A uniform thin rod \(AB\) of length \(L\) has linear mass density \(\mu \left( x \right) = a + \frac{{bx}}{L}\) , where \(x\) is measured from \(A\). If the \(CM\) of the rod lies at a distance of \(\left( {\frac{7}{12}} \right)L\) from \(A\), then \(a\) and \(b\) are related as

Two thin metallic spherical shells of radii \({r}_{1}\) and \({r}_{2}\) \(\left({r}_{1}<{r}_{2}\right)\) are placed with their centres coinciding. A material of thermal conductivity \({K}\) is filled in the space between the shells. The inner shell is maintained at temperature \(\theta_{1}\) and the outer shell at temperature \(\theta_{2}\left(\theta_{1}<\theta_{2}\right)\). The rate at which heat flows radially through the material is :-

A \(220 \; V , 50 \; Hz\) AC source is connected to a \(25 \; V\), \(5 \; W\) lamp and an additional resistance \(R\) in series (as shown in figure) to run the lamp at its peak brightness, then the value of \(R\) (in ohm) will be

In the following digital circuit, what will be the output at \('Z',\) when the input \((A, B)\) are \((1,0), (0,0),(1,1),(0,1)\)

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason \(R\). Assertion \(A:\) Alloys such as constantan and manganin are used in making standard resistance coils. Reason \(R:\) Constantan and manganin have very small value of temperature coefficient of resistance.In the light of the above statements, choose the correct answer from the options given below.

The length of a potentiometer wire is \(1200\; \mathrm{cm}\) and it carries a current of \(60 \;\mathrm{mA}\). For a cell of \(emf\;5\; \mathrm{V}\) and intemal resistance of \(20\; \Omega,\) the null point on it is found to be a \(1000\; \mathrm{cm} .\) The resistance of whole wire is .............. \(\Omega\)

The value of numerical aperature of the objective lens of a microscope is \(1.25\). If light of wavelength \(5000\,\mathop A\limits^o \) is used, the minimum separation between two points, to be seen as distinct, will be....\(\mu m\)

A cord is wound round the circumference of wheel of radius \(r\). The axis of the wheel is horizontal and the moment of inertia about it is \(I. \,A\) weight \(mg\) is attached to the cord at the end. The weight falls from rest. After falling through a distance \( 'h '\), the square of angular velocity of wheel will be ..... .

A network of four resistances is connected to \(9\,V\) battery, as shown in figure. The magnitude of voltage difference between the points \(A\) and \(B\) is .......... \(V.\)