JEE Main Pattern Practice Questions

Close the Notes. Start the Clock.

Sections 1 to 11 taught you this chapter: what a wave carries and what it does not, transverse against longitudinal motion, the displacement relation $y = a\sin(kx - \omega t + \phi)$ and everything you can read off it, the speed of a wave on a string and through a gas, superposition and interference, reflection at rigid and free boundaries, standing waves on strings and in air columns, beats, and the Doppler effect.

This section asks one different question: can you use any of it with a timer running?

There is no new theory below. There are 30 single-correct questions built to the exam pattern, and a marking scheme designed to punish the four habits this chapter rewards most cruelly: quoting a wave speed where a particle speed was asked for, putting an even harmonic into a pipe closed at one end, halving a beat frequency that was never doubled, and reversing a Doppler sign because the axis was never drawn.

The rules of engagement

Key Point: This is not a reading exercise. Blank sheet, pen, timer. Attempt all 30 questions in one unbroken sitting, and do not open a single explanation until the last answer is written.

Everything in this drill sits inside the JEE Main syllabus for waves; the handful of items tagged Advanced go a step past it and are there to stretch you, not to scare you.

Notation for This Drill

Three symbols in this chapter mean two different things each, and between them they account for more lost marks than any genuine gap in understanding.

Key Point — the tension trap. In $v = \sqrt{T/\mu}$ the symbol $T$ is the tension in the string, measured in newtons. Everywhere else in this chapter $T$ is the time period in seconds, $T = \dfrac{1}{\nu}$. They are unrelated quantities that share a letter. Every question below that uses a tension names it in newtons in the same line, so read the unit and not the letter.

Key Point — $k$ is the angular wave number. $k = \dfrac{2\pi}{\lambda}$, in radians per metre. In this chapter it is never a spring constant. If your answer to a wavelength question is $2\pi$ times somebody else's, one of you has dropped this factor.

Key Point — wave speed is not particle speed. $v = \nu\lambda = \dfrac{\omega}{k}$ is the speed at which the pattern advances. The particle speed is $\dfrac{\partial y}{\partial t}$, its greatest value is $\omega a$, and it has nothing whatever to do with $\nu\lambda$. Whenever a question says "speed", find the adjective in front of it before you compute anything.

$y = a\sin(kx - \omega t)$ travels towards $+x$; $y = a\sin(kx + \omega t)$ travels towards $-x$. Every angle inside a sine or a cosine is in radians.

The Doppler axis

Draw it before you substitute. Every Doppler question in this set is one line long once the axis is on the paper, and unreadable without it.

Key Point: Take the direction from the source towards the observer as positive. Then $v_o$ and $v_s$ are signed velocity components along that one line, and $$\nu^{\,\prime} = \nu\left(\frac{v - v_o}{v - v_s}\right)$$ covers all four cases at once. An observer moving towards the source is moving in the negative direction on that axis, so $v_o < 0$ and the numerator grows. With a wind, replace $v$ by $v + w$, where $w$ is the wind's component on the same axis.

For a reflecting surface apply the shift twice: the reflector is first an observer, then a source re-emitting what it received.

Harmonics and overtones

Half a dozen questions below turn on this table alone, and one of them is designed to catch you writing an even harmonic into a closed pipe.

The formulas that decide the most marks

$$y = a\sin(kx - \omega t + \phi), \qquad k = \frac{2\pi}{\lambda}, \qquad \omega = 2\pi\nu = \frac{2\pi}{T}, \qquad v = \nu\lambda = \frac{\omega}{k}$$

$$v_{\text{particle}} = \frac{\partial y}{\partial t} = -v\,\frac{\partial y}{\partial x}, \qquad \left(v_{\text{particle}}\right)_{\max} = \omega a$$

$$v = \sqrt{\frac{T}{\mu}} \ \ \text{on a string}, \qquad v = \sqrt{\frac{\gamma P}{\rho}} = \sqrt{\frac{\gamma R T_{\text{kelvin}}}{M_0}} \ \ \text{in a gas}, \qquad v \propto \sqrt{T_{\text{kelvin}}}$$

$$\Delta\phi = \frac{2\pi}{\lambda}\,\Delta x, \qquad A = \sqrt{a_1^2 + a_2^2 + 2a_1a_2\cos\phi}, \qquad \frac{I_{\max}}{I_{\min}} = \left(\frac{a_1 + a_2}{a_1 - a_2}\right)^2$$

$$y = 2a\sin kx\,\cos\omega t, \qquad \text{node spacing} = \frac{\lambda}{2}, \qquad \text{node to antinode} = \frac{\lambda}{4}$$

$$\nu_{\text{beat}} = \lvert \nu_1 - \nu_2 \rvert, \qquad \nu^{\,\prime} = \nu\left(\frac{v - v_o}{v - v_s}\right)$$

Useful numbers

Copy these to the top of your sheet before you start.

What this set covers

That spread mirrors how the paper actually samples this chapter. Wave speed, standing waves and pipes together carry 12 of the 30, because those are the blocks that carry the multi-step questions, and therefore the marks.

The difficulty mix is roughly 20% easy, 45% medium and 35% hard. A handful will feel brutal. They are meant to.

[Exam Tip] That $-1$ changes the arithmetic of guessing. A blind guess among four options returns $\dfrac{4}{4} - \dfrac{3 \times 1}{4} = +0.25$ marks on average, barely worth the minute it costs. A question narrowed to two options returns $\dfrac{4 - 1}{2} = +1.50$ marks on average, six times as much. Narrow first, then commit. Leave blank only what you could not narrow at all.

[Exam Tip] Before you start, write five lines at the top of your sheet: wave speed or particle speed?, is that $T$ a tension or a period?, is this pipe open or closed, and is that harmonic allowed?, is the beat rate the full difference?, which way does my Doppler axis point? Those five questions catch the overwhelming majority of the marks lost in this chapter.

Scoring Yourself Honestly

Mark your sheet with the real scheme, $+4$ and $-1$ and $0$, and total it. No half marks for "I knew that one really". The number you get is the number that matters.

The bands

Read your own answer sheet

Before you touch a single explanation, sort your mistakes into three piles. This is the most valuable ten minutes in the whole section.

1. Method errors. You used $v = \nu\lambda$ where the question wanted $\omega a$. You gave a closed pipe an even harmonic. You applied the Doppler shift once at a reflecting surface instead of twice. You took the beat rate to be half the frequency difference. These are the expensive ones, because the whole solution is wrong from line one.
2. Execution errors. Right method, wrong arithmetic. The classic four here: a $2\pi$ dropped between $k$ and $\lambda$; centimetres left unconverted inside a wave equation; a diameter used where the formula wanted a radius; a temperature ratio taken in Celsius instead of kelvin.
3. Reading errors. The question asked for the first overtone, not the first harmonic. For the particle speed, not the wave speed. For the frequency the observer hears, not the frequency the source emits. For the wavelength in the air, not the wavelength the source would have produced at rest.

Key Point: In this chapter pile 1 is unusually fat, because four of the chapter's standing traps are structural rather than arithmetical. An even harmonic in a closed pipe, a single Doppler shift off a reflector, a halved beat frequency and a particle speed read as a wave speed all produce numbers that look perfectly reasonable. Those are the ones that cost marks.

How badly each trap hurts

The middle three are the dangerous ones. Nobody ships an answer that is a hundred times too big; everybody ships one that is a factor of 2 out.

The eight habits this set is drilling

• Read the adjective in front of the word "speed". Wave speed is $\nu\lambda$; particle speed peaks at $\omega a$; source speed is the thing that shifts the frequency. Three different quantities, three different numbers, one word.
• Write down $k$, $\omega$, $\lambda$, $\nu$, $T$ and $v$ as six separate lines the moment you meet a wave equation. Then check them against one another: $k\lambda = 2\pi$, $\omega T = 2\pi$, $v = \nu\lambda = \dfrac{\omega}{k}$. Ten seconds, and it catches every dropped $2\pi$.
• Convert the units inside the bracket before anything else. A wave written with $x$ in centimetres and $t$ in seconds has a $k$ in rad/cm, and the speed that falls out is in cm/s.
• Name the pipe before you name the harmonic. Closed at one end means odd harmonics only, and the first overtone is the third harmonic. Open at both ends means every harmonic, and the first overtone is the second.
• A tension is in newtons. If your $T$ came from a hanging mass, it is a force; if it came from a frequency, it is a period. Write the unit next to it and the collision cannot bite you.
• Beats are the full difference. The signed envelope repeats $\dfrac{\lvert \nu_1 - \nu_2 \rvert}{2}$ times a second, but loudness depends on the magnitude of the envelope, and that peaks twice as often. The factor of two is the whole point of the derivation.
• Draw the Doppler axis, then say out loud whether the pitch should rise or fall, and only then substitute. If the algebra disagrees with the sentence, the algebra is wrong.
• Sanity-check every answer against the physics. A transverse wave on a laboratory wire runs at tens to hundreds of metres per second, sound in air at a few hundred, sound in steel at a few thousand. An audible pipe is centimetres to metres long. A Doppler shift from a road vehicle is a few per cent, not a few hundred.

[Exam Tip] Every explanation below is a full step-by-step solution, and each one ends by naming the mistake that produces each wrong option. Read the explanation even for the questions you got right - several of these have a two-line route and a two-page route, and it is the two-line route you will need in the hall.