For the given logic gate circuit, which of the following is the correct truth table?

A particle is projected with velocity \(v_{0}\) along \(x-\) axis. A damping force is acting on the particle which is proportional to the square of the distance from the origin i.e., \(ma =-\alpha x ^{2}.\) The distance at which the particle stops:

A radioactive nucleus decays by two different processes. The half life for the first process is \(10\, s\) and that for the second is \(100 s\). the effective half life of the nucleus is close to\(.....sec\)

A parallel plate capacitor whose capacitance \(C\) is \(14\, pF\) is charged by a battery to a potential difference \(V =12\, V\) between its plates. The charging battery is now disconnected and a porcelin plate with \(k =7\) is inserted between the plates, then the plate would oscillate back and forth between the plates with a constant mechanical energy of \(..........pJ\). (Assume no friction)

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 __________.

A horse rider covers half the distance with \(5\,m / s\) speed. The remaining part of the distance was travelled with speed \(10\,m / s\) for half the time and with speed \(15 m / s\) for other half of the time. The mean speed of the rider averaged over the whole time of motion is \(\frac{x}{7}\,m / s\). The value of \(x\) is

A radioactive nucleus \(A\) with a half life \(T\), decays into a nucleus \(B.\) At \(t = 0\), there is no nucleus \(B\). At sometime \(t\), the ratio of the number of \(B\) to that of \(A\) is \(0.3\). Then, \(t\) is given by

A beam of unpolarised light of intensity \(I_0\) is passed through a polaroid \(A\) and then through another polaroid \(B\) which is oriented so that its principal plane makes an angle of \(45^o \) relative to that of \(A\). The intensity of the emergent light is _______.

An \(\alpha\)-particle, a proton and an electron have the same kinetic energy. Which one of the following is correct in case of their De-Broglie wavelength:

The linear mass density of a thin rod \(A B\) of length \(L\) varies from \(A\) to \(B\) as \(\lambda( x )=\lambda_{0}\left(1+\frac{ x }{ L }\right),\) where \(x\) is the distance from A. If \(M\) is the mass of the rod then its moment of inertia about an axis passing through \(A\) and perpendicular to the rod is\(......ML^{2}\)

Two blocks of mass \(M_1 = 20\,kg\) and \(M_2 = 12\,kg\) are connected by a metal rod of mass \(8\,kg.\) The system is pulled vertically up by applying a force of \(480\,N\) as shown. The tension at the mid-point of the rod is ........ \(N\)

The relation between time ' \(t\) ' and distance ' \(x\) ' is \(t=\) \(\alpha x^2+\beta x\), where \(\alpha\) and \(\beta\) are constants. The relation between acceleration \((a)\) and velocity \((v)\) is _______.

A hole is drilled in a metal sheet. At \(27^{\circ} C\), the diameter of hole is 5 cm . When the sheet is heated to \(177^{\circ} C\), the change in the diameter of hole is \(d \times 10^{-3} cm\). The value of \(d\) will be ________ , if coefficient of linear expansion of the metal is \(1.6 \times 10^{-5} /{ }^{\circ} C\)