JEE Main 2023 (Online) 29th January Evening Shift

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is __________.

Let \(\mathrm{S} = \{ {w_1},{w_2},......\}\) be the sample space associated to a random experiment. Let \(P({w_n}) = {{P({w_{n - 1}})} \over 2},n \ge 2\). Let \(A = \{ 2k + 3l:k,l \in N\}\) and \(B = \{ {w_n}:n \in A\}\). Then P(B) is eq...

Let \(\alpha_1,\alpha_2,....,\alpha_7\) be the roots of the equation \({x^7} + 3{x^5} - 13{x^3} - 15x = 0\) and \(|{\alpha _1}| \ge |{\alpha _2}| \ge \,...\, \ge \,|{\alpha _7}|\). Then \(\alpha_1\alpha_2-\alpha_3\alpha_4+\alpha_5\alpha_6\)...

Let \(a_1=b_1=1\) and \({a_n} = {a_{n - 1}} + (n - 1),{b_n} = {b_{n - 1}} + {a_{n - 1}},\forall n \ge 2\). If \(S = \sum\limits_{n = 1}^{10} {{{{b_n}} \over {{2^n}}}}\) and \(T = \sum\limits_{n = 1}^8 {{n \over {{2^{n - 1}}}}}\), then $${...

Let \(\{ {a_k}\}\) and \(\{ {b_k}\} ,k \in N\), be two G.P.s with common ratios \({r_1}\) and \({r_2}\) respectively such that \({a_1} = {b_1} = 4\) and \({r_1} < {r_2}\). Let \({c_k} = {a_k} + {b_k},k \in N\). If \({c_2} = 5\) and $${c_3}...

Let R be a relation defined on \(\mathbb{N}\) as \(a\mathrm{R}b\) if \(2a+3b\) is a multiple of \(5,a,b\in \mathbb{N}\). Then R is

Let \(X=\{11,12,13,....,40,41\}\) and \(Y=\{61,62,63,....,90,91\}\) be the two sets of observations. If \(\overline x\) and \(\overline y\) are their respective means and \(\sigma^2\) is the variance of all the observations in $$\mathrm{X...

The set of all values of \(\lambda\) for which the equation \({\cos ^2}2x - 2{\sin ^4}x - 2{\cos ^2}x = \lambda\) has a real solution \(x\), is :

If $$\overrightarrow a = \widehat i + 2\widehat k,\overrightarrow b = \widehat i + \widehat j + \widehat k,\overrightarrow c = 7\widehat i - 3\widehat j + 4\widehat k,\overrightarrow r \times \overrightarrow b + \overrightarrow b \tim...

Let \(\overrightarrow a = 4\widehat i + 3\widehat j\) and \(\overrightarrow b = 3\widehat i - 4\widehat j + 5\widehat k\). If \(\overrightarrow c\) is a vector such that $$\overrightarrow c .\left( {\overrightarrow a \times \overrightar...

For the given figures, choose the correct options :

An inductor of inductance 2 \(\mathrm{\mu H}\) is connected in series with a resistance, a variable capacitor and an AC source of frequency 7 kHz. The value of capacitance for which maximum current is drawn into the circuit $$\frac{1}{x}\ma...

Substance A has atomic mass number 16 and half life of 1 day. Another substance B has atomic mass number 32 and half life of \(\frac{1}{2}\) day. If both A and B simultaneously start undergo radio activity at the same time with initial mass...

An object moves at a constant speed along a circular path in a horizontal plane with center at the origin. When the object is at \(x=+2~\mathrm{m}\), its velocity is \(\mathrm{ - 4\widehat j}\) m/s. The object's velocity (v) and acceleratio...

A car is moving on a circular path of radius 600 m such that the magnitudes of the tangential acceleration and centripetal acceleration are equal. The time taken by the car to complete first quarter of revolution, if it is moving with an in...

The modulation index for an A.M. wave having maximum and minimum peak-to-peak voltages of 14 mV and 6 mV respectively is-

With the help of potentiometer, we can determine the value of emf of a given cell. The sensitivity of the potentiometer is
(A) directly proportional to the length of the potentiometer wire
(B) directly proportional to the potential gradient...

A null point is found at 200 cm in potentiometer when cell in secondary circuit is shunted by 5\(\Omega\). When a resistance of 15\(\Omega\) is used for shunting, null point moves to 300 cm. The internal resistance of the cell is __________...

When two resistance \(\mathrm{R_1}\) and \(\mathrm{R_2}\) connected in series and introduced into the left gap of a meter bridge and a resistance of 10 \(\Omega\) is introduced into the right gap, a null point is found at 60 cm from left si...

The ratio of de-Broglie wavelength of an \(\alpha\) particle and a proton accelerated from rest by the same potential is \(\frac{1}{\sqrt m}\), the value of m is -

A square loop of area 25 cm\(^2\) has a resistance of 10 \(\Omega\). The loop is placed in uniform magnetic field of magnitude 40.0 T. The plane of loop is perpendicular to the magnetic field. The work done in pulling the loop out of the ma...

Given below are two statements :
Statement I : Electromagnetic waves are not deflected by electric and magnetic field.
Statement II : The amplitude of electric field and the magnetic field in electromagnetic waves are related to each other ...

For the given logic gates combination, the correct truth table will be

A point charge \(2\times10^{-2}~\mathrm{C}\) is moved from P to S in a uniform electric field of \(30~\mathrm{NC^{-1}}\) directed along positive x-axis. If coordinates of P and S are (1, 2, 0) m and (0, 0, 0) m respectively, the work done b...

For a charged spherical ball, electrostatic potential inside the ball varies with \(r\) as \(\mathrm{V}=2ar^2+b\).
Here, \(a\) and \(b\) are constant and r is the distance from the center. The volume charge density inside the ball is $$-\la...

A scientist is observing a bacteria through a compound microscope. For better analysis and to improve its resolving power he should. (Select the best option)

In an experiment of measuring the refractive index of a glass slab using travelling microscope in physics lab, a student measures real thickness of the glass slab as 5.25 mm and apparent thickness of the glass slab as 5.00 mm. Travelling mi...

The time period of a satellite of earth is 24 hours. If the separation between the earth and the satellite is decreased to one fourth of the previous value, then its new time period will become.

Heat energy of 184 kJ is given to ice of mass 600 g at \(-12^\circ \mathrm{C}\). Specific heat of ice is \(\mathrm{2222.3~J~kg^{-1^\circ}~C^{-1}}\) and latent heat of ice in 336 \(\mathrm{kJ/kg^{-1}}\)
A. Final temperature of system will be...

At 300 K, the rms speed of oxygen molecules is \(\sqrt {{{\alpha + 5} \over \alpha }}\) times to that of its average speed in the gas. Then, the value of \(\alpha\) will be
(used \(\pi = {{22} \over 7}\))

A force acts for 20 s on a body of mass 20 kg, starting from rest, after which the force ceases and then body describes 50 m in the next 10 s. The value of force will be:

The time taken by an object to slide down 45\(^\circ\) rough inclined plane is n times as it takes to slide down a perfectly smooth 45\(^\circ\) incline plane. The coefficient of kinetic friction between the object and the incline plane is ...

The electric current in a circular coil of four turns produces a magnetic induction 32 T at its centre. The coil is unwound and is rewound into a circular coil of single turn, the magnetic induction at the centre of the coil by the same cur...

A fully loaded boeing aircraft has a mass of \(5.4\times10^5\) kg. Its total wing area is 500 m\(^2\). It is in level flight with a speed of 1080 km/h. If the density of air \(\rho\) is 1.2 kg m\(^{-3}\), the fractional increase in the spe...

A metal block of base area 0.20 m\(^2\) is placed on a table, as shown in figure. A liquid film of thickness 0.25 mm is inserted between the block and the table. The block is pushed by a horizontal force of 0.1 N and moves with a constant s...

A particle of mass 100 g is projected at time t = 0 with a speed 20 ms\(^{-1}\) at an angle 45\(^\circ\) to the horizontal as given in the figure. The magnitude of the angular momentum of the particle about the starting point at time t = 2s...

A particle of mass 250 g executes a simple harmonic motion under a periodic force \(\mathrm{F}=(-25~x)\mathrm{N}\). The particle attains a maximum speed of 4 m/s during its oscillation. The amplitude of the motion is ___________ cm.

The equation of a circle is given by \(x^2+y^2=a^2\), where a is the radius. If the equation is modified to change the origin other than (0, 0), then find out the correct dimensions of A and B in a new equation : $${(x - At)^2} + {\left( {y...

Unpolarised light is incident on the boundary between two dielectric media, whose dielectric constants are 2.8 (medium \(-1\)) and 6.8 (medium \(-2\)), respectively. To satisfy the condition, so that the reflected and refracted rays are per...

Identify the correct statements from the following :
A. Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative.
B. Work done by gravitational force in lifting a bucket out of a well by a rope ...