Asseretion \(A:\) If in five complete rotations of the circular scale, the distance travelled on main scale of the screw gauge is \(5\, {mm}\) and there are \(50\) total divisions on circular scale, then least count is \(0.001\, {cm}\). Reason \(R:\) Least Count \(=\frac{\text { Pitch }}{\text { Total divisions on circular scale }}\) In the light of the above statements, choose the most appropriate answer from the options given below:

Two electric dipoles of dipole moments \(1.2 \times 10^{-30}\,cm\) and \(2.4 \times 10^{-30}\,cm\) are placed in two difference uniform electric fields of strengths \(5 \times 10^{4}\,NC ^{-1}\) and \(15 \times 10^{4}\,NC ^{-1}\) respectively. The ratio of maximum torque experienced by the electric dipoles will be \(\frac{1}{ x }\). The value of \(x\) is \(.....\)

In a Vernier Calipers. \(10\) divisions of Vernier scale is equal to the \(9\) divisions of main scale. When both jaws of Vernier calipers touch each other, the zero of the Vernier scale is shifted to the left of \(zero\) of the main scale and \(4^{\text {th }}\) Vernier scale division exactly coincides with the main scale reading. One main scale division is equal to \(1\,mm\). While measuring diameter of a spherical body, the body is held between two jaws. It is now observed that zero of the Vernier scale lies between \(30\) and \(31\) divisions of main scale reading and \(6^{\text {th }}\) Vernier scale division exactly. coincides with the main scale reading. The diameter of the spherical body will be \(.......cm\)

A force \(\left(3 x^2+2 x-5\right) N\) displaces a body from \(\mathrm{x}=2 \mathrm{~m}\) to \(\mathrm{x}=4 \mathrm{~m}\). Work done by this force is _______ J.

The amount of heat needed to raise the temperature of \(4\, moles\) of a rigid diatomic gas from \(0^{\circ} {C}\) to \(50^{\circ} {C}\) when no work is done is ......\({R}\) (\(R\) is the universal gas constant)

The volume charge density of a sphere of radius \(6 \,m\) is \(2 \,\mu cm ^{-3}\). The number of lines of force per unit surface area coming out from the surface of the sphere is \(....\times 10^{10}\, NC ^{-1}\).  [Given : Permittivity of vacuum  \(\left.\epsilon_{0}=8.85 \times 10^{-12} C ^{2} N ^{-1}- m ^{-2}\right]\)

In a vernier calliper, when both jaws touch each other, zero of the vernier scale shifts towards left and its \(4^{\text {th }}\) division coincides exactly with a certain division on main scale. If \(50\) vernier scale divisions equal to \(49\) main scale divisions and zero error in the instrument is \(0.04 \mathrm{~mm}\) then how many main scale divisions are there in \(1 \mathrm{~cm}\) ?

A data consists of \(20\) observations \(x_1, x_2, \ldots, x_{20}\). If \(\sum_{i=1}^{20}(x_i + 5)^2 = 2500\) and \(\sum_{i=1}^{20}(x_i - 5)^2 = 100\), then the ratio of mean to standard deviation of this data is:

Let \(\sum\limits_{k = 1}^{10} {f\,(a\, + \,k)} \, = \,16\,({2^{10}}\, - \,1),\) where the function \(f\) satisfies \(f(x + y) = f(x) f(y)\) for all natural numbers \(x, y\) and \(f(1) = 2.\) Then the natural number \(‘ a '\) is

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

Let \(B _{i}(i=1,2,3)\) be three independent events in a sample space. The probability that only \(B _{1}\) occur is \(\alpha,\) only \(B _{2}\) occurs is \(\beta\) and only \(B _{3}\) occurs is \(\gamma\). Let \(p\) be the probability that none of the events \(B _{i}\) occurs and these \(4\) probabilities satisfy the equations \((\alpha-2 \beta) p =\alpha \beta\) and \((\beta-3 \gamma) p =2 \beta \gamma\) (All the probabilities are assumed to lie in the interval \((0,1))\). Then \(\frac{ P \left( B _{1}\right)}{ P \left( B _{3}\right)}\) is equal to ..........

Let \(y=y(x)\) be the solution of the differential equation \(\left(x-x^{3}\right) d y=\left(y+y x^{2}-3 x^{4}\right) d x, x>2\). If \(y(3)=3\), then \(y(4)\) is equal to :

If the tangent to the curve \(y=x+\sin y\) at a point \((a, b)\) is parallel to the line joining \(\left(0, \frac{3}{2}\right)\) and \(\left(\frac{1}{2}, 2\right),\) then