List of top Questions asked in JEE Main

The value of \[ \sum_{r=1}^{20}\sqrt{\left|\pi\left(\int_0^r x|\sin \pi x|\,dx\right)\right|} \] is:

For some \( \theta\in\left(0,\frac{\pi}{2}\right) \), let the eccentricity and the length of the latus rectum of the hyperbola \[ x^2-y^2\sec^2\theta=8 \] be \( e_1 \) and \( l_1 \), respectively, and let the eccentricity and the length of the latus rectum of the ellipse \[ x^2\sec^2\theta+y^2=6 \] be \( e_2 \) and \( l_2 \), respectively. If \[ e_1^2=\frac{2}{e_2^2}\left(\sec^2\theta+1\right), \] then \[ \left(\frac{l_1l_2}{e_1^2e_2^2}\right)\tan^2\theta \] is equal to:

If \[ k=\tan\!\left(\frac{\pi}{4}+\frac{1}{2}\cos^{-1}\!\left(\frac{2}{3}\right)\right) +\tan\!\left(\frac{1}{2}\sin^{-1}\!\left(\frac{2}{3}\right)\right), \] then the number of solutions of the equation \[ \sin^{-1}(kx-1)=\sin^{-1}x-\cos^{-1}x \] is:

In a G.P., if the product of the first three terms is \(27\) and the set of all possible values for the sum of its first three terms is \( \mathbb{R} - (a,b) \), then \( a^2+b^2 \) is equal to:

Let \( f \) be a polynomial function such that \[ f(x^2+1)=x^4+5x^2+2,\quad \text{for all } x\in\mathbb{R}. \] Then \[ \int_0^3 f(x)\,dx \] is equal to:

If \[ \int \frac{1-5\cos^2 x}{\sin^5 x\cos^2 x}\,dx=f(x)+C, \] where \( C \) is the constant of integration, then \[ f\!\left(\frac{\pi}{6}\right)-f\!\left(\frac{\pi}{4}\right) \] is equal to:

Let \( ABC \) be an equilateral triangle with orthocenter at the origin and the side \( BC \) lying on the line \( x+2\sqrt{2}\,y=4 \). If the coordinates of the vertex \( A \) are \( (\alpha,\beta) \), then the greatest integer less than or equal to \( |\alpha+\sqrt{2}\beta| \) is:

Let \( A, B, C \) be three \( 2\times2 \) matrices with real entries such that \[ B=(I+A)^{-1} \quad \text{and} \quad A+C=I. \] If \[ BC=\begin{bmatrix}1 & -5 \\-1 & 2\end{bmatrix} \quad \text{and} \quad B\begin{bmatrix}x_1\\x_2\end{bmatrix} =\begin{bmatrix}12\\-6\end{bmatrix}, \] then \( x_1+x_2 \) is:

Let \( S=\{1,2,3,4,5,6,7,8,9\} \). Let \( x \) be the number of 9-digit numbers formed using the digits of the set \( S \) such that only one digit is repeated and it is repeated exactly twice. Let \( y \) be the number of 9-digit numbers formed using the digits of the set \( S \) such that only two digits are repeated and each of these is repeated exactly twice. Then:

If \( \alpha,\beta \) where \( \alpha<\beta \), are the roots of the equation \[ \lambda x^2-(\lambda+3)x+3=0 \] such that \[ \frac{1}{\alpha}-\frac{1}{\beta}=\frac{1}{3}, \] then the sum of all possible values of \( \lambda \) is:

The area of the region \[ R=\{(x,y): xy\le 8,\; 1\le y\le x^2,\; x\ge 0\} \] is:

Let \( z \) be a complex number such that \( |z-6|=5 \) and \( |z+2-6i|=5 \). Then the value of \( z^3+3z^2-15z+14 \) is equal to:

The value of \[ \lim_{x\to 0}\frac{\log_e\!\big(\sec(ex)\cdot \sec(e^2x)\cdots \sec(e^{10}x)\big)} {e^2-e^{2\cos x}} \] is equal to:

The common difference of the A.P.: \( a_1, a_2, \ldots, a_m \) is 13 more than the common difference of the A.P.: \( b_1, b_2, \ldots, b_n \). If \( b_{31} = -277 \), \( b_{43} = -385 \) and \( a_{78} = 327 \), then \( a_1 \) is equal to:

Let \( y = y(x) \) be the solution of the differential equation \[ x\frac{dy}{dx} - \sin 2y = x^3(2 - x^3)\cos^2 y,\; x \ne 0. \] If \( y(2) = 0 \), then \( \tan(y(1)) \) is equal to:

The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are \( 2, 3, 5, 10, 11, 13, 15, 21 \), then the mean deviation about the median of all the 10 observations is:

Let \( S = \{x^3 + ax^2 + bx + c : a, b, c \in \mathbb{N} \text{ and } a, b, c \le 20\} \) be a set of polynomials. Then the number of polynomials in \( S \), which are divisible by \( x^2 + 2 \), is:

Let \( y = x \) be the equation of a chord of the circle \( C_1 \) (in the closed half-plane \( x \ge 0 \)) of diameter 10 passing through the origin. Let \( C_2 \) be another circle described on the given chord as diameter. If the equation of the chord of the circle \( C_2 \), which passes through the point \( (2, 3) \) and is farthest from the center of \( C_2 \), is \( x + ay + b = 0 \), then \( b \) is equal to:

If \( g(x) = 3x^2 + 2x - 3 \), \( f(0) = -3 \) and \( 4g(f(x)) = 3x^2 - 32x + 72 \), then \( f(g(2)) \) is equal to:

After release, the blocks moves 81 cm in 9 seconds. Find moment of inertia of the pulley :
(Given $m_{1} = 400$ gm, $m_{2} = 350$ gm, R = 2 cm, g = 10 m/s$^2$)

For the following change: H$_2$O(l) $\rightarrow$ H$_2$O(g)
5$^\circ$C $\rightarrow$ 100$^\circ$C
Select the correct answer

The ratio of de Broglie wavelength of a deuteron with kinetic energy \(E\) to that of an alpha particle with kinetic energy \(2E\) is \(n:1\). (Assume mass of proton \(=\) mass of neutron.) Find the value of \(n\).