JEE Main 2022 (Online) 26th July Evening Shift

Let \(\mathrm{P}\) and \(\mathrm{Q}\) be any points on the curves \((x-1)^{2}+(y+1)^{2}=1\) and \(y=x^{2}\), respectively. The distance between \(P\) and \(Q\) is minimum for some value of the abscissa of \(P\) in the interval :

The equation of a common tangent to the parabolas \(y=x^{2}\) and \(y=-(x-2)^{2}\) is

Numbers are to be formed between 1000 and 3000 , which are divisible by 4 , using the digits \(1,2,3,4,5\) and 6 without repetition of digits. Then the total number of such numbers is ____________.

Let \(X\) be a binomially distributed random variable with mean 4 and variance \(\frac{4}{3}\). Then, \(54 \,P(X \leq 2)\) is equal to :

The minimum value of the sum of the squares of the roots of \(x^{2}+(3-a) x+1=2 a\) is:

If \(\sum\limits_{k=1}^{10} \frac{k}{k^{4}+k^{2}+1}=\frac{m}{n}\), where m and n are co-prime, then \(m+n\) is equal to _____________.

Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is ___________.

Let \(A=\{1,2,3,4,5,6,7\}\) and \(B=\{3,6,7,9\}\). Then the number of elements in the set \(\{C \subseteq A: C \cap B \neq \phi\}\) is ___________.

The mean and standard deviation of 40 observations are 30 and 5 respectively. It was noticed that two of these observations 12 and 10 were wrongly recorded. If \(\sigma\) is the standard deviation of the data after omitting the two wrong ob...

If the sum of solutions of the system of equations \(2 \sin ^{2} \theta-\cos 2 \theta=0\) and \(2 \cos ^{2} \theta+3 \sin \theta=0\) in the interval \([0,2 \pi]\) is \(k \pi\), then \(k\) is equal to __________.

A nucleus of mass \(M\) at rest splits into two parts having masses \(\frac{M^{\prime}}{3}\) and \({{2M'} \over 3}(M' < M)\). The ratio of de Broglie wavelength of two parts will be :

Mass numbers of two nuclei are in the ratio of \(4: 3\). Their nuclear densities will be in the ratio of

Two lighter nuclei combine to form a comparatively heavier nucleus by the relation given below :
\({ }_{1}^{2} X+{ }_{1}^{2} X={ }_{2}^{4} Y\)
The binding energies per nucleon for \(\frac{2}{1} X\) and \({ }_{2}^{4} Y\) are $$1.1 \,\mathrm{...

A source of potential difference \(V\) is connected to the combination of two identical capacitors as shown in the figure. When key '\(K\)' is closed, the total energy stored across the combination is \(E_{1}\). Now key '\(K\)' is opened an...

A ball of mass \(0.15 \mathrm{~kg}\) hits the wall with its initial speed of \(12 \mathrm{~ms}^{-1}\) and bounces back without changing its initial speed. If the force applied by the wall on the ball during the contact is $$100 \mathrm{~N}$...

A body of mass \(8 \mathrm{~kg}\) and another of mass \(2 \mathrm{~kg}\) are moving with equal kinetic energy. The ratio of their respective momentum will be :

The maximum and minimum voltage of an amplitude modulated signal are \(60 \mathrm{~V}\) and \(20 \mathrm{~V}\) respectively. The percentage modulation index will be :

A battery of \(6 \mathrm{~V}\) is connected to the circuit as shown below. The current I drawn from the battery is :

A potentiometer wire of length \(300 \mathrm{~cm}\) is connected in series with a resistance 780 \(\Omega\) and a standard cell of emf \(4 \mathrm{V}\). A constant current flows through potentiometer wire. The length of the null point for c...

In a coil of resistance \(8 \,\Omega\), the magnetic flux due to an external magnetic field varies with time as \(\phi=\frac{2}{3}\left(9-t^{2}\right)\). The value of total heat produced in the coil, till the flux becomes zero, will be ____...

The oscillating magnetic field in a plane electromagnetic wave is given by \(B_{y}=5 \times 10^{-6} \sin 1000 \pi\left(5 x-4 \times 10^{8} t\right) T\). The amplitude of electric field will be :

A velocity selector consists of electric field \(\vec{E}=E \,\hat{k}\) and magnetic field \(\vec{B}=B \,\hat{j}\) with \(B=12 \,m T\). The value of \(E\) required for an electron of energy \(728 \,\mathrm{e} V\) moving along the positive $$...

The typical transfer characteristics of a transistor in CE configuration is shown in figure. A load resistor of \(2 \,k \Omega\) is connected in the collector branch of the circuit used. The input resistance of the transistor is $$0.50 \,\m...

Two uniformly charged spherical conductors \(A\) and \(B\) of radii \(5 \mathrm{~mm}\) and \(10 \mathrm{~mm}\) are separated by a distance of \(2 \mathrm{~cm}\). If the spheres are connected by a conducting wire, then in equilibrium conditi...

Three point charges of magnitude \(5 \mu \mathrm{C}, 0.16 \mu \mathrm{C}\) and \(0.3 \mu \mathrm{C}\) are located at the vertices \(A, B, C\) of a right angled triangle whose sides are \(A B=3 \mathrm{~cm}, B C=3 \sqrt{2} \mathrm{~cm}\) and...

Light travels in two media \(M_{1}\) and \(M_{2}\) with speeds \(1.5 \times 10^{8} \mathrm{~ms}^{-1}\) and \(2.0 \times 10^{8} \mathrm{~ms}^{-1}\) respectively. The critical angle between them is :

In the given figure, the face \(A C\) of the equilateral prism is immersed in a liquid of refractive index '\(n\)'. For incident angle \(60^{\circ}\) at the side \(A C\), the refractive light beam just grazes along face \(A C\). The refract...

A body is projected vertically upwards from the surface of earth with a velocity equal to one third of escape velocity. The maximum height attained by the body will be :
(Take radius of earth \(=6400 \mathrm{~km}\) and $$\mathrm{g}=10 \math...

An ice cube of dimensions \(60 \mathrm{~cm} \times 50 \mathrm{~cm} \times 20 \mathrm{~cm}\) is placed in an insulation box of wall thickness \(1 \mathrm{~cm}\). The box keeping the ice cube at \(0^{\circ} \mathrm{C}\) of temperature is brou...

A gas has \(n\) degrees of freedom. The ratio of specific heat of gas at constant volume to the specific heat of gas at constant pressure will be :

Two masses \(M_{1}\) and \(M_{2}\) are tied together at the two ends of a light inextensible string that passes over a frictionless pulley. When the mass \(M_{2}\) is twice that of \(M_{1}\), the acceleration of the system is \(a_{1}\). Whe...

Two concentric circular loops of radii \(r_{1}=30 \mathrm{~cm}\) and \(r_{2}=50 \mathrm{~cm}\) are placed in \(\mathrm{X}-\mathrm{Y}\) plane as shown in the figure. A current \(I=7 \mathrm{~A}\) is flowing through them in the direction as s...

Two projectiles are thrown with same initial velocity making an angle of \(45^{\circ}\) and \(30^{\circ}\) with the horizontal respectively. The ratio of their respective ranges will be :

The area of cross section of the rope used to lift a load by a crane is \(2.5 \times 10^{-4} \mathrm{~m}^{2}\). The maximum lifting capacity of the crane is 10 metric tons. To increase the lifting capacity of the crane to 25 metric tons, th...

A uniform heavy rod of mass \(20 \mathrm{~kg}\), cross sectional area \(0.4 \mathrm{~m}^{2}\) and length \(20 \mathrm{~m}\) is hanging from a fixed support. Neglecting the lateral contraction, the elongation in the rod due to its own weight...

The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through the center will be ___________ \(\mathrm{m}\).
Given, the length of the rod is \(10 \sqrt{3} \mathrm{~m}\).

As per given figures, two springs of spring constants \(k\) and \(2 k\) are connected to mass \(m\). If the period of oscillation in figure (a) is \(3 \mathrm{s}\), then the period of oscillation in figure (b) will be \(\sqrt{x}~ s\). The v...

In a Vernier Calipers, 10 divisions of Vernier scale is equal to the 9 divisions of main scale. When both jaws of Vernier calipers touch each other, the zero of the Vernier scale is shifted to the left of zero of the main scale and $$4^{\te...

If \(\vec{A}=(2 \hat{i}+3 \hat{j}-\hat{k})\, \mathrm{m}\) and \(\vec{B}=(\hat{i}+2 \hat{j}+2 \hat{k}) \,\mathrm{m}\). The magnitude of component of vector \(\vec{A}\) along vector \(\vec{B}\) will be ____________ \(\mathrm{m}\).

A transverse wave is represented by \(y=2 \sin (\omega t-k x)\, \mathrm{cm}\). The value of wavelength (in \(\mathrm{cm}\)) for which the wave velocity becomes equal to the maximum particle velocity, will be :