Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as: (where \(\lambda\) is the decay constant)

The earth's magnetic field lines resemble that of a dipole at the centre of the earth. If the magnetic moment of this dipole is close to \(8 \times 10^{22}\, Am^2\), the value of earth 's magnetic field near the equator is close to....\(Gauss\) (radius of the earth \(= 6.4 \times 10^6\, m\))

Two cells of emf 1 V and 2 V and internal resistance \(2 \Omega\) and \(1 \Omega\), respectively, are connected in series with an external resistance of \(6 \Omega\). The total current in the circuit is \(\mathrm{I}_1\). Now the same two cells in parallel configuration are connected to same external resistance. In this case, the total current drawn is \(I_2\). The value of \(\left(\frac{I_1}{I_2}\right)\) is \(\frac{x}{3}\). The value of \(x\) is _____.

The current \(i\) in the network is.....\(A\)

Match List-I with List-II.

List-I	List-II
A. \(\sin^2 \omega t\)	I. Periodic with time period \(T = \dfrac{\pi}{\omega}\) but not simple harmonic motion (SHM)
B. \(\sin^3(2\omega t)\)	II. Periodic with time period \(T = \dfrac{2\pi}{\omega}\) but Not SHM
C. \(\sin(\omega t) + \cos(\pi \omega t)\)	III. Periodic with time period \(T = \dfrac{\pi}{\omega}\) and SHM
D. \(\cos\omega t + \cos 2\omega t\)	IV. Non-periodic

Choose the correct answer from the options given below :

Electric field in a certain region is given by \(\overrightarrow{ E }=\left(\frac{ A }{ x ^2} \hat{ i }+\frac{ B }{ y ^3} \hat{ j }\right)\). The \(SI\) unit of \(A\) and \(B\) are

A simple pendulum made of mass 10 g and a metallic wire of length 10 cm is suspended vertically in a uniform magnetic field of 2 T . The magnetic field direction is perpendicular to the plane of oscillations of the pendulum. If the pendulum is released from an angle of \(60^{\circ}\) with vertical, then maximum induced EMF between the point of suspension and point of oscillation is ___________ mV. Take \(\left.g =10 m / s ^2\right)\)

An oil drop of radius \(2\, mm\) with a density \(3\, g\) \(cm ^{-3}\) is held stationary under a constant electric field \(3.55 \times 10^{5}\, V\, m ^{-1}\) in the Millikan's oil drop experiment. What is the number of excess electrons that the oil drop will possess ? (consider \(\left. g =9.81\, m / s ^{2}\right)\)

Six equal resistances are connected between points \(P, Q\) and \(R\) as shown in figure. Then net resistance will be maximum between

A source of sound \(S\) is moving with the velocity of \(50\,m/s\) towards a stationary observer. The observer measures the frequency of the sound as \(1000\,Hz.\) What will be the apparent frequency of the source when it is moving away from the observer after crossing him ... \(Hz\) ? (Take velocity of sound in air is \(350\,m/s\) )

Two wires \(A\) and \(B\) made of different materials of length 6.0 cm and 5.4 cm, respectively and area of cross sections \(3.0 \times 10^{-5} m^2\) and \(4.5 \times 10^{-5} m^2\), respectively are stretched by the same magnitude under a given load. The ratio of the Young's modulus of \(A\) to that of \(B\) is \(x : 3\). The value of x is ___________.

Two planets \(A\) and \(B\) having masses \(m_1\) and \(m_2\) move around the sun in circular orbits of \(r_1\) and \(r_2\) radii respectively. If angular momentum of \(\mathrm{A}\) is \(\mathrm{L}\) and that of \(B\) is \(3L\), the ratio of time period \(\left(\frac{T_A}{T_B}\right)\) is _______.

A vector has magnitude same as that of \(\overrightarrow{\mathrm{A}}-=3 \hat{\mathrm{j}}+4 \hat{\mathrm{j}}\) and is parallel to \(\overrightarrow{\mathrm{B}}=4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}\). The \(\mathrm{x}\) and \(y\) components of this vector in first quadrant are \(\mathrm{x}\) and \(3\) respectively where  \(x\)= _______.