JEE Main 2022 (Online) 27th July Evening Shift

Consider a matrix $$A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta\end{array}\right]$$, where \(\alpha, \beta, \gamma\) are three distinct natural numb...

If the length of the latus rectum of a parabola, whose focus is \((a, a)\) and the tangent at its vertex is \(x+y=a\), is 16, then \(|a|\) is equal to :

Let X have a binomial distribution B(n, p) such that the sum and the product of the mean and variance of X are 24 and 128 respectively. If \(P(X>n-3)=\frac{k}{2^{n}}\), then k is equal to :

A six faced die is biased such that
\(3 \times \mathrm{P}(\)a prime number\()\,=6 \times \mathrm{P}(\)a composite number\()\,=2 \times \mathrm{P}(1)\).
Let X be a random variable that counts the number of times one gets a perfect square on ...

If \(\alpha, \beta\) are the roots of the equation
\(x^{2}-\left(5+3^{\sqrt{\log _{3} 5}}-5^{\sqrt{\log _{5} 3}}\right)x+3\left(3^{\left(\log _{3} 5\right)^{\frac{1}{3}}}-5^{\left(\log _{5} 3\right)^{\frac{2}{3}}}-1\right)=0\),
then the e...

Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5 . Let the sum of its first five terms be \(\frac{98}{25}\). Then the sum of the first 21 terms of an AP, whose first term is $$10\mathrm{a r}, \mathrm{n}...

\(\frac{2^{3}-1^{3}}{1 \times 7}+\frac{4^{3}-3^{3}+2^{3}-1^{3}}{2 \times 11}+\frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}{3 \times 15}+\cdots+ \frac{30^{3}-29^{3}+28^{3}-27^{3}+\ldots+2^{3}-1^{3}}{15 \times 63}\) is equal to _____________.

The equations of the sides \(\mathrm{AB}, \mathrm{BC}\) and CA of a triangle ABC are \(2 x+y=0, x+\mathrm{p} y=39\) and \(x-y=3\) respectively and \(\mathrm{P}(2,3)\) is its circumcentre. Then which of the following is NOT true?

Let \(S=\left\{\theta \in\left(0, \frac{\pi}{2}\right): \sum\limits_{m=1}^{9} \sec \left(\theta+(m-1) \frac{\pi}{6}\right) \sec \left(\theta+\frac{m \pi}{6}\right)=-\frac{8}{\sqrt{3}}\right\}\). Then

Let \(\overrightarrow a\), \(\overrightarrow b\), \(\overrightarrow c\) be three non-coplanar vectors such that \(\overrightarrow a\) \(\times\) \(\overrightarrow b\) = 4\(\overrightarrow c\), \(\overrightarrow b\) \(\times\) $$\over...

A series LCR circuit has \(\mathrm{L}=0.01\, \mathrm{H}, \mathrm{R}=10\, \Omega\) and \(\mathrm{C}=1 \mu \mathrm{F}\) and it is connected to ac voltage of amplitude \(\left(\mathrm{V}_{\mathrm{m}}\right) 50 \mathrm{~V}\). At frequency $$60 ...

The activity of a radioactive material is \(6.4 \times 10^{-4}\) curie. Its half life is 5 days. The activity will become \(5 \times 10^{-6}\) curie after :

As show in the figure, in steady state, the charge stored in the capacitor is ____________ \(\times\, 10^{-6}\) C.

A parallel plate capacitor with width \(4 \mathrm{~cm}\), length \(8 \mathrm{~cm}\) and separation between the plates of \(4 \mathrm{~mm}\) is connected to a battery of \(20 \mathrm{~V}\). A dielectric slab of dielectric constant 5 having l...

A body of mass \(10 \mathrm{~kg}\) is projected at an angle of \(45^{\circ}\) with the horizontal. The trajectory of the body is observed to pass through a point \((20,10)\). If \(\mathrm{T}\) is the time of flight, then its momentum vector...

A square wave of the modulating signal is shown in the figure. The carrier wave is given by \(C(t)=5 \sin (8 \pi t)\) Volt. The modulation index is :

(A) The drift velocity of electrons decreases with the increase in the temperature of conductor.
(B) The drift velocity is inversely proportional to the area of cross-section of given conductor.
(C) The drift velocity does not depend on the...

In the given figure of meter bridge experiment, the balancing length AC corresponding to null deflection of the galvanometer is \(40 \mathrm{~cm}\). The balancing length, if the radius of the wire \(\mathrm{AB}\) is doubled, will be _______...

With reference to the observations in photo-electric effect, identify the correct statements from below :
(A) The square of maximum velocity of photoelectrons varies linearly with frequency of incident light.
(B) The value of saturation cur...

A conducting circular loop is placed in \(X-Y\) plane in presence of magnetic field \(\overrightarrow{\mathrm{B}}=\left(3 \mathrm{t}^{3} \,\hat{j}+3 \mathrm{t}^{2}\, \hat{k}\right)\) in SI unit. If the radius of the loop is $$1 \mathrm{~m}$...

Identify the correct statements from the following descriptions of various properties of electromagnetic waves.
(A) In a plane electromagnetic wave electric field and magnetic field must be perpendicular to each other and direction of propa...

For a constant collector-emitter voltage of \(8 \mathrm{~V}\), the collector current of a transistor reached to the value of \(6 \mathrm{~mA}\) from \(4 \mathrm{~mA}\), whereas base current changed from \(20 \,\mu \mathrm{A}\) to $$25 \,\mu...

A charge of \(4 \,\mu \mathrm{C}\) is to be divided into two. The distance between the two divided charges is constant. The magnitude of the divided charges so that the force between them is maximum, will be :

A thin prism of angle \(6^{\circ}\) and refractive index for yellow light \(\left(\mathrm{n}_{\mathrm{Y}}\right) 1.5\) is combined with another prism of angle \(5^{\circ}\) and \(\mathrm{n}_{\mathrm{Y}}=1.55\). The combination produces no d...

A body of mass \(\mathrm{m}\) is projected with velocity \(\lambda \,v_{\mathrm{e}}\) in vertically upward direction from the surface of the earth into space. It is given that \(v_{\mathrm{e}}\) is escape velocity and \(\lambda<1\). If air ...

In \(1^{\text {st }}\) case, Carnot engine operates between temperatures \(300 \mathrm{~K}\) and \(100 \mathrm{~K}\). In \(2^{\text {nd }}\) case, as shown in the figure, a combination of two engines is used. The efficiency of this combinat...

Which statements are correct about degrees of freedom ?
(A) A molecule with n degrees of freedom has n\(^{2}\) different ways of storing energy.
(B) Each degree of freedom is associated with \(\frac{1}{2}\) RT average energy per mole.
(C) A...

A block of mass M slides down on a rough inclined plane with constant velocity. The angle made by the incline plane with horizontal is \(\theta\). The magnitude of the contact force will be :

A block 'A' takes 2 s to slide down a frictionless incline of 30\(^\circ\) and length 'l', kept inside a lift going up with uniform velocity 'v'. If the incline is changed to 45\(^\circ\), the time taken by the block, to slide down the incl...

Two inclined planes are placed as shown in figure. A block is projected from the Point A of inclined plane AB along its surface with a velocity just sufficient to carry it to the top Point B at a height 10 m. After reaching the Point B the ...

A compass needle of oscillation magnetometer oscillates 20 times per minute at a place \(\mathrm{P}\) of \(\operatorname{dip} 30^{\circ}\). The number of oscillations per minute become 10 at another place \(\mathrm{Q}\) of \(60^{\circ}\) di...

A cyclotron is used to accelerate protons. If the operating magnetic field is \(1.0 \mathrm{~T}\) and the radius of the cyclotron 'dees' is \(60 \mathrm{~cm}\), the kinetic energy of the accelerated protons in MeV will be :
$$[\mathrm{use} ...

The velocity of the bullet becomes one third after it penetrates 4 cm in a wooden block. Assuming that bullet is facing a constant resistance during its motion in the block. The bullet stops completely after travelling at (4 + x) cm inside ...

A steel wire of length \(3.2 \mathrm{~m}\left(\mathrm{Y}_{\mathrm{s}}=2.0 \times 10^{11} \,\mathrm{Nm}^{-2}\right)\) and a copper wire of length \(4.4 \mathrm{~m}\left(\mathrm{Y}_{\mathrm{c}}=1.1 \times 10^{11} \,\mathrm{Nm}^{-2}\right)\), ...

In an experiment to determine the Young's modulus, steel wires of five different lengths \((1,2,3,4\), and \(5 \mathrm{~m})\) but of same cross section \(\left(2 \mathrm{~mm}^{2}\right)\) were taken and curves between extension and load wer...

A spherical soap bubble of radius 3 cm is formed inside another spherical soap bubble of radius 6 cm. If the internal pressure of the smaller bubble of radius 3 cm in the above system is equal to the internal pressure of the another single ...

A solid cylinder length is suspended symmetrically through two massless strings, as shown in the figure. The distance from the initial rest position, the cylinder should be unbinding the strings to achieve a speed of \(4 \mathrm{~ms}^{-1}\)...

An expression of energy density is given by \(u=\frac{\alpha}{\beta} \sin \left(\frac{\alpha x}{k t}\right)\), where \(\alpha, \beta\) are constants, \(x\) is displacement, \(k\) is Boltzmann constant and t is the temperature. The dimension...

Two coherent sources of light interfere. The intensity ratio of two sources is \(1: 4\). For this interference pattern if the value of \(\frac{I_{\max }+I_{\min }}{I_{\max }-I_{\min }}\) is equal to \(\frac{2 \alpha+1}{\beta+3}\), then $$\f...

A wire of length 30 cm, stretched between rigid supports, has it's nth and (n + 1)th harmonics at 400 Hz and 450 Hz, respectively. If tension in the string is 2700 N, it's linear mass density is ____________ kg/m.