List of top Questions asked in JEE Main

Four resistances of \(15\,\Omega\), \(12\,\Omega\), \(4\,\Omega\) and \(10\,\Omega\) respectively in cyclic order to form Wheatstone's network. The resistance that is to be connected in parallel with the resistance of \(10\,\Omega\) to balance the network is ______ \(\Omega\).

Consider a uniform rod of mass M = 4m and length \(\ell\) pivoted about its centre. A mass m moving with velocity v making angle \(\theta = \frac{\pi}{4}\) to the rod's long axis collides with one end of the rod and sticks to it. The angular speed of the rod-mass system just after the collision is :

The magnifying power of a telescope with tube 60 cm is 5. What is the focal length of its eye piece ?

A leak proof cylinder of length 1 m, made of a metal which has very low coefficient of expansion is floating vertically in water at \(0^\circ\mathrm{C}\) such that its height above the water surface is 20 cm. When the temperature of water is increased to \(4^\circ\mathrm{C}\), the height of the cylinder above the water surface becomes 21 cm. The density of water at \(T=4^\circ\mathrm{C}\), relative to the density at \(T=0^\circ\mathrm{C}\) is close to :

Effective capacitance of parallel combination of two capacitors \(C_1\) and \(C_2\) is 10 \(\mu\)F. When these capacitors are individually connected to a voltage source of 1V, the energy stored in the capacitor \(C_2\) is 4 times that of \(C_1\). If these capacitors are connected in series, their effective capacitance will be :

An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at the most three of them are red is ___________.

Consider a solid sphere of radius R and mass density
\(\rho(r) = \rho_0\left(1 - \frac{r^2}{R^2}\right)\), \(0 \lt r \le R\)
The minimum density of a liquid in which it will float is :

The sum \(\displaystyle\sum_{k=1}^{20} (1 + 2 + 3 + \ldots + k)\) is :

Let the normal at a point P on the curve \(y^2 - 3x^2 + y + 10 = 0\) intersect the y-axis at \(\left(0, \frac{3}{2}\right)\). If m is the slope of the tangent at P to the curve, then \(|m|\) is equal to

Let the line \(y=mx\) and the ellipse \(2x^2+y^2=1\) intersect at a ponit \(P\) in the first quadrant. If the normal to this ellipse at \(P\) meets the co-ordinate axes at
\(\left(-\frac{1}{3\sqrt2},0\right)\) and \((0,\beta)\), then \(\beta\) is equal to :

The least positive value of 'a' for which the equation \(2x^2 + (a - 10)x + \frac{33}{2} = 2a\) has real roots is

The number of all \(3 \times 3\) matrices A, with entries from the set \(\{-1, 0, 1\}\) such that the sum of the diagonal elements of \(AA^T\) is 3, is

If \(c\) is a point at which Rolle's theorem holds for the function,
\(f(x)=\log_e\left(\frac{x^2+\alpha}{7x}\right)\)
in the interval \([3,4]\), where \(\alpha\in R\), then \(f''(c)\) is equal to

If the equation, \(x^2+bx+45=0\) (\(b\in R\)) has conjugate complex roots and they satisfy \(|z+1|=2\sqrt{10}\), then :

Let \(f(x) = (\sin(\tan^{-1}x)+\sin(\cot^{-1}x))^2-1\), \(|x|>1\).
If \(\frac{dy}{dx}=\frac12\frac{d}{dx}\left(\sin^{-1}(f(x))\right)\) and \(y(\sqrt3)=\frac{\pi}{6}\), then \(y(-\sqrt3)\) is equal to :

Let \(f:\mathbf{R}\to\mathbf{R}\) be such that for all \(x\in\mathbf{R}\)
\((2^{1+x}+2^{1-x})\), \(f(x)\) and \((3^x+3^{-x})\) are in A.P.,
then the minimum value of \(f(x)\) is

Let \(y=y(x)\) be a solution of the differential equation,
\(\sqrt{1-x^2}\,\frac{dy}{dx}+\sqrt{1-y^2}=0\), \(|x|<1\).
If \(y\left(\frac12\right)=\frac{\sqrt3}{2}\), then \(y\left(-\frac{1}{\sqrt2}\right)\) is equal to :

The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 resepectively. Each of these 10 observations is multiplied by \(p\) and then reduced by \(q\), where \(p\ne0\) and \(q\ne0\). If the new mean and new s.d. become half of their original values, then \(q\) is equal to

The shortest distance between the lines
\(\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}\)
and
\(\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}\)
is :

Let \(f(x) = x\cos^{-1}(-\sin|x|)\), \(x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\), then which of the following is true?

For which of the following ordered pairs \((\mu, \delta)\), the system of linear equations
\(x + 2y + 3z = 1\)
\(3x + 4y + 5z = \mu\)
\(4x + 4y + 4z = \delta\)
is inconsistent ?

If \(\displaystyle\int \frac{\cos x\,dx}{\sin^3 x\,(1+\sin^6 x)^{2/3}} = f(x)\,(1+\sin^6 x)^{1/\lambda} + c\)
where \(c\) is a constant of integration, then \(\lambda f\left(\frac{\pi}{3}\right)\) is equal to

Let two points be \(A(1,-1)\) and \(B(0,2)\). If a point \(P(x',y')\) be such that the area of \(\Delta PAB = 5\) sq. units and it lies on the line, \(3x+y-4\lambda=0\), then a value of \(\lambda\) is :

The locus of the mid-points of the perpendiculars drawn from points on the line, x = 2y to the line x = y is :

Let A(1, 0), B(6, 2) and C \( \left( {{3 \over 2},6} \right) \) be the vertices of a triangle ABC. If P is a Point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ, where Q is the point \( \left( { - {7 \over 6}, - {1 \over 3}} \right) \), is ________.