List of top Questions asked in JEE Main

For changing the capacitance of a given parallel plate capacitor, a dielectric material of dielectric constant K is used, which has the same area as the plates of the capacitor. The thickness of the dielectric slab is \(\frac{3}{4}d\), where 'd' is the separation between the plates of parallel plate capacitor. The new capacitance (C') in terms of original capacitance (\(C_0\)) is given by the following relation :

Four equal masses, m each are placed at the comers of a square of length (\(l\)) as shown in the figure. The moment of inertia of the system about an axis passing through A and parallel to DB would be :

A conducting wire of length '\(l\)', area of cross-section A and electric resistivity \(\rho\) is connected between the terminals of a battery. A potential difference V is developed between its ends, causing an electric current.

If the length of the wire of the same material is doubled and the area of cross-section is halved, the resultant current would be :

In thermodynamics, heat and work are :

The maximum and minimum distances of a comet from the Sun are \(1.6 \times 10^{12}\) m and \(8.0 \times 10^{10}\) m respectively. If the speed of the comet at the nearest point is \(6 \times 10^{4}\ \text{ms}^{-1}\), the speed at the farthest point is :

The angle of deviation through a prism is minimum when

(A) Incident ray and emergent ray are symmetric to the prism.
(B) The refracted ray inside the prism becomes parallel to its base
(C) Angle of incidence is equal to that of the angle of emergence
(D) When angle of emergence is double the angle of incidence
Choose the correct answer from the options given below :

Let \(f : (0, 2) \to R\) be defined as \(f(x) = \log_2\left(1 + \tan\left(\frac{\pi x}{4}\right)\right)\). Then, \(\lim\limits_{n \to \infty} \frac{2}{n}\left(f\left(\frac1n\right) + f\left(\frac2n\right) + \dots + f(1)\right)\) is equal to ____________.

The pressure acting on a submarine is \(3 \times 10^5\) Pa at a certain depth. If the depth is doubled, the percentage increase in the pressure acting on the submarine would be :
(Assume that atmospheric pressure is \(1 \times 10^5\) Pa density of water is \(10^3\) kg m^-3, g = 10 ms^-2)

Let \(P = \begin{bmatrix} -30 \& 20 \& 56 \\ 90 \& 140 \& 112 \\ 120 \& 60 \& 14 \end{bmatrix}\) and \(A = \begin{bmatrix} 2 \& 7 \& \omega^2 \\ -1 \& -\omega \& 1 \\ 0 \& -\omega \& -\omega + 1 \end{bmatrix}\) where \(\omega = \frac{-1 + i\sqrt3}{2}\), and \(I_3\) be the identity matrix of order 3. If the determinant of the matrix \((P^{-1}AP - I_3)^2\) is \(\alpha\omega^2\), then the value of \(\alpha\) is equal to ____________.

The velocity-displacement graph describing the motion of bicycle is shown in the figure.

The acceleration-displacement graph of the bicycle's motion is best described by :

A block of 200 g mass moves with a uniform speed in a horizontal circular groove, with vertical side walls of radius 20 cm. If the block takes 40 s to complete one round, the normal force by the side walls of the groove is :

Let ABCD be a square of side of unit length. Let a circle \(C_1\) centered at A with unit radius is drawn. Another circle \(C_2\) which touches \(C_1\) and the lines AD and AB are tangent to it, is also drawn. Let a tangent line from the point C to the circle \(C_2\) meet the side AB at E. If the length of EB is \(\alpha + \sqrt3\,\beta\), where \(\alpha, \beta\) are integers, then \(\alpha + \beta\) is equal to ____________.

Consider an arithmetic series and a geometric series having four initial terms from the set \(\{11, 8, 21, 16, 26, 32, 4\}\). If the last terms of these series are the maximum possible four digit numbers, then the number of common terms in these two series is equal to ____________.

If \(\lim\limits_{x \to 0} \frac{a e^x - b\cos x + c e^{-x}}{x \sin x} = 2\), then \(a + b + c\) is equal to ____________.

If the normal to the curve \(y(x) = \int\limits_0^x (2t^2 - 15t + 10)dt\) at a point \((a, b)\) is parallel to the line \(x + 3y = -5\), \(a \gt 1\), then the value of \(|\,a + 6b\,|\) is equal to ____________.

Let z and \(\omega\) be two complex numbers such that \(\omega = z\overline{z} - 2z + 2\), \(\left|\frac{z+i}{z-3i}\right| = 1\) and Re(\(\omega\)) has minimum value. Then, the minimum value of \(n \in N\) for which \(\omega^n\) is real, is equal to ____________.

Let \(f : R \to R\) be a continuous function such that \(f(x) + f(x+1) = 2\), for all \(x \in R\). If \(I_1 = \int\limits_0^8 f(x)dx\) and \(I_2 = \int\limits_{-1}^3 f(x)dx\), then the value of \(I_1 + 2I_2\) is equal to ____________.

Let P be a plane \(lx + my + nz = 0\) containing the line, \(\frac{1-x}{1} = \frac{y+4}{2} = \frac{z+2}{3}\). If plane P divides the line segment AB joining points A(\(-3, -6, 1\)) and B(\(2, 4, -3\)) in ratio k : 1 then the value of k is equal to :

The number of roots of the equation, \((81)^{\sin^2 x} + (81)^{\cos^2 x} = 30\) in the interval \([\,0, \pi\,]\) is equal to :

If the three normals drawn to the parabola, \(y^2=2x\) pass through the point \((a,0)\) \(a\ne0\), then 'a' must be greater than :

If for \(a \gt 0\), the feet of perpendiculars from the points A(\(a, -2a, 3\)) and B(\(0, 4, 5\)) on the plane \(lx + my + nz = 0\) are points C(\(0, -a, -1\)) and D respectively, then the length of line segment CD is equal to :

Let [ x ] denote greatest integer less than or equal to x. If for \(n \in N\), \((1 - x + x^3)^n = \sum\limits_{j=0}^{3n} a_j x^j\), then \(\sum\limits_{j=0}^{\left[\frac{3n}{2}\right]} a_{2j} + 4 \sum\limits_{j=0}^{\left[\frac{3n-1}{2}\right]} a_{2j+1}\) is equal to :

Let the curve \(y = y(x)\) be the solution of the differential equation, \(\frac{dy}{dx} = 2(x+1)\). If the numerical value of area bounded by the curve \(y = y(x)\) and x-axis is \(\frac{4\sqrt8}{3}\), then the value of \(y(1)\) is equal to ____________.

If \(y=y(x)\) is the solution of the differential equation, \(\frac{dy}{dx}+2y\tan x=\sin x,\ y\left(\frac{\pi}{3}\right)=0\), then the maximum value of the function \(y(x)\) over \(\mathbb R\) is equal to:

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :