JEE Main 2023 (Online) 12th April Morning Shift

Two dice A and B are rolled. Let the numbers obtained on A and B be \(\alpha\) and \(\beta\) respectively. If the variance of \(\alpha-\beta\) is \(\frac{p}{q}\), where \(p\) and \(q\) are co-prime, then the sum of the positive divisors of...

A fair \(n(n > 1)\) faces die is rolled repeatedly until a number less than \(n\) appears. If the mean of the number of tosses required is \(\frac{n}{9}\), then \(n\) is equal to ____________.

In a triangle ABC, if \(\cos \mathrm{A}+2 \cos \mathrm{B}+\cos C=2\) and the lengths of the sides opposite to the angles A and C are 3 and 7 respectively, then \(\mathrm{\cos A-\cos C}\) is equal to

Let \(\alpha, \beta\) be the roots of the quadratic equation \(x^{2}+\sqrt{6} x+3=0\). Then \(\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}\) is equal to :

Let \(< a_{\mathrm{n}} >\) be a sequence such that \(a_{1}+a_{2}+\ldots+a_{n}=\frac{n^{2}+3 n}{(n+1)(n+2)}\). If \(28 \sum_\limits{k=1}^{10} \frac{1}{a_{k}}=p_{1} p_{2} p_{3} \ldots p_{m}\), where $$\mathrm{p}_{1}, \mathrm{p}_{2}, \ldots ....

The number of relations, on the set \(\{1,2,3\}\) containing \((1,2)\) and \((2,3)\), which are reflexive and transitive but not symmetric, is __________.

Let the positive numbers \(a_{1}, a_{2}, a_{3}, a_{4}\) and \(a_{5}\) be in a G.P. Let their mean and variance be \(\frac{31}{10}\) and \(\frac{m}{n}\) respectively, where \(m\) and \(n\) are co-prime. If the mean of their reciprocals is $$...

If the point \(\left(\alpha, \frac{7 \sqrt{3}}{3}\right)\) lies on the curve traced by the mid-points of the line segments of the lines \(x \cos \theta+y \sin \theta=7, \theta \in\left(0, \frac{\pi}{2}\right)\) between the co-ordinates axes...

Let \(a, b, c\) be three distinct real numbers, none equal to one. If the vectors \(a \hat{i}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+b \hat{j}+\hat{\mathrm{k}}\) and \(\hat{\mathrm{i}}+\hat{\mathrm{j}}+c \hat{\mathrm{k}}\) are ...

Let \(\lambda \in \mathbb{Z}, \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}\) and \(\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}\). Let \(\vec{c}\) be a vector such that $$(\vec{a}+\vec{b}+\vec{c}) \times \vec{c}=\overrightarrow{0}, \vec{a} \cdot \vec{c}=...

Given below are two statements:
Statement I : When the frequency of an a.c source in a series LCR circuit increases, the current in the circuit first increases, attains a maximum value and then decreases.
Statement II : In a series LCR circ...

A \(12.5 \mathrm{~eV}\) electron beam is used to bombard gaseous hydrogen at room temperature. The number of spectral lines emitted will be:

A common example of alpha decay is \({ }_{92}^{238} \mathrm{U} \longrightarrow{ }_{90}^{234} \mathrm{Th}+{ }_{2} \mathrm{He}^{4}+\mathrm{Q}\)
Given :
\({ }_{92}^{238} \mathrm{U}=238.05060 ~\mathrm{u}\),
$${ }_{90}^{234} \mathrm{Th}=234.0436...

The amplitude of \(15 \sin (1000 \pi \mathrm{t})\) is modulated by \(10 \sin (4 \pi \mathrm{t})\) signal. The amplitude modulated signal contains frequencies of
A. \(500 \mathrm{~Hz}\)
B. \(2 \mathrm{~Hz}\)
C. \(250 \mathrm{~Hz}\)
D. $$498 ...

A wire of resistance \(160 ~\Omega\) is melted and drawn in a wire of one-fourth of its length. The new resistance of the wire will be

The current flowing through a conductor connected across a source is \(2 \mathrm{~A}\) and 1.2 \(\mathrm{A}\) at \(0^{\circ} \mathrm{C}\) and \(100^{\circ} \mathrm{C}\) respectively. The current flowing through the conductor at $$50^{\circ}...

A proton and an \(\alpha\)-particle are accelerated from rest by \(2 \mathrm{~V}\) and \(4 \mathrm{~V}\) potentials, respectively. The ratio of their de-Broglie wavelength is :

A conducting circular loop is placed in a uniform magnetic field of \(0.4 \mathrm{~T}\) with its plane perpendicular to the field. Somehow, the radius of the loop starts expanding at a constant rate of \(1 \mathrm{~mm} / \mathrm{s}\). The m...

Given below are two statements: one is labelled as Assertion \(\mathbf{A}\) and the other is labelled as Reason \(\mathbf{R}\)
Assertion A : EM waves used for optical communication have longer wavelengths than that of microwave, employed in...

In an n-p-n common emitter (CE) transistor the collector current changes from 5 \(\mathrm{mA}\) to \(16 \mathrm{~mA}\) for the change in base current from \(100~ \mu \mathrm{A}\) and \(200 ~\mu \mathrm{A}\), respectively. The current gain o...

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : If an electric dipole of dipole moment \(30 \times 10^{-5} ~\mathrm{C} ~\mathrm{m}\) is enclosed by a closed surface, the net...

64 identical drops each charged upto potential of \(10 ~\mathrm{mV}\) are combined to form a bigger drop. The potential of the bigger drop will be __________ \(\mathrm{mV}\).

An ice cube has a bubble inside. When viewed from one side the apparent distance of the bubble is \(12 \mathrm{~cm}\). When viewed from the opposite side, the apparent distance of the bubble is observed as \(4 \mathrm{~cm}\). If the side of...

Two convex lenses of focal length \(20 \mathrm{~cm}\) each are placed coaxially with a separation of \(60 \mathrm{~cm}\) between them. The image of the distant object formed by the combination is at _____________ \(\mathrm{cm}\) from the fi...

The ratio of escape velocity of a planet to the escape velocity of earth will be:-
Given: Mass of the planet is 16 times mass of earth and radius of the planet is 4 times the radius of earth.

Two satellites \(\mathrm{A}\) and \(\mathrm{B}\) move round the earth in the same orbit. The mass of \(\mathrm{A}\) is twice the mass of \(\mathrm{B}\). The quantity which is same for the two satellites will be

An engine operating between the boiling and freezing points of water will have
A. efficiency more than 27%.
B. efficiency less than the efficiency of a Carnot engine operating between the same two temperatures.
C. efficiency equal to $$27 \...

If the r. m.s speed of chlorine molecule is \(490 \mathrm{~m} / \mathrm{s}\) at \(27^{\circ} \mathrm{C}\), the r. m. s speed of argon molecules at the same temperature will be (Atomic mass of argon \(=39.9 \mathrm{u}\), molecular mass of ch...

Three forces \(F_{1}=10 \mathrm{~N}, F_{2}=8 \mathrm{~N}, \mathrm{~F}_{3}=6 \mathrm{~N}\) are acting on a particle of mass \(5 \mathrm{~kg}\). The forces \(\mathrm{F}_{2}\) and \(\mathrm{F}_{3}\) are applied perpendicularly so that particle...

Given below are two statements:
Statement I : The diamagnetic property depends on temperature.
Statement II : The induced magnetic dipole moment in a diamagnetic sample is always opposite to the magnetizing field.
In the light of given stat...

A compass needle oscillates 20 times per minute at a place where the dip is \(30^{\circ}\) and 30 times per minute where the dip is \(60^{\circ}\). The ratio of total magnetic field due to the earth at two places respectively is $$\frac{4}{...

A ball is thrown vertically upward with an initial velocity of \(150 \mathrm{~m} / \mathrm{s}\). The ratio of velocity after \(3 \mathrm{~s}\) and \(5 \mathrm{~s}\) is \(\frac{x+1}{x}\). The value of \(x\) is ___________.
\(\left\{\right.\)...

A body cools from \(80^{\circ} \mathrm{C}\) to \(60^{\circ} \mathrm{C}\) in 5 minutes. The temperature of the surrounding is \(20^{\circ} \mathrm{C}\). The time it takes to cool from \(60^{\circ} \mathrm{C}\) to \(40^{\circ} \mathrm{C}\) is...

Glycerin of density \(1.25 \times 10^{3} \mathrm{~kg} \mathrm{~m}^{-3}\) is flowing through the conical section of pipe The area of cross-section of the pipe at its ends are \(10 \mathrm{~cm}^{2}\) and \(5 \mathrm{~cm}^{2}\) and pressure dr...

For a rolling spherical shell, the ratio of rotational kinetic energy and total kinetic energy is \(\frac{x}{5}\). The value of \(x\) is ___________.

A particle is executing simple harmonic motion (SHM). The ratio of potential energy and kinetic energy of the particle when its displacement is half of its amplitude will be

Match List I with List II

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For a certain organ pipe, the first three resonance frequencies are in the ratio of \(1:3:5\) respectively. If the frequency of fifth harmonic is \(405 \mathrm{~Hz}\) and the speed of sound in air is \(324 \mathrm{~ms}^{-1}\) the length of ...

Given below are two statements:
Statement I : A truck and a car moving with same kinetic energy are brought to rest by applying breaks which provide equal retarding forces. Both come to rest in equal distance.
Statement II : A car moving to...

To maintain a speed of 80 km/h by a bus of mass 500 kg on a plane rough road for 4 km distance, the work done by the engine of the bus will be ____________ KJ. [The coefficient of friction between tyre of bus and road is 0.04.]