A particle moves along the \(x\)-axis and has its displacement \(x\) varying with time \(t\) according to the equation \(\mathrm{x}=\mathrm{c}_0\left(\mathrm{t}^2-2\right)+\mathrm{c}(\mathrm{t}-2)^2\) where \(c_0\) and \(c\) are constants of appropriate dimensions. Then, which of the following statements is correct?

Two identical spherical balls of mass \(M\) and radius \(R\) each are stuck on two ends of a rod of length \(2R\) and mass \(M\) (see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod is

The electromagnetic waves travel in a medium at a speed of \(2.0 \times 10^{8}\, m / s\). The relative permeability of the medium is \(1.0.\) The relative permittivity of the medium will be

In an ac circuit, an inductor, a capacitor and a resistor are connected in series with \({X}_{{L}}={R}={X}_{{C}}\). Impedance of this circuit is:

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.

Identify the correct statements :
A. Effective capacitance of a series combination of capacitors is always smaller than the smallest capacitance of the capacitor in the combination.
B. When a dielectric medium is placed between the charged plates of a capacitor, displacement of charges cannot occur due to insulation property of dielectric.
C. Increasing of area of capacitor plate or decreasing of thickness of dielectric is an alternate method to increase the capacitance.
D. For a point charge, concentric spherical shells centered at the location of the charge are equipotential surfaces.
Choose the correct answer from the options given below.

Consider a Disc of mass \(5 \mathrm{~kg}\), radius \(2 \mathrm{~m}\), rotating with angular velocity of \(10 \mathrm{rad} / \mathrm{s}\) about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is _______ \(J\).

Let \(a\), \(b\) be two non-zero real numbers. If \(p\) and \(r\) are the roots of the equation \(x ^{2}-8 ax +2 a =0\) and \(q\) and \(s\) are the roots of the equation \(x^{2}+12 b x+6 b\) \(=0\), such that \(\frac{1}{ p }, \frac{1}{ q }, \frac{1}{ r }, \frac{1}{ s }\) are in A.P., then \(a ^{-1}- b ^{-1}\) is equal to \(......\)

If \(x,y,z\)  are in \(A.P.\) and  \({\tan ^{ - 1}}x,{\tan ^{ - 1}}y\) and \({\tan ^{ - 1}}z\) are also in other \(A.P.\) then  . . . 

If the system of equations \(x-2 y+3 z=9\) \(2 x+y+z=b\) \(x-7 y+a z=24\) has infinitely many solutions, then \(a - b\) is equal to

Let \(f:(1,\infty)\to\mathbb{R}\) be a function defined as \(f(x) = \dfrac{x-1}{x+1}\). Let \(f^{i+1}(x) = f(f^i(x))\), \(i=1, 2, \ldots, 25\), where \(f^1(x)=f(x)\). If \(g(x) + f^{26}(x) = 0\), \(x \in (1, \infty)\), then the area of the region bounded by the curves \(y=g(x)\), \(2y=2x-3\), \(y=0\) and \(x=4\) is:

Let \(f:[0, \infty) \rightarrow[0,3]\) be a function defined by \(f(x)=\max \{\sin t: 0 \leq t \leq x\}, \quad 0 \leq x \leq \pi\) \(\quad \quad \quad \quad \quad \quad 2+\cos x,\quad \quad \quad \quad x>\pi\) Then which of the following is true?

Let \(f:R \to R\) be a continuously differentiable function such that \(f\left( 2 \right) = 6\) and \(f'\left( 2 \right) = \frac{1}{{48}}\). If \(\int_6^{f\left( x \right)} {4{t^3}} \,dt = \left( {x - 2} \right)\,g\left( x \right)\), then \(\mathop {\lim }\limits_{x \to 2} \,g\left( x \right)\) is equal to