressive waves y₁ = 4 sin (2x - 6t) and  are superimposed, the amplitude of the resultant wave is

Ans. 5

Solution. 

Q.4. Four harmonic waves of equal frequencies and equal intensities I₀ have phase angles   When theyare superposed, the intensity of the resulting wave is nI₀. The value of n is             (JEE Adv. 2015)

Ans. 3

Solution.