List of top Questions asked in JEE Main- 2024

A small liquid drop of radius R is divided into 27 identical liquid drops. If the surface tension is T, then the work done in the process will be :

In Young’s double slit experiment, light from two identical sources is superimposing on a screen. The path difference between the two lights reaching a point on the screen is \( \frac{7\lambda}{4} \). The ratio of the intensity of the fringe at this point with respect to the maximum intensity of the fringe is:

A stone of mass 900 g is tied to a string and moved in a vertical circle of radius 1 m making 10 rpm. The tension in the string, when the stone is at the lowest point, is (if \( \pi^2 = 9.8 \) and \( g = 9.8 \, \text{m/s}^2 \))

In an a.c. circuit, voltage and current are given by:\[V = 100 \sin (100 \, t) \, \text{V}\]and\[I = 100 \sin \left(100 \, t + \frac{\pi}{3}\right) \, \text{mA}\]respectively. The average power dissipated in one cycle is:

The truth table for this given circuit is :The truth table for this given circuit is :

Two sources of light emit with a power of 200 W.The ratio of number of photons of visible light emitted by each source having wavelengths 300 nm and 500 nm respectively, will be :

Let the slope of the line \( 45x + 5y + 3 = 0 \) be \( 27r_1 + \frac{9r_2}{2} \) for some \( r_1, r_2 \in \mathbb{R} \).Then\[\lim_{x \to 3} \left( \int_3^x \frac{8t^2}{\frac{3r_1 x}{2} - r_2 x^2 - r_1 x^3 - 3x} \, dt \right)\]is equal to _____.

Let the set \( C = \{(x, y) \mid x^2 - 2^y = 2023, x, y \in \mathbb{N}\} \).Then\[\sum_{(x, y) \in C} (x + y)\]is equal to ______.

Remainder when \( 64^{32^{32}} \) is divided by 9 is equal to ______.

Let\[f(x) = \sqrt{\lim_{r \to x} \left( \frac{2r^2 \left[ (f(r))^2 - f(x) f(r) \right]}{r^2 - x^2} - r^3 e^{\frac{f(r)}{r}} \right)}\]be differentiable in \( (-\infty, 0) \cup (0, \infty) \) and \( f(1) = 1 \). Then the value of \( ea \), such that \( f(a) = 0 \), is equal to ______.

Let \( O \) be the origin, and \( M \) and \( N \) be the points on the lines \[\frac{x - 5}{4} = \frac{y - 4}{1} = \frac{z - 5}{3} \]and\[\frac{x + 8}{12} = \frac{y + 2}{5} = \frac{z + 11}{9} \]respectively, such that \( MN \) is the shortest distance between the given lines. Then \( OM \cdot ON \) is equal to ______.

Let the area of the region \( \{(x, y): 0 \leq x \leq 3, 0 \leq y \leq \min\{x^2 + 2, 2x + 2\}\} \) be \( A \). Then \( 12A \) is equal to ______.

If\[\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1 - \sin 2x} \, dx = \alpha + \beta \sqrt{2} + \gamma \sqrt{3},\]where \( \alpha \), \( \beta \), and \( \gamma \) are rational numbers, then \( 3\alpha + 4\beta - \gamma \) is equal to ______.

Let \( P(\alpha, \beta) \) be a point on the parabola \( y^2 = 4x \). If \( P \) also lies on the chord of the parabola \( x^2 = 8y \) whose midpoint is \( \left( 1, \frac{5}{4} \right) \), then \( (\alpha - 28)(\beta - 8) \) is equal to ______.

Let for any three distinct consecutive terms \( a, b, c \) of an A.P., the lines \( ax + by + c = 0 \) be concurrent at the point \( P \) and \( Q (\alpha, \beta) \) be a point such that the system of equations \[x + y + z = 6,\]\[2x + 5y + \alpha z = \beta,\]\[x + 2y + 3z = 4,\]has infinitely many solutions. Then \( (PQ)^2 \) is equal to ______.

If \( R \) is the smallest equivalence relation on the set \( \{1, 2, 3, 4\} \) such that \( \{(1,2), (1,3)\} \subseteq R \), then the number of elements in \( R \) is ______.

Let \( x = \frac{m}{n} \) ( \( m, n \) are co-prime natural numbers) be a solution of the equation \( \cos \left( 2 \sin^{-1} x \right) = \frac{1}{9} \) and let \( \alpha, \beta (\alpha > \beta) \) be the roots of the equation \( mx^2 - nx - m + n = 0 \). Then the point \( (\alpha, \beta) \) lies on the line

Let \( A \) be the point of intersection of the lines \( 3x + 2y = 14 \), \( 5x - y = 6 \) and \( B \) be the point of intersection of the lines \( 4x + 3y = 8 \), \( 6x + y = 5 \). The distance of the point \( P(5, -2) \) from the line \( AB \) is

If each term of a geometric progression \( a_1, a_2, a_3, \dots \) with \( a_1 = \frac{1}{8} \) and \( a_2 \neq a_1 \), is the arithmetic mean of the next two terms and \( S_n = a_1 + a_2 + \dots + a_n \), then \( S_{20} - S_{18} \) is equal to

If\[\int \frac{\sin^{\frac{2}{3}} x + \cos^{\frac{2}{3}} x}{\sqrt{\sin^{\frac{1}{3}} x \cos^{\frac{1}{3}} x \sin(x - \theta)}} \, dx = A \sqrt{\cos \theta \tan x - \sin \theta} + B \sqrt{\cos \theta \cot x + \sin(x - \theta)} + C,\]where \( C \) is the integration constant, then \( AB \) is equal to ______.

If \( \log_e a, \log_e b, \log_e c \) are in an A.P. and \( \log_e a - \log_e 2b, \log_e 2b - \log_e 3c, \log_e 3c - \log_e a \) are also in an A.P., then \( a : b : c \) is equal to

Let \( \overrightarrow{OA} = \vec{a}, \overrightarrow{OB} = 12\vec{a} + 4\vec{b} \) and \( \overrightarrow{OC} = \vec{b} \), where \( O \) is the origin. If \( S \) is the parallelogram with adjacent sides \( OA \) and \( OC \), then\[\frac{\text{area of the quadrilateral OABC}}{\text{area of } S}\]is equal to ___.

The sum of the solutions \( x \in \mathbb{R} \) of the equation\[\frac{3 \cos 2x + \cos^3 2x}{\cos^6 x - \sin^6 x} = x^3 - x^2 + 6\]is

If the mean and variance of five observations are \( \frac{24}{5} \) and \( \frac{194}{25} \) respectively and the mean of first four observations is \( \frac{7}{2} \), then the variance of the first four observations is equal to

Let \( P(3, 2, 3) \), \( Q(4, 6, 2) \), and \( R(7, 3, 2) \) be the vertices of \( \triangle PQR \). Then, the angle \( \angle QPR \) is