JEE Main 2022 (Online) 26th July Morning Shift

The mean and variance of a binomial distribution are \(\alpha\) and \(\frac{\alpha}{3}\) respectively. If \(\mathrm{P}(X=1)=\frac{4}{243}\), then \(\mathrm{P}(X=4\) or 5\()\) is equal to :

Let \(\mathrm{E}_{1}, \mathrm{E}_{2}, \mathrm{E}_{3}\) be three mutually exclusive events such that \(\mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{2+3 \mathrm{p}}{6}, \mathrm{P}\left(\mathrm{E}_{2}\right)=\frac{2-\mathrm{p}}{8}\) and $$\math...

If for some \(\mathrm{p}, \mathrm{q}, \mathrm{r} \in \mathbf{R}\), not all have same sign, one of the roots of the equation $$\left(\mathrm{p}^{2}+\mathrm{q}^{2}\right) x^{2}-2 \mathrm{q}(\mathrm{p}+\mathrm{r}) x+\mathrm{q}^{2}+\mathrm{r}^{...

The number of distinct real roots of the equation \(x^{5}\left(x^{3}-x^{2}-x+1\right)+x\left(3 x^{3}-4 x^{2}-2 x+4\right)-1=0\) is ______________.

Consider two G.Ps. 2, 22, 23, ..... and 4, 42, 43, .... of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is \({(2)^{{{225} \over 8}}}\), then \(\sum\limits_{k = 1}^n {k(n - k)}\) is equal to :

The series of positive multiples of 3 is divided into sets : \(\{3\},\{6,9,12\},\{15,18,21,24,27\}, \ldots\) Then the sum of the elements in the \(11^{\text {th }}\) set is equal to ____________.

A point \(P\) moves so that the sum of squares of its distances from the points \((1,2)\) and \((-2,1)\) is 14. Let \(f(x, y)=0\) be the locus of \(\mathrm{P}\), which intersects the \(x\)-axis at the points \(\mathrm{A}\), \(\mathrm{B}\) a...

The equations of the sides \(\mathrm{AB}, \mathrm{BC}\) and \(\mathrm{CA}\) of a triangle \(\mathrm{ABC}\) are \(2 x+y=0, x+\mathrm{p} y=15 \mathrm{a}\) and \(x-y=3\) respectively. If its orthocentre is \((2, a),-\frac{1}{2}<\mathrm{a}<2\),...

Let \(S=\left\{\theta \in[0,2 \pi]: 8^{2 \sin ^{2} \theta}+8^{2 \cos ^{2} \theta}=16\right\} .\) Then $$n(s) + \sum\limits_{\theta \in S}^{} {\left( {\sec \left( {{\pi \over 4} + 2\theta } \right)\cos ec\left( {{\pi \over 4} + 2\theta } ...

Let \(\overrightarrow{\mathrm{a}}=\alpha \hat{i}+\hat{j}-\hat{k}\) and \(\overrightarrow{\mathrm{b}}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0\). If the projection of \(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\) on th...

In a series \(L R\) circuit \(X_{L}=R\) and power factor of the circuit is \(P_{1}\). When capacitor with capacitance \(C\) such that \(X_{L}=X_{C}\) is put in series, the power factor becomes \(P_{2}\). The ratio \(\frac{P_{1}}{P_{2}}\) is...

The effective current I in the given circuit at very high frequencies will be ___________ A.

The disintegration rate of a certain radioactive sample at any instant is 4250 disintegrations per minute. 10 minutes later, the rate becomes 2250 disintegrations per minute. The approximate decay constant is :
\(\left(\right.\)Take $$\left...

In the hydrogen spectrum, \(\lambda\) be the wavelength of first transition line of Lyman series. The wavelength difference will be "a\(\lambda\)'' between the wavelength of \(3^{\text {rd }}\) transition line of Paschen series and that of ...

The total charge on the system of capacitors \(C_{1}=1 \mu \mathrm{F}, C_{2}=2 \mu \mathrm{F}, \mathrm{C}_{3}=4 \mu \mathrm{F}\) and \(\mathrm{C}_{4}=3 \mu \mathrm{F}\) connected in parallel is :
(Assume a battery of \(20 \mathrm{~V}\) is c...

A composite parallel plate capacitor is made up of two different dielectric materials with different thickness \(\left(t_{1}\right.\) and \(\left.t_{2}\right)\) as shown in figure. The two different dielectric materials are separated by a c...

A radio can tune to any station in \(6 \,\mathrm{MHz}\) to \(10 \,\mathrm{MHz}\) band. The value of corresponding wavelength bandwidth will be :

The current I in the given circuit will be :

Resistances are connected in a meter bridge circuit as shown in the figure. The balancing length \(l_{1}\) is \(40 \mathrm{~cm}\). Now an unknown resistance \(x\) is connected in series with \(\mathrm{P}\) and new balancing length is found ...

A parallel beam of light of wavelength \(900 \mathrm{~nm}\) and intensity \(100 \,\mathrm{Wm}^{-2}\) is incident on a surface perpendicular to the beam. The number of photons crossing \(1 \mathrm{~cm}^{2}\) area perpendicular to the beam in...

The magnetic field of a plane electromagnetic wave is given by :
\(\overrightarrow{\mathrm{B}}=2 \times 10^{-8} \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} \mathrm{t}\right) \,\hat{j} \mathrm{~T}\).
The amplitude of the electric fiel...

In the circuit shown below, maximum zener diode current will be _________ \(\mathrm{mA}\).

The graph between \(\frac{1}{u}\) and \(\frac{1}{v}\) for a thin convex lens in order to determine its focal length is plotted as shown in the figure. The refractive index of lens is \(1.5\) and its both the surfaces have same radius of cur...

The percentage decrease in the weight of a rocket, when taken to a height of \(32 \mathrm{~km}\) above the surface of earth will, be :
\((\) Radius of earth \(=6400 \mathrm{~km})\)

7 mol of a certain monoatomic ideal gas undergoes a temperature increase of \(40 \mathrm{~K}\) at constant pressure. The increase in the internal energy of the gas in this process is :
(Given $$\mathrm{R}=8.3 \,\mathrm{JK}^{-1} \mathrm{~mol...

A monoatomic gas at pressure \(\mathrm{P}\) and volume \(\mathrm{V}\) is suddenly compressed to one eighth of its original volume. The final pressure at constant entropy will be :

Three masses \(M=100 \mathrm{~kg}, \mathrm{~m}_{1}=10 \mathrm{~kg}\) and \(\mathrm{m}_{2}=20 \mathrm{~kg}\) are arranged in a system as shown in figure. All the surfaces are frictionless and strings are inextensible and weightless. The pull...

A monkey of mass \(50 \mathrm{~kg}\) climbs on a rope which can withstand the tension (T) of \(350 \mathrm{~N}\). If monkey initially climbs down with an acceleration of \(4 \mathrm{~m} / \mathrm{s}^{2}\) and then climbs up with an accelera...

A charge particle is moving in a uniform magnetic field \((2 \hat{i}+3 \hat{j}) \,\mathrm{T}\). If it has an acceleration of \((\alpha \hat{i}-4 \hat{j})\, \mathrm{m} / \mathrm{s}^{2}\), then the value of \(\alpha\) will be :

\(\mathrm{B}_{X}\) and \(\mathrm{B}_{\mathrm{Y}}\) are the magnetic fields at the centre of two coils \(\mathrm{X}\) and \(\mathrm{Y}\) respectively each carrying equal current. If coil \(X\) has 200 turns and \(20 \mathrm{~cm}\) radius and...

Two projectiles thrown at \(30^{\circ}\) and \(45^{\circ}\) with the horizontal respectively, reach the maximum height in same time. The ratio of their initial velocities is :

If the initial velocity in horizontal direction of a projectile is unit vector \(\hat{i}\) and the equation of trajectory is \(y=5 x(1-x)\). The \(y\) component vector of the initial velocity is ______________ \(\hat{j}\). ($$\mathrm{Take}$...

A water drop of radius \(1 \mathrm{~cm}\) is broken into 729 equal droplets. If surface tension of water is 75 dyne/ \(\mathrm{cm}\), then the gain in surface energy upto first decimal place will be :
(Given \(\pi=3.14\) )

In an experiment to determine the Young's modulus of wire of a length exactly \(1 \mathrm{~m}\), the extension in the length of the wire is measured as \(0.4 \mathrm{~mm}\) with an uncertainty of \(\pm\, 0.02 \mathrm{~mm}\) when a load of $...

A disc of mass \(1 \mathrm{~kg}\) and radius \(\mathrm{R}\) is free to rotate about a horizontal axis passing through its centre and perpendicular to the plane of disc. A body of same mass as that of disc is fixed at the highest point of th...

When a particle executes Simple Hormonic Motion, the nature of graph of velocity as a function of displacement will be :

A screw gauge of pitch \(0.5 \mathrm{~mm}\) is used to measure the diameter of uniform wire of length \(6.8 \mathrm{~cm}\), the main scale reading is \(1.5 \mathrm{~mm}\) and circular scale reading is 7 . The calculated curved surface area ...

In Young's double slit experiment, the fringe width is \(12 \mathrm{~mm}\). If the entire arrangement is placed in water of refractive index \(\frac{4}{3}\), then the fringe width becomes (in mm):

When a car is approaching the observer, the frequency of horn is \(100 \mathrm{~Hz}\). After passing the observer, it is \(50 \mathrm{~Hz}\). If the observer moves with the car, the frequency will be \(\frac{x}{3} \mathrm{~Hz}\) where $$x=$...

As per the given figure, two blocks each of mass \(250 \mathrm{~g}\) are connected to a spring of spring constant \(2 \,\mathrm{Nm}^{-1}\). If both are given velocity \(v\) in opposite directions, then maximum elongation of the spring is :