JEE Main 2023 (Online) 1st February Morning Shift

For a triangle \(ABC\), the value of \(\cos 2A + \cos 2B + \cos 2C\) is least. If its inradius is 3 and incentre is M, then which of the following is NOT correct?

Let \(S = \left\{ {x:x \in \mathbb{R}\,\mathrm{and}\,{{(\sqrt 3 + \sqrt 2 )}^{{x^2} - 4}} + {{(\sqrt 3 - \sqrt 2 )}^{{x^2} - 4}} = 10} \right\}\). Then \(n(S)\) is equal to

The sum of 10 terms of the series
\({1 \over {1 + {1^2} + {1^4}}} + {2 \over {1 + {2^2} + {2^4}}} + {3 \over {1 + {3^2} + {3^4}}}\, + \,....\) is

Let \(a_{1}=8, a_{2}, a_{3}, \ldots, a_{n}\) be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170 , then the product of its middle two terms is ___________.

Let \(R\) be a relation on \(\mathbb{R}\), given by \(R=\{(a, b): 3 a-3 b+\sqrt{7}\) is an irrational number \(\}\). Then \(R\) is

The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes of the remaining two observations is :

The combined equation of the two lines \(ax+by+c=0\) and \(a'x+b'y+c'=0\) can be written as
\((ax+by+c)(a'x+b'y+c')=0\).
The equation of the angle bisectors of the lines represented by the equation \(2x^2+xy-3y^2=0\) is :

If the orthocentre of the triangle, whose vertices are (1, 2), (2, 3) and (3, 1) is \((\alpha,\beta)\), then the quadratic equation whose roots are \(\alpha+4\beta\) and \(4\alpha+\beta\), is :

Let \(\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}, \vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}\) and \(\vec{u}\) be a vector such that \(|\vec{u}|=\alpha>0\). If the minimum value of the scalar triple product $$\left[ {\matrix{
{\overrightar...

\(A(2,6,2), B(-4,0, \lambda), C(2,3,-1)\) and \(D(4,5,0),|\lambda| \leq 5\) are the vertices of a quadrilateral \(A B C D\). If its area is 18 square units, then \(5-6 \lambda\) is equal to __________.

Match List - I with List - II :

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:nor...

A series LCR circuit is connected to an ac source of \(220 \mathrm{~V}, 50 \mathrm{~Hz}\). The circuit contain a resistance \(\mathrm{R}=100 ~\Omega\) and an inductor of inductive reactance \(\mathrm{X}_{\mathrm{L}}=79.6 ~\Omega\). The capa...

The mass of proton, neutron and helium nucleus are respectively \(1.0073~u,1.0087~u\) and \(4.0015~u\). The binding energy of helium nucleus is :

A light of energy \(12.75 ~\mathrm{eV}\) is incident on a hydrogen atom in its ground state. The atom absorbs the radiation and reaches to one of its excited states. The angular momentum of the atom in the excited state is $$\frac{x}{\pi} \...

Which of the following frequencies does not belong to FM broadcast ?

The equivalent resistance between \(A\) and \(B\) of the network shown in figure;

In an experiment to find emf of a cell using potentiometer, the length of null point for a cell of emf \(1.5 \mathrm{~V}\) is found to be \(60 \mathrm{~cm}\). If this cell is replaced by another cell of emf E, the length-of null point incre...

A proton moving with one tenth of velocity of light has a certain de Broglie wavelength of \(\lambda\). An alpha particle having certain kinetic energy has the same de-Brogle wavelength \(\lambda\). The ratio of kinetic energy of proton and...

Match List I with List II :

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:normal;...

Match List I with List II:

.tg {border-collapse:collapse;border-spacing:0;}
.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
overflow:hidden;padding:10px 5px;word-break:normal;}...

Let \(\sigma\) be the uniform surface charge density of two infinite thin plane sheets shown in figure. Then the electric fields in three different region \(E_{I}, E_{I I}\) and \(E_{I I I}\) are:

Two equal positive point charges are separated by a distance \(2 a\). The distance of a point from the centre of the line joining two charges on the equatorial line (perpendicular bisector) at which force experienced by a test charge $$\mat...

A thin cylindrical rod of length \(10 \mathrm{~cm}\) is placed horizontally on the principle axis of a concave mirror of focal length \(20 \mathrm{~cm}\). The rod is placed in a such a way that mid point of the rod is at \(40 \mathrm{~cm}\)...

If earth has a mass nine times and radius twice to that of a planet P. Then \(\frac{v_{e}}{3} \sqrt{x} \mathrm{~ms}^{-1}\) will be the minimum velocity required by a rocket to pull out of gravitational force of \(\mathrm{P}\), where $$v_{e}...

Given below are two statements:
Statement I: Acceleration due to gravity is different at different places on the surface of earth.
Statement II: Acceleration due to gravity increases as we go down below the earth's surface.
In the light of ...

A sample of gas at temperature \(T\) is adiabatically expanded to double its volume. The work done by the gas in the process is \(\left(\mathrm{given}, \gamma=\frac{3}{2}\right)\) :

\(\left(P+\frac{a}{V^{2}}\right)(V-b)=R T\) represents the equation of state of some gases. Where \(P\) is the pressure, \(V\) is the volume, \(T\) is the temperature and \(a, b, R\) are the constants. The physical quantity, which has dimen...

The average kinetic energy of a molecule of the gas is

A block of mass \(5 \mathrm{~kg}\) is placed at rest on a table of rough surface. Now, if a force of \(30 \mathrm{~N}\) is applied in the direction parallel to surface of the table, the block slides through a distance of \(50 \mathrm{~m}\) ...

Find the magnetic field at the point \(\mathrm{P}\) in figure. The curved portion is a semicircle connected to two long straight wires.

A charge particle of \(2 ~\mu \mathrm{C}\) accelerated by a potential difference of \(100 \mathrm{~V}\) enters a region of uniform magnetic field of magnitude \(4 ~\mathrm{mT}\) at right angle to the direction of field. The charge particle ...

A child stands on the edge of the cliff \(10 \mathrm{~m}\) above the ground and throws a stone horizontally with an initial speed of \(5 \mathrm{~ms}^{-1}\). Neglecting the air resistance, the speed with which the stone hits the ground will...

An object moves with speed \(v_1,v_2\) and \(v_3\) along a line segment AB, BC and CD respectively as shown in figure. Where AB = BC and AD = 3AB, then average speed of the object will be:

A mercury drop of radius \(10^{-3}~\mathrm{m}\) is broken into 125 equal size droplets. Surface tension of mercury is \(0.45~\mathrm{Nm}^{-1}\). The gain in surface energy is :

A certain pressure '\(\mathrm{P}\)' is applied to 1 litre of water and 2 litre of a liquid separately. Water gets compressed to \(0.01 \%\) whereas the liquid gets compressed to \(0.03 \%\). The ratio of Bulk modulus of water to that of the...

A solid cylinder is released from rest from the top of an inclined plane of inclination \(30^{\circ}\) and length \(60 \mathrm{~cm}\). If the cylinder rolls without slipping, its speed upon reaching the bottom of the inclined plane is _____...

The amplitude of a particle executing SHM is \(3 \mathrm{~cm}\). The displacement at which its kinetic energy will be \(25 \%\) more than the potential energy is: __________ \(\mathrm{cm}\)

'\(n\)' polarizing sheets are arranged such that each makes an angle \(45^{\circ}\) with the preceeding sheet. An unpolarized light of intensity I is incident into this arrangement. The output intensity is found to be \(I / 64\). The value ...

A steel wire with mass per unit length \(7.0 \times 10^{-3} \mathrm{~kg} \mathrm{~m}^{-1}\) is under tension of \(70 \mathrm{~N}\). The speed of transverse waves in the wire will be:

A small particle moves to position \(5 \hat{i}-2 \hat{j}+\hat{k}\) from its initial position \(2 \hat{i}+3 \hat{j}-4 \hat{k}\) under the action of force \(5 \hat{i}+2 \hat{j}+7 \hat{k} \mathrm{~N}\). The value of work done will be _________...