List of top Questions asked in JEE Main- 2014

A radioactive nuclei with decay constant \(0.5/\mathrm{s}\) is being produced at a constant rate of 100 nuclei/s. If at \(t=0\) there were no nuclei, the time when there are 50 nuclei is:

In a compound microscope the focal length of objective lens is \(1.2\ \mathrm{cm}\) and focal length of eye piece is \(3.0\ \mathrm{cm}\). When object is kept at \(1.25\ \mathrm{cm}\) in front of objective, final image is formed at infinity. Magnifying power of the compound microscope should be:

A photon of wavelength \(\lambda\) is scattered from an electron, which was at rest. The wavelength shift \(\Delta\lambda\) is three times of \(\lambda\) and the angle of scattering \(\theta\) is \(60^\circ\). The angle at which the electron recoiled is \(\phi\). The value of \(\tan\phi\) is : (electron speed is much smaller than the speed of light)

A coil of circular cross-section having 1000 turns and \(4\ \mathrm{cm^2}\) face area is placed with its axis parallel to a magnetic field which decreases by \(10^{-2}\ \mathrm{Wb\,m^{-2}}\) in 0.01 s. The e.m.f. induced in the coil is:

Three identical bars A, B and C are made of different magnetic materials. When kept in a uniform magnetic field, the field lines around them look as follows:

Make the correspondence of these bars with their material being diamagnetic ( D), ferromagnetic (F) and paramagnetic (P) :

Three straight parallel current carrying conductors are shown in the figure. The force experienced by the middle conductor of length 25 cm is:

In the circuit diagrams (A, B, C and D) shown below, R is a high resistance and S is a resistance of the order of galvanometer resistance G. The correct circuit, corresponding to the half deflection method for finding the resistance and figure of merit of the galvanometer, is the circuit labelled as:

An electromagnetic wave of frequency \(1\times10^{14}\) hertz is propagating along z-axis. The amplitude of electric field is 4 V/m. If \(\varepsilon_0=8.8\times10^{-12}\ \mathrm{C^2/N\text{-}m^2}\), then average energy density of electric field will be:

The bulk moduli of ethanol, mercury and water are given as \(0.9,\ 25\) and \(2.2\) respectively in units of \(10^9\,\mathrm{Nm^{-2}}\). For a given value of pressure, the fractional compression in volume is \(\frac{\Delta V}{V}\). Which of the following statements about \(\frac{\Delta V}{V}\) for these three liquids is correct ?

From a sphere of mass \(M\) and radius R, a smaller sphere of radius \(\frac{R}{2}\) is carved out such that the cavity made in the original sphere is between its centre and the periphery (See figure). For the configuration in the figure where the distance between the centre of the original sphere and the removed sphere is 3R, the gravitational force between the two sphere is:

A tank with a small hole at the bottom has been filled with water and kerosene (specific gravity 0.8 ). The height of water is 3 m and that of kerosene 2 m. When the hole is opened the velocity of fluid coming out from it is nearly: (take \(g=10\ \mathrm{ms^{-2}}\) and density of water \(=10^3\ \mathrm{kg\,m^{-3}}\))

A hot body, obeying Newton's law of cooling is cooling down from its peak value \(80^\circ\mathrm{C}\) to an ambient temperature of \(30^\circ\mathrm{C}\). It takes 5 minutes in cooling down from \(80^\circ\mathrm{C}\) to \(40^\circ\mathrm{C}\). How much time will it take to cool down from \(62^\circ\mathrm{C}\) to \(32^\circ\mathrm{C}\) ? (Given \(\ln 2=0.693,\ \ln 5=1.609\) )

A parallel plate capacitor is made of two plates of length \(l\), width \(w\) and separated by distance \(d\). A dielectric slab (dielectric constant K ) that fits exactly between the plates is held near the edge of the plates. It is pulled into the capacitor by a force \(F=-\frac{\partial U}{\partial x}\) where \(U\) is the energy of the capacitor when dielectric is inside the capacitor up to distance \(x\) (See figure). If the charge on the capacitor is Q then the force on the dielectric when it is near the edge is:

If \(X\) has a binomial distribution, \(B(n,p)\) with parameters \(n\) and \(p\) such that \(P(X=2)=P(X=3)\), then \(E(X)\), the mean of variable \(X\), is

If the general solution of the differential equation \(y' = \frac{y}{x} + \Phi\left(\frac{x}{y}\right)\), for some function \(\Phi\), is given by \(y\ln|cx| = x\), where c is an arbitrary constant, then \(\Phi(2)\) is equal to:

If \(|\vec{c}|^2 = 60\) and \(\vec{c}\times(\hat{i}+2\hat{j}+5\hat{k}) = \vec{0}\), then a value of \(\vec{c}\cdot(-7\hat{i}+2\hat{j}+3\hat{k})\) is:

The plane containing the line \(\frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3}\) and parallel to the line \(\frac{x}{1} = \frac{y}{1} = \frac{z}{4}\) passes through the point:

A set S contains 7 elements. A non-empty subset A of S and an element x of S are chosen at random. Then the probability that \(x \in A\) is:

A small ball of mass m starts at a point A with speed \(v_o\) and moves along a frictionless track AB as shown. The track BC has coefficient of friction \(\mu\). The ball comes to stop at C after travelling a distance \(L\) which is:

Let f be an odd function defined on the set of real numbers such that for \(x \ge 0\), \(f(x) = 3\sin x + 4\cos x\). Then f(x) at \(x = -\frac{11\pi}{6}\) is equal to:

For the curve \(y = 3\sin\theta\cos\theta\), \(x = e^{\theta}\sin\theta\), \(0 \le \theta \le \pi\), the tangent is parallel to x-axis when \(\theta\) is:

If \(n \ge 1\), \(P_n = \int_1^e (\log x)^n\,dx\), then \(P_{10} - 90P_8\) is equal to:

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius \(= \sqrt{3}\) is:

Let \(f(x) = x|x|\), \(g(x) = \sin x\) and \(h(x) = (g\circ f)(x)\). Then

Let for \(i = 1,2,3\), \(p_i(x)\) be a polynomial of degree 2 in x, \(p_i'(x)\) and \(p_i''(x)\) be the first and second order derivatives of \(p_i(x)\) respectively. Let, \(A(x) = \begin{bmatrix}p_1(x)&p_1'(x)&p_1''(x)\\p_2(x)&p_2'(x)&p_2''(x)\\p_3(x)&p_3'(x)&p_3''(x)\end{bmatrix}\) and \(B(x) = [A(x)]^{T}A(x)\). Then determinant of B(x):