A Carnot's engine working between \(400\, K\) and \(800\, K\) has a work output of \(1200\, J\) per cycle. The amount of heat energy supplied to the engine from the source in each cycle is ........... \(J\)

A cube having a side of 10 cm with unknown mass and 200 gm mass were hung at two ends of an uniform rigid rod of 27 cm long. The rod along with masses was placed on a wedge keeping the distance between wedge point and 200 gm weight as 25 cm. Initially the masses were not at balance. A beaker is placed beneath the unknown mass and water is added slowly to it. At given point the masses were in balance and half volume of the unknown mass was inside the water.
(Take the density of unknown mass is more than that of the water, the mass did not absorb water and water density is \(1 \mathrm{gm} / \mathrm{cm}^3\).) The unknown mass is ________ kg.

In a Young's double slit experiment, the intensity at some point on the screen is found to be \(\dfrac{3}{4}\) times of the maximum of the interference pattern. The path difference between the interfering waves at this point is \(\dfrac{\lambda}{x}\) where \(\lambda\) is wavelength of the incident light. The value of \(x\) is _______.

The image of an illuminated square is obtained on a screen with the help of a converging lens. The distance of the square from the lens is \(40\,cm.\) The area of the image is \(9\,times\) that of the square. The focal length of the lens is........\(cm\)

The time dependence of the position of a particle of mass \(m = 2\) is given by \(\vec r\,(t)\, = \,2t\,\hat i\, - 3{t^2}\hat j\) Its angular momentum with respect to the origin at time \(t = 2\) is

The de Broglie wavelength of a proton and \(\alpha\) -particle are equal. The ratio of their velocities is ...... .

An unstable heavy nucleus at rest breaks into two nuclei which move away with velocities in the ratio of \(8:27\). The ratio of the radii of the nuclei (assumed to be spherical ) is

Let \(\overrightarrow{ a }\) be a non-zero vector parallel to the line of intersection of the two planes described by \(\hat{i}+\hat{j}, \hat{i}+\hat{k}\) and \(\hat{i}-\hat{j}, \hat{j}-\hat{k}\). If \(\theta\) is the angle between the vector \(\vec{a}\) and the vector \(\vec{b}=2 \hat{i}-2 \hat{j}+\hat{k}\) and \(\vec{a} \cdot \vec{b}=6\) then the ordered pair \((\theta,|\vec{a} \times \vec{b}|)\) is equal to

A uniform rectangular thin sheet \(ABCD\) of mass \(M\) has length \(a\) and breadth \(b\), as shown in the figure. If the shaded portion \(HBGO\) is cut off, the coordinates of the centre of mass of the remaining portion will be

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

The value of \(\left|\begin{array}{lll}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{array}\right|\) is 

Two harmonic waves moving in the same direction superimpose to form a wave \(\mathrm{x}=\mathrm{a} \cos (1.5 \mathrm{t}) \cos (50.5 \mathrm{t})\) where \(t\) is in seconds. Find the period with which they beat (close to nearest integer)

Let a tangent to the curve \(y^2=24 x\) meet the curve \(xy =2\) at the points \(A\) and \(B\). Then the mid points of such line segments \(A B\) lie on a parabola with the