JEE Main 2024 (Online) 8th April Morning Shift

Let the sum of two positive integers be 24 . If the probability, that their product is not less than \(\frac{3}{4}\) times their greatest possible product, is \(\frac{m}{n}\), where \(\operatorname{gcd}(m, n)=1\), then \(n\)-\(m\) equals

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables \(X\) and \(Y\) respectively denote the number of blue and yellow balls. If \(\bar{X}\) and \(\bar{Y}\) are the means of \(X\) and $$Y...

The sum of all the solutions of the equation \((8)^{2 x}-16 \cdot(8)^x+48=0\) is :

Let \(\alpha=\sum_\limits{r=0}^n\left(4 r^2+2 r+1\right){ }^n C_r\) and \(\beta=\left(\sum_\limits{r=0}^n \frac{{ }^n C_r}{r+1}\right)+\frac{1}{n+1}\). If \(140<\frac{2 \alpha}{\beta}<281\), then the value of \(n\) is _________.

Let the positive integers be written in the form :

If the \(k^{\text {th }}\) row contains exactly \(k\) numbers for every natural number \(k\), then the row in which the number 5310 will be, is __________.

If the orthocentre of the triangle formed by the lines \(2 x+3 y-1=0, x+2 y-1=0\) and \(a x+b y-1=0\), is the centroid of another triangle, whose circumcentre and orthocentre respectively are \((3,4)\) and \((-6,-8)\), then the value of $$|...

The equations of two sides \(\mathrm{AB}\) and \(\mathrm{AC}\) of a triangle \(\mathrm{ABC}\) are \(4 x+y=14\) and \(3 x-2 y=5\), respectively. The point \(\left(2,-\frac{4}{3}\right)\) divides the third side \(\mathrm{BC}\) internally in t...

If \(\sin x=-\frac{3}{5}\), where \(\pi< x <\frac{3 \pi}{2}\), then \(80\left(\tan ^2 x-\cos x\right)\) is equal to

The set of all \(\alpha\), for which the vectors \(\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}\) and \(\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}\) are inclined at an obtuse angle for all \(t \in \mathbb{R}\), is

Let \(\vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}, \vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k}\) and \(\vec{c}=17 \hat{i}-2 \hat{j}+\hat{k}\) be three given vectors. If \(\vec{r}\) is a vector such that $$\vec{r} \times \vec{a}=(\vec{b}+\vec{c}) \ti...

A LCR circuit is at resonance for a capacitor C, inductance L and resistance R. Now the value of resistance is halved keeping all other parameters same. The current amplitude at resonance will be now:

In an alpha particle scattering experiment distance of closest approach for the \(\alpha\) particle is \(4.5 \times 10^{-14} \mathrm{~m}\). If target nucleus has atomic number 80 , then maximum velocity of \(\alpha\)-particle is __________ ...

Binding energy of a certain nucleus is \(18 \times 10^8 \mathrm{~J}\). How much is the difference between total mass of all the nucleons and nuclear mass of the given nucleus:

A stationary particle breaks into two parts of masses \(m_A\) and \(m_B\) which move with velocities \(v_A\) and \(v_B\) respectively. The ratio of their kinetic energies \(\left(K_B: K_A\right)\) is :

A uniform thin metal plate of mass \(10 \mathrm{~kg}\) with dimensions is shown. The ratio of \(\mathrm{x}\) and y coordinates of center of mass of plate in \(\frac{n}{9}\). The value of \(n\) is ________.

A clock has \(75 \mathrm{~cm}, 60 \mathrm{~cm}\) long second hand and minute hand respectively. In 30 minutes duration the tip of second hand will travel \(x\) distance more than the tip of minute hand. The value of \(x\) in meter is nearly...

Resistance of a wire at \(0^{\circ} \mathrm{C}, 100^{\circ} \mathrm{C}\) and \(t^{\circ} \mathrm{C}\) is found to be \(10 \Omega, 10.2 \Omega\) and \(10.95 \Omega\) respectively. The temperature \(t\) in Kelvin scale is _________.

In the given circuit, the terminal potential difference of the cell is :

A proton and an electron are associated with same de-Broglie wavelength. The ratio of their kinetic energies is:
(Assume h = 6.63 \(\times 10^{-34} \mathrm{~J} \mathrm{~s}, \mathrm{~m}_{\mathrm{e}}=9.0 \times 10^{-31} \mathrm{~kg}\) and $$\...

Average force exerted on a non-reflecting surface at normal incidence is \(2.4 \times 10^{-4} \mathrm{~N}\). If \(360 \mathrm{~W} / \mathrm{cm}^2\) is the light energy flux during span of 1 hour 30 minutes, Then the area of the surface is:

The output \(\mathrm{Y}\) of following circuit for given inputs is :

Two charged conducting spheres of radii \(a\) and \(b\) are connected to each other by a conducting wire. The ratio of charges of the two spheres respectively is:

An electric field, \(\overrightarrow{\mathrm{E}}=\frac{2 \hat{i}+6 \hat{j}+8 \hat{k}}{\sqrt{6}}\) passes through the surface of \(4 \mathrm{~m}^2\) area having unit vector \(\hat{n}=\left(\frac{2 \hat{i}+\hat{j}+\hat{k}}{\sqrt{6}}\right)\)....

Critical angle of incidence for a pair of optical media is \(45^{\circ}\). The refractive indices of first and second media are in the ratio:

Two planets \(A\) and \(B\) having masses \(m_1\) and \(m_2\) move around the sun in circular orbits of \(r_1\) and \(r_2\) radii respectively. If angular momentum of \(A\) is \(L\) and that of \(B\) is \(3 \mathrm{~L}\), the ratio of time ...

A mixture of one mole of monoatomic gas and one mole of a diatomic gas (rigid) are kept at room temperature \((27^{\circ} \mathrm{C})\). The ratio of specific heat of gases at constant volume respectively is:

Two different adiabatic paths for the same gas intersect two isothermal curves as shown in P-V diagram. The relation between the ratio \(\frac{V_a}{V_d}\) and the ratio \(\frac{V_b}{V_c}\) is:

A player caught a cricket ball of mass \(150 \mathrm{~g}\) moving at a speed of \(20 \mathrm{~m} / \mathrm{s}\). If the catching process is completed in \(0.1 \mathrm{~s}\), the magnitude of force exerted by the ball on the hand of the play...

Paramagnetic substances:
A. align themselves along the directions of external magnetic field.
B. attract strongly towards external magnetic field.
C. has susceptibility little more than zero.
D. move from a region of strong magnetic field t...

An electron with kinetic energy \(5 \mathrm{~eV}\) enters a region of uniform magnetic field of 3 \(\mu \mathrm{T}\) perpendicular to its direction. An electric field \(\mathrm{E}\) is applied perpendicular to the direction of velocity and ...

A square loop PQRS having 10 turns, area \(3.6 \times 10^{-3} \mathrm{~m}^2\) and resistance \(100 \Omega\) is slowly and uniformly being pulled out of a uniform magnetic field of magnitude \(\mathrm{B}=0.5 \mathrm{~T}\) as shown. Work done...

Correct Bernoulli's equation is (symbols have their usual meaning) :

Young's modulus is determined by the equation given by \(\mathrm{Y}=49000 \frac{\mathrm{m}}{\mathrm{l}} \frac{\mathrm{dyne}}{\mathrm{cm}^2}\) where \(M\) is the mass and \(l\) is the extension of wire used in the experiment. Now error in Yo...

A liquid column of height \(0.04 \mathrm{~cm}\) balances excess pressure of a soap bubble of certain radius. If density of liquid is \(8 \times 10^3 \mathrm{~kg} \mathrm{~m}^{-3}\) and surface tension of soap solution is $$0.28 \mathrm{~Nm}...

The diameter of a sphere is measured using a vernier caliper whose 9 divisions of main scale are equal to 10 divisions of vernier scale. The shortest division on the main scale is equal to \(1 \mathrm{~mm}\). The main scale reading is $$2 \...

In an expression \(a \times 10^b\) :

Three vectors \(\overrightarrow{\mathrm{OP}}, \overrightarrow{\mathrm{OQ}}\) and \(\overrightarrow{\mathrm{OR}}\) each of magnitude \(\mathrm{A}\) are acting as shown in figure. The resultant of the three vectors is \(\mathrm{A} \sqrt{x}\)....

A parallel beam of monochromatic light of wavelength \(600 \mathrm{~nm}\) passes through single slit of \(0.4 \mathrm{~mm}\) width. Angular divergence corresponding to second order minima would be _________ \(\times 10^{-3} \mathrm{~rad}\).

A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is \(\left(\frac{a-1}{a}\right)\) then the value of \(a\) is _________.

Three bodies A, B and C have equal kinetic energies and their masses are \(400 \mathrm{~g}, 1.2 \mathrm{~kg}\) and \(1.6 \mathrm{~kg}\) respectively. The ratio of their linear momenta is :