Consider two uniform discs of the same thickness and different radii \(R _{1}= R\) and \(R _{2}=\alpha R\) made of the same material. If the ratio of their moments of inertia \(I_{1}\) and \(I_{2}\), respectively, about their axes is \(I _{1}: I _{2}=1: 16\) then the value of \(\alpha\) is

A null point is found at \(200\,cm\) in potentiometer when cell in secondary circuit is shunted by \(5\,\Omega\). When a resistance of \(15\,\Omega\) is used for shunting null point moves to \(300\,cm\). The internal resistance of the cell is \(..............\,\Omega\).

Heat energy of \(184\,kJ\) is given to ice of mass \(600\,g\) at \(-12^{\circ}\,C\), Specific heat of ice is \(2222.3\,J\,kg ^{-1^{\circ}}\,C ^{-1}\) and latent heat of ice in \(336\,kJ / kg ^{-1}\) \((A)\) Final temperature of system will be \(0^{\circ} C\). \((B)\) Final temperature of the system will be greater than \(0^{\circ} C\). \((C)\) The final system will have a mixture of ice and water in the ratio of \(5: 1\). \((D)\) The final system will have a mixture of ice and water in the ratio of \(1: 5\). \((E)\) The final system will have water only. Choose the correct answer from the options given below:

The ratio of surface tensions of mercury and water is given to be \(7.5\) while the ratio of their densities is \(13.6\). Their contact angles, with glass, are close to \(135^o\) and \(0^o\), respectively. It is observed that mercury gets depressed by an amount \(h\) in a capillary tube of radius \(r_1\), while water rises by the same amount \(h\) in a capillary tube of radius \(r_2\). The ratio, \((r_1/r_2)\), is then close to

If wattless current flows in the \(AC\) circuit, then the circuit is

Given below are two statements: one is labelled as Assertion\((A)\) and the other is labelled as Reason \((R).\) \(Assertion\) \((A)\) : In Vernier calliper if positive zero error exists, then while taking measurements, the reading taken will be more than the actual reading. \(Reason\) \((R)\) : The zero error in Vernier Calliper might have happened due to manufacturing defect or due to rough handling. In the light of the above statements, choose the correct answer from the options given below :

A wind-10.Powered generator converts wind energy into electrical energy. Assume that the generator converts a fixed fraction of the wind energy intercepted by its blades into electrical energy. For wind speed \(v\), the electrical 10.Power output will be most likely proportional to

A square loop of side \(2\, a ,\) and carrying current I, is kept in \(XZ\) plane with its centre at origin. A long wire carrying the same current \(I\) is placed parallel to the \(z-\)axis and passing through the point \((0, b, 0),(b>>a)\). The magnitude of the torque on the loop about \(z-\)axis is given by:

Slope of a line passing through \(P(2, 3)\) and intersecting the line, \(x + y = 7\) at a distance of \(4\) units from \(P,\) is

Let \(A=\{0,1,2,3,4,5\}\). Let \(R\) be a relation on A defined by \((x, y) \in R\) if and only if max \(\{x, y\} \in\{3,4\}\). Then among the statements \(\left(\mathrm{S}_1\right)\) : The number of elements in R is 18 , and \(\left(\mathrm{S}_2\right)\) : The relation R is symmetric but neither reflexive nor transitive

A bullet is fired into a fixed target looses one third of its velocity after travelling \(4 \mathrm{~cm}\). It penetrates further \(\mathrm{D} \times 10^{-3} \mathrm{~m}\) before coming to rest. The value of \(D\) is:

Let z be the complex number satisfying \( |z-5|\le3 \) and having maximum positive principal argument. Then \( 34|\frac{5z-12}{5iz+16}|^{2} \) is equal to :

If the radius of the largest circle with centre \((2,0)\) inscribed in the ellipse \(x^2+4 y^2=36\) is \(r\), then \(12 r^2\) is equal to