A Carnot engine operating between two reservoirs has efficiency \(\frac{1}{3}\). When the temperature of cold reservoir raised by \(x\), its efficiency decreases to \(\frac{1}{6}\). The value of \(x\), if the temperature of hot reservoir is \(99^{\circ}\,C\), will be \(........\,K\)

A mass \(\mathrm{m}\) is suspended from a spring of negligible mass and the system oscillates with a frequency \(f_1\). The frequency of oscillations if a mass \(9 \mathrm{~m}\) is suspended from the same spring is \(f_2\). The value of \(\frac{f_1}{f_{.2}}\) is _______.

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A\) : The photoelectric effect does not take place, if the energy of the incident radiation is less than the work function of a metal. Reason \(R\) : Kinetic energy of the photoelectrons is zero, if the energy of the incident radiation is equal to the work function of a metal. In the light of the above statements, choose the most appropriate answer from the options given below

A rectangular parallelopiped is measured as \(1\,cm \times 1\,cm \times 100\,cm\). If its specific resistance is \(3 \times 10^{-7}\,\Omega\,m\), then the resistance between its two opposite rectangular faces will be \(..........x^{-7} \Omega\).

In an experiment to find out the diameter of wire using screw gauge, the following observation were noted. \((a)\) Screw moves \(0.5\,mm\) on main scale in one complete rotation \((b)\) Total divisions on circular scale \(=50\) \((c)\) Main scale reading is \(2.5\,mm\) \((d)\) \(45^{\text {th }}\) division of circular scale is in the pitch line \((e)\) Instrument has \(0.03 \;mm\) negative error Then the diameter of wire is \(...........\,mm\)

Consider two blocks A and B of masses \(m_1=10 \mathrm{~kg}\) and \(m_2=5 \mathrm{~kg}\) that are placed on a frictionless table. The block A moves with a constant speed \(v=3 \mathrm{~m} / \mathrm{s}\) towards the block B kept at rest. A spring with spring constant \(\mathrm{k}=3000 \mathrm{~N} / \mathrm{m}\) is attached with the block B as shown in the figure. After the collision, suppose that the blocks A and B, along with the spring in constant compression state, move together, then the compression in the spring is _______. (Neglect the mass of the spring)

In Young's double slit experiment, carried out with light of wavelength \(5000\) \(A\), the distance between the slits is \(0.3 \mathrm{~mm}\) and the screen is at \(200 \mathrm{~cm}\) from the slits. The central maximum is at \(x=0 \mathrm{~cm}\). The value of \(\mathrm{x}\) for third maxima is _______ \(\mathrm{mm}\).

in circular plate of mass \(M\) and radius \(R\) has its density varying as \(p\left( r \right) = {p_0}\,r\) with \(P_0\) as constant and \(r\) is the distance from its center. The moment of Inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is \(I = aMR^2\) . The value of the coefficient \(a\) is

A uniform cylinder of mass \(M\) and radius \(R\) is to be pulled over a step of height \(a (a < R\)) by applying a force \(F\) at its centre \('O'\) perpendicular to the plane through the axes of the cylinder on the edge of the step ( see figure). The minimum value of \(F\) required is 

For a free body diagram shown in the figure, the four forces are applied in the ' \(x\) ' and ' \(y\) ' directions. What additional force must be applied and at what angle with positive \(x\)-axis so that the net acceleration of body is zero?

A 400 g solid cube having an edge of length 10 cm floats in water. How much volume of the cube is outside the water ?
(Given : density of water \(=1000 \mathrm{~kg} \mathrm{~m}^{-3}\) )

A person is standing in an elevator. In which situation, he experiences weight loss?