JEE Main 2022 (Online) 27th July Morning Shift

Let $$A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$$. Let \(\alpha, \beta \in \mathbb{R}\) be such that \(\alpha A^{2}+\beta A=2 I\). Then \(\alpha+\beta\) is equal to

Let \(S\) be the set containing all \(3 \times 3\) matrices with entries from \(\{-1,0,1\}\). The total number of matrices \(A \in S\) such that the sum of all the diagonal elements of \(A^{\mathrm{T}} A\) is 6 is ____________.

Let \(P(a, b)\) be a point on the parabola \(y^{2}=8 x\) such that the tangent at \(P\) passes through the centre of the circle \(x^{2}+y^{2}-10 x-14 y+65=0\). Let \(A\) be the product of all possible values of \(a\) and \(B\) be the produc...

Let \(S\) be the sample space of all five digit numbers. It \(p\) is the probability that a randomly selected number from \(S\), is a multiple of 7 but not divisible by 5 , then \(9 p\) is equal to :

Suppose \(a_{1}, a_{2}, \ldots, a_{n}\), .. be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of first nine terms of the progression is \(5: 17\) and , \(110 < {a_{15}} < 120\), then the...

Let \(R_{1}\) and \(R_{2}\) be two relations defined on \(\mathbb{R}\) by
\(a \,R_{1} \,b \Leftrightarrow a b \geq 0\) and \(a \,R_{2} \,b \Leftrightarrow a \geq b\)
Then,

The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is _____________.

Let \(A(1,1), B(-4,3), C(-2,-5)\) be vertices of a triangle \(A B C, P\) be a point on side \(B C\), and \(\Delta_{1}\) and \(\Delta_{2}\) be the areas of triangles \(A P B\) and \(A B C\), respectively. If \(\Delta_{1}: \Delta_{2}=4: 7\), ...

Let \(\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}\) and \(\vec{b}=3 \hat{i}-5 \hat{j}+4 \hat{k}\) be two vectors, such that \(\vec{a} \times \vec{b}=-\hat{i}+9 \hat{j}+12 \hat{k}\). Then the projection of \(\vec{b}-2 \vec{a}\) on $$\vec{b}...

\(\text { Let } \vec{a}=2 \hat{i}-\hat{j}+5 \hat{k} \text { and } \vec{b}=\alpha \hat{i}+\beta \hat{j}+2 \hat{k} \text {. If }((\vec{a} \times \vec{b}) \times \hat{i}) \cdot \hat{k}=\frac{23}{2} \text {, then }|\vec{b} \times 2 \hat{j}|\)...

A direct current of \(4 \mathrm{~A}\) and an alternating current of peak value \(4 \mathrm{~A}\) flow through resistance of \(3\, \Omega\) and \(2\,\Omega\) respectively. The ratio of heat produced in the two resistances in same interval of...

To light, a \(50 \mathrm{~W}, 100 \mathrm{~V}\) lamp is connected, in series with a capacitor of capacitance \(\frac{50}{\pi \sqrt{x}} \mu F\), with \(200 \mathrm{~V}, 50 \mathrm{~Hz} \,\mathrm{AC}\) source. The value of \(x\) will be _____...

What is the half-life period of a radioactive material if its activity drops to \(1 / 16^{\text {th }}\) of its initial value in 30 years?

At a particular station, the TV transmission tower has a height of 100 m. To triple its coverage range, height of the tower should be increased to

Two sources of equal emfs are connected in series. This combination is connected to an external resistance R. The internal resistances of the two sources are \(r_{1}\) and \(r_{2}\) \(\left(r_{1}>r_{2}\right)\). If the potential difference ...

In a meter bridge experiment, for measuring unknown resistance 'S', the null point is obtained at a distance \(30 \mathrm{~cm}\) from the left side as shown at point D. If R is \(5.6\) \(\mathrm{k} \Omega\), then the value of unknown resist...

A \(1 \mathrm{~m}\) long copper wire carries a current of \(1 \mathrm{~A}\). If the cross section of the wire is \(2.0 \mathrm{~mm}^{2}\) and the resistivity of copper is \(1.7 \times 10^{-8}\, \Omega \mathrm{m}\), the force experienced by ...

An electron (mass \(\mathrm{m}\)) with an initial velocity \(\vec{v}=v_{0} \hat{i}\left(v_{0}>0\right)\) is moving in an electric field \(\vec{E}=-E_{0} \hat{i}\left(E_{0}>0\right)\) where \(E_{0}\) is constant. If at \(\mathrm{t}=0\) de Br...

A beam of light travelling along \(X\)-axis is described by the electric field \(E_{y}=900 \sin \omega(\mathrm{t}-x / c)\). The ratio of electric force to magnetic force on a charge \(\mathrm{q}\) moving along \(Y\)-axis with a speed of $$3...

A logic gate circuit has two inputs A and B and output Y. The voltage waveforms of A, B and Y are shown below.

The logic gate circuit is :

Two identical positive charges \(Q\) each are fixed at a distance of '2a' apart from each other. Another point charge \(q_{0}\) with mass 'm' is placed at midpoint between two fixed charges. For a small displacement along the line joining t...

A long cylindrical volume contains a uniformly distributed charge of density \(\rho \,\mathrm{Cm}^{-3}\). The electric field inside the cylindrical volume at a distance \(x=\frac{2 \varepsilon_{0}}{\rho} \mathrm{m}\) from its axis is ______...

A microscope was initially placed in air (refractive index 1). It is then immersed in oil (refractive index 2). For a light whose wavelength in air is \(\lambda\), calculate the change of microscope's resolving power due to oil and choose t...

Two satellites \(\mathrm{A}\) and \(\mathrm{B}\), having masses in the ratio \(4: 3\), are revolving in circular orbits of radii \(3 \mathrm{r}\) and \(4 \mathrm{r}\) respectively around the earth. The ratio of total mechanical energy of $$...

If \(K_{1}\) and \(K_{2}\) are the thermal conductivities, \(L_{1}\) and \(L_{2}\) are the lengths and \(A_{1}\) and \(A_{2}\) are the cross sectional areas of steel and copper rods respectively such that $$\frac{K_{2}}{K_{1}}=9, \frac{A_{1...

Read the following statements :
A. When small temperature difference between a liquid and its surrounding is doubled, the rate of loss of heat of the liquid becomes twice.
B. Two bodies \(P\) and \(Q\) having equal surface areas are maintai...

Same gas is filled in two vessels of the same volume at the same temperature. If the ratio of the number of molecules is \(1: 4\), then
A. The r.m.s. velocity of gas molecules in two vessels will be the same.
B. The ratio of pressure in the...

A bag is gently dropped on a conveyor belt moving at a speed of \(2 \mathrm{~m} / \mathrm{s}\). The coefficient of friction between the conveyor belt and bag is \(0.4\). Initially, the bag slips on the belt before it stops due to friction. ...

Two bar magnets oscillate in a horizontal plane in earth's magnetic field with time periods of \(3 \mathrm{~s}\) and \(4 \mathrm{~s}\) respectively. If their moments of inertia are in the ratio of \(3: 2\), then the ratio of their magnetic ...

A magnet hung at \(45^{\circ}\) with magnetic meridian makes an angle of \(60^{\circ}\) with the horizontal. The actual value of the angle of dip is -

A ball of mass m is thrown vertically upward. Another ball of mass \(2 \mathrm{~m}\) is thrown at an angle \(\theta\) with the vertical. Both the balls stay in air for the same period of time. The ratio of the heights attained by the two ba...

A bullet is shot vertically downwards with an initial velocity of \(100 \mathrm{~m} / \mathrm{s}\) from a certain height. Within 10 s, the bullet reaches the ground and instantaneously comes to rest due to the perfectly inelastic collision....

Two cylindrical vessels of equal cross-sectional area \(16 \mathrm{~cm}^{2}\) contain water upto heights \(100 \mathrm{~cm}\) and \(150 \mathrm{~cm}\) respectively. The vessels are interconnected so that the water levels in them become equa...

A square aluminum (shear modulus is \(25 \times 10^{9}\, \mathrm{Nm}^{-2}\)) slab of side \(60 \mathrm{~cm}\) and thickness \(15 \mathrm{~cm}\) is subjected to a shearing force (on its narrow face) of \(18.0 \times 10^{4}\) \(\mathrm{N}\). ...

A pulley of radius \(1.5 \mathrm{~m}\) is rotated about its axis by a force \(F=\left(12 \mathrm{t}-3 \mathrm{t}^{2}\right) N\) applied tangentially (while t is measured in seconds). If moment of inertia of the pulley about its axis of rota...

A mass \(0.9 \mathrm{~kg}\), attached to a horizontal spring, executes SHM with an amplitude \(\mathrm{A}_{1}\). When this mass passes through its mean position, then a smaller mass of \(124 \mathrm{~g}\) is placed over it and both masses m...

A torque meter is calibrated to reference standards of mass, length and time each with \(5 \%\) accuracy. After calibration, the measured torque with this torque meter will have net accuracy of :

The one division of main scale of Vernier callipers reads \(1 \mathrm{~mm}\) and 10 divisions of Vernier scale is equal to the 9 divisions on main scale. When the two jaws of the instrument touch each other, the zero of the Vernier lies to ...

Two beams of light having intensities I and 4I interfere to produce a fringe pattern on a screen. The phase difference between the two beams are \(\pi / 2\) and \(\pi / 3\) at points \(\mathrm{A}\) and \(\mathrm{B}\) respectively. The diffe...

Sand is being dropped from a stationary dropper at a rate of \(0.5 \,\mathrm{kgs}^{-1}\) on a conveyor belt moving with a velocity of \(5 \mathrm{~ms}^{-1}\). The power needed to keep the belt moving with the same velocity will be :