The centre of mass of a solid hemisphere of radius \(8\, cm\) is \(X \,cm\) from the centre of the flat surface. Then value of \(x\) is\(......\)

Thermodynamic process is shown below on a \(P-V\) diagram for one mole of an ideal gas. If \(V _{2}=2 V _{1}\) then the ratio of temperature \(T _{2} / T _{1}\) is ...... .

Assuming human pupil to have a radius of \(0.25\ cm\) and a comfortable viewing distance of \(25\ cm\), the minimum separation between two objects that human eye can resolve at \(500\ nm\) wavelength is.....\(\mu m\)

A Carnot engine has efficiency of \(50 \%\). If the temperature of sink is reduced by \(40^{\circ} C\), its efficiency increases by \(30 \%\). The temperature of the source will be\(....K\)

Choose the correct relationship between Poisson ratio \((\sigma)\). bulk modulus \(( K )\) and modulus of rigidity \((\eta)\) of a given solid object:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R .
Assertion A : If oxygen ion \(\left(\mathrm{O}^{-2}\right)\) and Hydrogen ion \(\left(\mathrm{H}^{+}\right)\)enter normal to the magnetic field with equal momentum, then the path of \(\mathrm{O}^{-2}\) ion has a smaller curvature than that of \(\mathrm{H}^{+}\).
Reason R : A proton with same linear momentum as an electron will form a path of smaller radius of curvature on entering a uniform magnetic field perpendicularly.
In the light of the above statements, choose the correct answer from the options given below :

A string of area of cross-section \(4\,mm ^{2}\) and length \(0.5\) is connected with a rigid body of mass \(2\,kg\). The body is rotated in a vertical circular path of radius \(0.5\,m\). The body acquires a speed of \(5\,m / s\) at the bottom of the circular path. Strain produced in the string when the body is at the bottom of the circle is \(\ldots . . \times 10^{-5}\). (Use Young's modulus \(10^{11}\,N / m ^{2}\) and \(g =10\,m / s ^{2}\) )

If momentum \([ P ]\), area \([ A ]\) and time \([ T ]\) are taken as fundamental quantities, then the dimensional formula for coefficient of viscosity is :

A light wave travelling linearly in a medium of dielectric constant \(4,\) incident on the horizontal interface separating medium with air. The angle of incidence for which the total intensity of incident wave will be reflected back into the same medium will be (Given : relative permeability of medium \(\mu_{ s }=1)\) 

A mass \(m\) is attached to two springs as shown in figure. The spring constants of two springs are \(K _1\) and \(K _2\). For the frictionless surface, the time period of oscillation of mass \(m\) is

A cylinder of mass \(M_c\) and sphere of mass \(M_s\) are placed at points \(A\) and \(B\) of two inclines, respectively (See Figure). If they roll on the incline without sipping such that their accelerations are the same, then the ratio \(\frac{{\sin \,{\theta _c}}}{{\sin \,{\theta _s}}}\) is

The differential equation of the family of circles passing the origin and having center at the line \(y=x\) is :

Let \(\vec{a}=\hat{i}+\hat{j}+2 \hat{k}, \vec{b}=2 \hat{i}-3 \hat{j}+\hat{k}\) and \(\overrightarrow{ c }=\hat{ i }-\hat{ j }+\hat{ k }\) be three given vectors. Let \(\vec{v}\) be a vector in the plane of \(\vec{a}\) and \(\overrightarrow{ b }\) whose projection on \(\overrightarrow{ c }\) is \(\frac{2}{\sqrt{3}}\). If \(\overrightarrow{ v } . \hat{ j }=7\), then \(\overrightarrow{ v } \cdot(\hat{ i }+\hat{ k })\) is equal to