Identify the correct truth table of the given logical circuit.

A radioactive element \({ }_{92}^{242} X\) emits two \(\alpha\)-particles, one electron and two positrons. The product nucleus is represented by \({ }_{ P }^{234} Y\). The value of \(P\) is \(..................\)

A block of mass \(m\) is placed on a surface having vertical cross section given by \(y=x^2 / 4\). If coefficient of friction is \(0.5\) , the maximum height above the ground at which block can be placed without slipping is _______.

Three students \(S_{1}, S_{2}\) and \(S_{3}\) perform an experiment for determining the acceleration due to gravity \((g)\) using a simple pendulum. They use different lengths of pendulum and record time for different number of oscillations. The observations are as shown in the table.

Student No.	Length of pendulum \((cm)\)	No. of osci \((n)\)	Total time for osci	Time period \((s)\)
\(1.\)	\(64.0\)	\(8\)	\(128.0\)	\(16.0\)
\(2.\)	\(64.0\)	\(4\)	\(64.0\)	\(16.0\)
\(3.\)	\(20.0\)	\(4\)	\(36.0\)	\(9.0\)

(Least count of length \(=0.1 \,{m}\), least count for time \(=0.1\, {s}\) ) If \(E_{1}, E_{2}\) and \(E_{3}\) are the percentage errors in \('g'\) for students \(1,2\) and \(3\) respectively, then the minimum percentage error is obtained by student no. ....... .

Given below are two statements : \(Statement\) \((I)\) :Viscosity of gases is greater than that of liquids. \(Statement\) \((II)\) : Surface tension of a liquid decreases due to the presence of insoluble impurities. In the light of the above statements, choose the most appropriate answer from the options given below :

For the given logic circuit, which of the following inputs combination will make both LED\(-1\) and LED\(-2\) to glow?

Two identical positive charges \(Q\) each are fixed at a distance of ' \(2 a\) ' apart from each other. Another point charge qo with mass ' \(m\) ' is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge \(q_{0}\) executes \(SHM\). The time period of oscillation of charge \(q_{0}\) will be.

Mass numbers of two nuclei are in the ratio of \(4: 3\). Their nuclear densities will be in the ratio of

A solid cylinder of mass \(m\) is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is: [The coefficient of static friction, \(\mu_{ s },\) is \(\left.0.4\right]\)

A particle of mass \(m\) is moving with speed \(2\, v\) collides with a mass \(2\,m\) moving with speed \(v\) in the same direction. After collision, the first mass is stopped completely while the second one splits into two particles each of mass \(m\), which move at angle \(45^o\) with respect to the original direction. The speed of each of the moving particle will be

silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of \(10^{12}  /sec\). What is the force constant of the bonds connecting one atom with the other? ................ \(\mathrm{N/m}\)     (Mole wt. of silver \(= 108 \) andAvagadro number \(= 6.02 \times 10^{23}\) \(gm \ mole^{ -1}\) )