A mass of \(10\; kg\) is suspended by a rope of length \(4 \;\mathrm{m},\) from the ceiling. A force \(\mathrm{F}\) is applied horizontally at the mid-point of the rope such that the top half of the rope makes an angle of \(45^{\circ}\) with the vertical. Then \(\mathrm{F}\) equals ........... \(N\) (Take \(\mathrm{g}=10 \;\mathrm{ms}^{-2}\) and the rope to be massless)

In a radioactive decay chain, the initial nucleus is \({}_{90}^{232}Th\).  At the end there are \(6\,\,\alpha -\) particles and \(4\,\,\beta -\) particles with are emitted. If the end nucleus is \({}_Z^AX\,,\,A\) and \(Z\) are given by

In a screw gauge, \(5\) complete rotations of the screw cause it to move a linear distance of \(0.25\, cm\). There are \(100\) circular scale divisions. The thickness of a wire measured by this screw gauge gives a reading of \(4\) main scale divisions and \(30\) circular scale divisions . Assuming negligible zero error, the thickness of the wire is

Two identical heater filaments are connected first in parallel and then in series. At the same applied voltage, the ratio of heat produced in same time for parallel to series will be:

Four resistances \(40 \ \Omega, 60\  \Omega, 90\  \Omega\) and \(110\  \Omega\) make the arms of a quadrilateral \(A,B,C,D\). Across \(AC\) is a battery of emf \(40\, V\) and internal resistance negligible. The potential difference across \(BD\) is \(V\) is...... 

Five equal resistances are connected in a network as shown in figure. The net resistance between the points \(A\) and \(B\) is

In a Young's double slit experiment, three polarizers are kept as shown in the figure. The transmission axes of \(P_1\) and \(P_2\) are orthogonal to each other. The polarizer \(P_3\) covers both the slits with its transmission axis at \(45^{\circ}\) to those of \(P_1\) and \(P_2\). An unpolarized light of wavelength \(\lambda\) and intensity \(I_0\) is incident on \(P_1\) and \(P_2\). The intensity at a point after \(P_3\) where the path difference between the light waves from \(s_1\) and \(s_2\) is \(\frac{\lambda}{3}\), is _______.

A monochromatic light of frequency \(5 \times 10^{14} \mathrm{~Hz}\) travelling through air, is incident on a medium of refractive index ' 2 '. Wavelength of the refracted light will be :

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed \(v\). The sphere and the cylinder reaches upto maximum heights \(h_1\) and \(h_2\), respectively, above the initial level. The ratio \(h_1: h_2\) is \(\frac{n}{10}\). The value of \(\mathrm{n}\) is __________.

The figure shows a liquid of given density flowing steadily in horizontal tube of varying cross-section. Cross sectional areas at \(A\) is \(1.5\,cm ^2\), and \(B\) is \(25\,mm ^2\), if the speed of liquid at \(B\) is \(60\,cm / s\) then \(\left( P _{ A }- P _{ B }\right)\) is :(Given \(P _{ A }\) and \(P _{ B }\) are liquid pressures at \(A\) and \(B\) points.Density \(\rho=1000\,kg\,m ^{-3}\) \(A\) and \(B\) are on the axis of tube \(............\,Pa\)

An ideal gas undergoes a process maintaining relation between pressure \((P)\) and volume \((V)\) as \(P = P_o\left(1 + \left(\dfrac{V_o}{V}\right)^2\right)^{-1}\), where \(P_o\) and \(V_o\) are constants. If two samples \(A\) and \(B\) (two moles each) with initial volumes \(V_o\) and \(3V_o\) respectively undergo above mentioned process and attain same pressure, then the difference at the temperatures of these samples, \(T_B - T_A\) is _____. (\(R = \) gas constant)

The de Broglie wavelength of an oxygen molecule at \(27^{\circ} C\) is \(x \times 10^{-12} m\). The value of \(x\) is (take Planck's constant \(=6.63 \times 10^{-34} J . s\), Boltzmann constant \(=1.38 \times 10^{-23} J / K\), mass of oxygen. Molecule \(\left.=5.31 \times 10^{-26} kg\right)\).