JEE Main 2024 (Online) 27th January Morning Shift

If the shortest distance of the parabola $y^2=4 x$ from the centre of the circle $x^2+y^2-4 x-16 y+64=0$ is $\mathrm{d}$, then $\mathrm{d}^2$ is equal to :

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $a=P(X=3), b=P(X \geqslant 3)$ and $c=P(X \geqslant 6 \mid X>3)$. Then $\frac{b+c}{a}$ is equal to __________.

The number of common terms in the progressions $4,9,14,19, \ldots \ldots$, up to $25^{\text {th }}$ term and $3,6,9,12, \ldots \ldots$, up to $37^{\text {th }}$ term is :

If $8=3+\frac{1}{4}(3+p)+\frac{1}{4^2}(3+2 p)+\frac{1}{4^3}(3+3 p)+\cdots \cdots \infty$, then the value of $p$ is ____________.

Let $S=\{1,2,3, \ldots, 10\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $\mathrm{R}=\{(\mathrm{A}, \mathrm{B}): \mathrm{A} \cap \mathrm{B} \neq \phi ; \mathrm{A}, \mathrm{B} \in \mathrm{M}\}$ is :

Let $\mathrm{a}_1, \mathrm{a}_2, \ldots \mathrm{a}_{10}$ be 10 observations such that $\sum\limits_{\mathrm{k}=1}^{10} \mathrm{a}_{\mathrm{k}}=50$ and $\sum\limits_{\forall \mathrm{k} < \mathrm{j}} \mathrm{a}_{\mathrm{k}} \cdot \mathrm{a}_{...

The portion of the line $4 x+5 y=20$ in the first quadrant is trisected by the lines $\mathrm{L}_1$ and $\mathrm{L}_2$ passing through the origin. The tangent of an angle between the lines $\mathrm{L}_1$ and $\mathrm{L}_2$ is :

Let the set of all $a \in \mathbf{R}$ such that the equation $\cos 2 x+a \sin x=2 a-7$ has a solution be $[p, q]$ and $r=\tan 9^{\circ}-\tan 27^{\circ}-\frac{1}{\cot 63^{\circ}}+\tan 81^{\circ}$, then pqr is equal to ____________.

Let $\overrightarrow{\mathrm{a}}=\hat{i}+2 \hat{j}+\hat{k}, $
$\overrightarrow{\mathrm{b}}=3(\hat{i}-\hat{j}+\hat{k})$.
Let $\overrightarrow{\mathrm{c}}$ be the vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c...

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{i}-2 \hat{j}+2 \hat{k}$ and $\alpha \hat{i}+2 \alpha \hat{j}-2 \hat{k}$ is acute, is ___________.

The radius of third stationary orbit of electron for Bohr's atom is R. The radius of fourth stationary orbit will be:

In a nuclear fission process, a high mass nuclide \((A \approx 236)\) with binding energy \(7.6 \mathrm{~MeV} /\) Nucleon dissociated into middle mass nuclides \((\mathrm{A} \approx 118)\), having binding energy of $$8.6 \mathrm{~MeV} / \ma...

The charge accumulated on the capacitor connected in the following circuit is _______ \(\mu \mathrm{C}\) (Given \(\mathrm{C}=150 \mu \mathrm{F})\)

Two bodies of mass \(4 \mathrm{~g}\) and \(25 \mathrm{~g}\) are moving with equal kinetic energies. The ratio of magnitude of their linear momentum is :

A body of mass \(1000 \mathrm{~kg}\) is moving horizontally with a velocity \(6 \mathrm{~m} / \mathrm{s}\). If \(200 \mathrm{~kg}\) extra mass is added, the final velocity (in \(\mathrm{m} / \mathrm{s}\)) is:

A train is moving with a speed of \(12 \mathrm{~m} / \mathrm{s}\) on rails which are \(1.5 \mathrm{~m}\) apart. To negotiate a curve radius \(400 \mathrm{~m}\), the height by which the outer rail should be raised with respect to the inner r...

A wire of length \(10 \mathrm{~cm}\) and radius \(\sqrt{7} \times 10^{-4} \mathrm{~m}\) connected across the right gap of a meter bridge. When a resistance of \(4.5 \Omega\) is connected on the left gap by using a resistance box, the balanc...

A wire of resistance \(\mathrm{R}\) and length \(\mathrm{L}\) is cut into 5 equal parts. If these parts are joined parallely, then resultant resistance will be :

A rectangular loop of length \(2.5 \mathrm{~m}\) and width \(2 \mathrm{~m}\) is placed at \(60^{\circ}\) to a magnetic field of \(4 \mathrm{~T}\). The loop is removed from the field in \(10 \mathrm{~sec}\). The average emf induced in the lo...

Two coils have mutual inductance \(0.002 \mathrm{~H}\). The current changes in the first coil according to the relation \(\mathrm{i}=\mathrm{i}_0 \sin \omega \mathrm{t}\), where \(\mathrm{i}_0=5 \mathrm{~A}\) and \(\omega=50 \pi\) rad/s. Th...

A plane electromagnetic wave propagating in \(\mathrm{x}\)-direction is described by
\(E_y=\left(200 \mathrm{Vm}^{-1}\right) \sin \left[1.5 \times 10^7 t-0.05 x\right] \text {; }\)
The intensity of the wave is :
(Use $$\epsilon_0=8.85 \time...

Which of the following circuits is reverse - biased?

An electric charge \(10^{-6} \mu \mathrm{C}\) is placed at origin \((0,0)\) \(\mathrm{m}\) of \(\mathrm{X}-\mathrm{Y}\) co-ordinate system. Two points \(\mathrm{P}\) and \(\mathrm{Q}\) are situated at \((\sqrt{3}, \sqrt{3}) \mathrm{m}\) and...

A thin metallic wire having cross sectional area of \(10^{-4} \mathrm{~m}^2\) is used to make a ring of radius \(30 \mathrm{~cm}\). A positive charge of \(2 \pi \mathrm{~C}\) is uniformly distributed over the ring, while another positive ch...

If the refractive index of the material of a prism is \(\cot \left(\frac{A}{2}\right)\), where \(A\) is the angle of prism then the angle of minimum deviation will be

Identify the physical quantity that cannot be measured using spherometer :

A convex lens of focal length \(40 \mathrm{~cm}\) forms an image of an extended source of light on a photoelectric cell. A current I is produced. The lens is replaced by another convex lens having the same diameter but focal length $$20 \ma...

Two immiscible liquids of refractive indices \(\frac{8}{5}\) and \(\frac{3}{2}\) respectively are put in a beaker as shown in the figure. The height of each column is \(6 \mathrm{~cm}\). A coin is placed at the bottom of the beaker. For nea...

The acceleration due to gravity on the surface of earth is \(\mathrm{g}\). If the diameter of earth reduces to half of its original value and mass remains constant, then acceleration due to gravity on the surface of earth would be :

\(0.08 \mathrm{~kg}\) air is heated at constant volume through \(5^{\circ} \mathrm{C}\). The specific heat of air at constant volume is \(0.17 \mathrm{~kcal} / \mathrm{kg}^{\circ} \mathrm{C}\) and \(\mathrm{J}=4.18\) joule/\(\mathrm{~cal}\)...

The average kinetic energy of a monatomic molecule is \(0.414 \mathrm{~eV}\) at temperature :
(Use \(K_B=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{mol}-\mathrm{K}\))

A proton moving with a constant velocity passes through a region of space without any change in its velocity. If \(\overrightarrow{\mathrm{E}}\) and \(\overrightarrow{\mathrm{B}}\) represent the electric and magnetic fields respectively, th...

Two long, straight wires carry equal currents in opposite directions as shown in figure. The separation between the wires is \(5.0 \mathrm{~cm}\). The magnitude of the magnetic field at a point \(\mathrm{P}\) midway between the wires is ___...

Position of an ant (\(\mathrm{S}\) in metres) moving in \(\mathrm{Y}\)-\(\mathrm{Z}\) plane is given by \(S=2 t^2 \hat{j}+5 \hat{k}\) (where \(t\) is in second). The magnitude and direction of velocity of the ant at $$\mathrm{t}=1 \mathrm{~...

A particle starts from origin at \(t=0\) with a velocity \(5 \hat{i} \mathrm{~m} / \mathrm{s}\) and moves in \(x-y\) plane under action of a force which produces a constant acceleration of \((3 \hat{i}+2 \hat{j}) \mathrm{m} / \mathrm{s}^2\)...

Given below are two statements :
Statement (I) :Viscosity of gases is greater than that of liquids.
Statement (II) : Surface tension of a liquid decreases due to the presence of insoluble impurities.
In the light of the above statements, ch...

If average depth of an ocean is \(4000 \mathrm{~m}\) and the bulk modulus of water is \(2 \times 10^9 \mathrm{~Nm}^{-2}\), then fractional compression \(\frac{\Delta V}{V}\) of water at the bottom of ocean is \(\alpha \times 10^{-2}\). The ...

Four particles each of mass \(1 \mathrm{~kg}\) are placed at four corners of a square of side \(2 \mathrm{~m}\). Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is _____ $$\mathrm{k...

A particle executes simple harmonic motion with an amplitude of \(4 \mathrm{~cm}\). At the mean position, velocity of the particle is \(10 \mathrm{~cm} / \mathrm{s}\). The distance of the particle from the mean position when its speed becom...

Given below are two statements :
Statement (I) : Planck's constant and angular momentum have same dimensions.
Statement (II) : Linear momentum and moment of force have same dimensions.
In the light of the above statements, choose the correc...