List of top Questions asked in JEE Main

The bob of a simple pendulum has mass 2g and a charge of \(5.0\ \mu\mathrm{C}\). It is at rest in a uniform horizontal electric field of intensity 2000 V/m. At equilibrium, the angle that the pendulum makes with the vertical is : (take \(g = 10\ \mathrm{m/s^2}\))

Four particles A, B, C and D with masses \(m_A = m\), \(m_B = 2m\), \(m_C = 3m\) and \(m_D = 4m\) are at the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is :

A thin strip 10 cm long is on a U shaped wire of negligible resistance and it is connected to a spring of spring constant \(0.5\ \mathrm{N\,m^{-1}}\) (see figure). The assembly is kept in a uniform magnetic field of 0.1 T. If the strip is pulled from its equilibrium position and released, the number of oscillation it performs before its amplitude decreases by a factor of \(e\) is \(N\). If the mass of the strip is 50 grams, its resistance \(10\ \Omega\) and air drag negligible, \(N\) will be close to :

If the tangents on the ellipse \(4x^2 + y^2 = 8\) at the points \((1, 2)\) and \((a, b)\) are perpendicular to each other, then \(a^2\) is equal to :

The shortest distance between the line \(y = x\) and the curve \(y^2 = x - 2\) is :

If \(2y = \left(\cot^{-1}\left(\frac{\sqrt{3}\cos x + \sin x}{\cos x - \sqrt{3}\sin x}\right)\right)^2,\ x \in \left(0, \frac{\pi}{2}\right)\), then \(\frac{dy}{dx}\) is equal to :

The magnitude of the projection of the vector \(2\hat{i} + 3\hat{j} + \hat{k}\) on the vector perpendicular to the plane containing the vectors \(\hat{i} + \hat{j} + \hat{k}\) and \(\hat{i} + 2\hat{j} + 3\hat{k}\), is :

If \(S_1\) and \(S_2\) are respectively the sets of local minimum and local maximum points of the function \(f(x) = 9x^4 + 12x^3 - 36x^2 + 25,\ x \in R\), then :

The equation of a plane containing the line of intersection of the planes \(2x - y - 4 = 0\) and \(y + 2z - 4 = 0\) and passing through the point \((1, 1, 0)\) is :

The length of the perpendicular from the point \((2, -1, 4)\) on the straight line,
\(\frac{x + 3}{10} = \frac{y - 2}{-7} = \frac{z}{1}\) is :

The area (in sq. units) of the region \(A = \{(x, y) \in R \times R \mid 0 \le x \le 3,\ 0 \le y \le 4,\ y \le x^2 + 3x\}\) is :

If \(\alpha = \cos^{-1}\left(\frac{3}{5}\right)\), \(\beta = \tan^{-1}\left(\frac{1}{3}\right)\), where \(0 \lt \alpha, \beta \lt \frac{\pi}{2}\), then \(\alpha - \beta\) is equal to :

Let \(A = \begin{pmatrix}\cos\alpha & -\sin\alpha\\ \sin\alpha & \cos\alpha\end{pmatrix}\), \((\alpha \in R)\) such that \(A^{32} = \begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}\) then a value of \(\alpha\) is

Let A and B be two non-null events such that \(A \subset B\). Then, which of the following statements is always correct?

Let \(O(0, 0)\) and \(A(0, 1)\) be two fixed points. Then the locus of a point P such that the perimeter of \(\Delta AOP\) is 4, is :

If \(f(x) = \log_e\left(\frac{1 - x}{1 + x}\right)\), \(|x| \lt 1\), then \(f\left(\frac{2x}{1 + x^2}\right)\) is equal to

If \(\alpha\) and \(\beta\) be the roots of the equation \(x^2 - 2x + 2 = 0\), then the least value of n for which \(\left(\frac{\alpha}{\beta}\right)^n = 1\) is :

Let \(f : [0, 2] \to R\) be a twice differentiable function such that \(f''(x) \gt 0\), for all \(x \in (0, 2)\). If \(\phi(x) = f(x) + f(2 - x)\), then \(\phi\) is :

The sum of the solutions of the equation
\(\left|\sqrt{x} - 2\right| + \sqrt{x}\left(\sqrt{x} - 4\right) + 2 = 0\)
\((x \gt 0)\) is equal to:

The sum of the squares of the lengths of the chords intercepted on the circle, \(x^2 + y^2 = 16\), by the lines, \(x + y = n\), \(n \in N\), where N is the set of all natural numbers, is :

The sum of all natural numbers 'n' such that \(100 \lt n \lt 200\) and H.C.F. \((91, n) \gt 1\) is :

If \(\cos(\alpha + \beta) = 3/5\), \(\sin(\alpha - \beta) = 5/13\) and \(0 \lt \alpha, \beta \lt \frac{\pi}{4}\), then \(\tan(2\alpha)\) is equal to :

Let \(y = y(x)\) be the solution of the differential equation,
\((x^2 + 1)^2\frac{dy}{dx} + 2x(x^2 + 1)y = 1\)
such that \(y(0) = 0\). If \(\sqrt{a}\,y(1) = \frac{\pi}{32}\), then the value of 'a' is :

A point on the straight line, \(3x + 5y = 15\) which is equidistant from the coordinate axes will lie only in :

The greatest value of \(c \in R\) for which the system of linear equations
\(x - cy - cz = 0\)
\(cx - y + cz = 0\)
\(cx + cy - z = 0\)
has a non-trivial solution, is :