JEE Main 2022 (Online) 28th July Morning Shift

Let $$A=\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right]$$ and $$B=\left[\begin{array}{cc}\beta & 1 \\ 1 & 0\end{array}\right], \alpha, \beta \in \mathbf{R}$$. Let \(\alpha_{1}\) be the value of \(\alpha\) which satisfies $$(\m...

If the tangents drawn at the points \(\mathrm{P}\) and \(\mathrm{Q}\) on the parabola \(y^{2}=2 x-3\) intersect at the point \(R(0,1)\), then the orthocentre of the triangle \(P Q R\) is :

Let \(S\) be the set of all passwords which are six to eight characters long, where each character is either an alphabet from \(\{A, B, C, D, E\}\) or a number from \(\{1,2,3,4,5\}\) with the repetition of characters allowed. If the number ...

Out of \(60 \%\) female and \(40 \%\) male candidates appearing in an exam, \(60 \%\) candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the quali...

The sum of all real values of \(x\) for which \(\frac{3 x^{2}-9 x+17}{x^{2}+3 x+10}=\frac{5 x^{2}-7 x+19}{3 x^{2}+5 x+12}\) is equal to __________.

Consider the sequence \(a_{1}, a_{2}, a_{3}, \ldots\) such that \(a_{1}=1, a_{2}=2\) and \(a_{n+2}=\frac{2}{a_{n+1}}+a_{n}\) for \(\mathrm{n}=1,2,3, \ldots .\) If $$\left(\frac{\mathrm{a}_{1}+\frac{1}{\mathrm{a}_{2}}}{\mathrm{a}_{3}}\right)...

For \(\alpha \in \mathbf{N}\), consider a relation \(\mathrm{R}\) on \(\mathbf{N}\) given by \(\mathrm{R}=\{(x, y): 3 x+\alpha y\) is a multiple of 7\(\}\). The relation \(R\) is an equivalence relation if and only if :

Let \(x_{1}, x_{2}, x_{3}, \ldots, x_{20}\) be in geometric progression with \(x_{1}=3\) and the common ratio \(\frac{1}{2}\). A new data is constructed replacing each \(x_{i}\) by \(\left(x_{i}-i\right)^{2}\). If \(\bar{x}\) is the mean of...

Let the vectors \(\vec{a}=(1+t) \hat{i}+(1-t) \hat{j}+\hat{k}, \vec{b}=(1-t) \hat{i}+(1+t) \hat{j}+2 \hat{k}\) and \(\vec{c}=t \hat{i}-t \hat{j}+\hat{k}, t \in \mathbf{R}\) be such that for $$\alpha, \beta, \gamma \in \mathbf{R}, \alpha \ve...

Let a vector \(\vec{a}\) has magnitude 9. Let a vector \(\vec{b}\) be such that for every \((x, y) \in \mathbf{R} \times \mathbf{R}-\{(0,0)\}\), the vector \((x \vec{a}+y \vec{b})\) is perpendicular to the vector $$(6 y \vec{a}-18 x \vec{b}...

The equation of current in a purely inductive circuit is \(5 \sin \left(49\, \pi t-30^{\circ}\right)\). If the inductance is \(30 \,\mathrm{mH}\) then the equation for the voltage across the inductor, will be :
\(\left\{\right.\) Let $$\lef...

The frequencies at which the current amplitude in an LCR series circuit becomes \(\frac{1}{\sqrt{2}}\) times its maximum value, are \(212\,\mathrm{rad} \,\mathrm{s}^{-1}\) and \(232 \,\mathrm{rad} \,\mathrm{s}^{-1}\). The value of resistanc...

The half life period of a radioactive substance is 60 days. The time taken for \(\frac{7}{8}\)th of its original mass to disintegrate will be :

A freshly prepared radioactive source of half life 2 hours 30 minutes emits radiation which is 64 times the permissible safe level. The minimum time, after which it would be possible to work safely with source, will be _________ hours.

Two capacitors, each having capacitance \(40 \,\mu \mathrm{F}\) are connected in series. The space between one of the capacitors is filled with dielectric material of dielectric constant \(\mathrm{K}\) such that the equivalence capacitance ...

In two different experiments, an object of mass \(5 \mathrm{~kg}\) moving with a speed of \(25 \mathrm{~ms}^{-1}\) hits two different walls and comes to rest within (i) 3 second, (ii) 5 seconds, respectively. Choose the correct option out o...

In the case of amplitude modulation to avoid distortion the modulation index \((\mu)\) should be :

A wire of resistance R1 is drawn out so that its length is increased by twice of its original length. The ratio of new resistance to original resistance is :

As shown in the figure, a potentiometer wire of resistance \(20 \,\Omega\) and length \(300 \mathrm{~cm}\) is connected with resistance box (R.B.) and a standard cell of emf \(4 \mathrm{~V}\). For a resistance '\(R\)' of resistance box intr...

The equation \(\lambda=\frac{1.227}{x} \mathrm{~nm}\) can be used to find the de-Brogli wavelength of an electron. In this equation \(x\) stands for :
Where
\(\mathrm{m}=\) mass of electron
\(\mathrm{P}=\) momentum of electron
$$\mathrm{K}=...

Identify the solar cell characteristics from the following options :

Two electric dipoles of dipole moments \(1.2 \times 10^{-30} \,\mathrm{Cm}\) and \(2.4 \times 10^{-30} \,\mathrm{Cm}\) are placed in two different uniform electric fields of strengths \(5 \times 10^{4} \,\mathrm{NC}^{-1}\) and $$15 \times 1...

As shown in the figure, after passing through the medium 1 . The speed of light \(v_{2}\) in medium 2 will be :
\(\left(\right.\) Given \(\mathrm{c}=3 \times 10^{8} \mathrm{~ms}^{-1}\) )

In normal adujstment, for a refracting telescope, the distance between objective and eye piece is \(30 \mathrm{~cm}\). The focal length of the objective, when the angular magnification of the telescope is 2 , will be :

If the radius of earth shrinks by \(2 \%\) while its mass remains same. The acceleration due to gravity on the earth's surface will approximately :

A Carnot engine has efficiency of \(50 \%\). If the temperature of sink is reduced by \(40^{\circ} \mathrm{C}\), its efficiency increases by \(30 \%\). The temperature of the source will be:

Given below are two statements :
Statement I : The average momentum of a molecule in a sample of an ideal gas depends on temperature.
Statement II : The rms speed of oxygen molecules in a gas is \(v\). If the temperature is doubled and the ...

The current sensitivity of a galvanometer can be increased by :
(A) decreasing the number of turns
(B) increasing the magnetic field
(C) decreasing the area of the coil
(D) decreasing the torsional constant of the spring
Choose the most app...

As shown in the figure, a metallic rod of linear density \(0.45 \mathrm{~kg} \mathrm{~m}^{-1}\) is lying horizontally on a smooth inclined plane which makes an angle of \(45^{\circ}\) with the horizontal. The minimum current flowing in the ...

A NCC parade is going at a uniform speed of \(9 \mathrm{~km} / \mathrm{h}\) under a mango tree on which a monkey is sitting at a height of \(19.6 \mathrm{~m}\). At any particular instant, the monkey drops a mango. A cadet will receive the m...

A balloon has mass of \(10 \mathrm{~g}\) in air. The air escapes from the balloon at a uniform rate with velocity \(4.5 \mathrm{~cm} / \mathrm{s}\). If the balloon shrinks in \(5 \mathrm{~s}\) completely. Then, the average force acting on t...

The force required to stretch a wire of cross-section \(1 \mathrm{~cm}^{2}\) to double its length will be : (Given Yong's modulus of the wire \(=2 \times 10^{11} \mathrm{~N} / \mathrm{m}^{2}\))

The diameter of an air bubble which was initially \(2 \mathrm{~mm}\), rises steadily through a solution of density \(1750 \mathrm{~kg} \mathrm{~m}^{-3}\) at the rate of \(0.35 \,\mathrm{cms}^{-1}\). The coefficient of viscosity of the solut...

Four identical discs each of mass '\(\mathrm{M}\)' and diameter '\(\mathrm{a}\)' are arranged in a small plane as shown in figure. If the moment of inertia of the system about \(\mathrm{OO}^{\prime}\) is \(\frac{x}{4} \,\mathrm{Ma}^{2}\). T...

The dimensions of \(\left(\frac{\mathrm{B}^{2}}{\mu_{0}}\right)\) will be :
(if \(\mu_{0}\) : permeability of free space and \(B\) : magnetic field)

If the projection of \(2 \hat{i}+4 \hat{j}-2 \hat{k}\) on \(\hat{i}+2 \hat{j}+\alpha \hat{k}\) is zero. Then, the value of \(\alpha\) will be ___________.

In a Young's double slit experiment, a laser light of 560 nm produces an interference pattern with consecutive bright fringes' separation of 7.2 mm. Now another light is used to produce an interference pattern with consecutive bright fringe...

In the wave equation
\(y=0.5 \sin \frac{2 \pi}{\lambda}(400 \mathrm{t}-x) \,\mathrm{m}\)
the velocity of the wave will be:

The frequency of echo will be __________ Hz if the train blowing a whistle of frequency 320 Hz is moving with a velocity of 36 km/h towards a hill from which an echo is heard by the train driver. Velocity of sound in air is 330 m/s.

A block of mass '\(\mathrm{m}\)' (as shown in figure) moving with kinetic energy E compresses a spring through a distance \(25 \mathrm{~cm}\) when, its speed is halved. The value of spring constant of used spring will be $$\mathrm{nE} \,\,\...