A parallel plate capacitor with plate separation 5 mm is charged by a battery. On introducing a mica sheet of 2 mm and maintaining the connections of the plates with the terminals of the battery, it is found that it draws \(25 \%\) more charge from the battery. The dielectric constant of mica is ___________ .

Which of the following are correct expression for torque acting on a body?
A. \(\vec{\tau}=\vec{r}\times\vec{L}\)
B. \(\vec{\tau}=\frac{\mathrm{d}}{\mathrm{dt}}(\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{p}})\)
C. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \frac{\mathrm{d} \mathrm{p}}{\mathrm{dt}}\)
D. \(\vec{\tau}=\mathrm{I} \vec{\alpha}\)
E. \(\vec{\tau}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}\)
( \(\overrightarrow{\mathrm{r}}=\) position vector; \(\overrightarrow{\mathrm{p}}=\) linear momentum;
\(\overrightarrow{\mathrm{L}}=\) angular momentum; \(\vec{\alpha}=\) angular acceleration;
\(I=\) moment of inertia; \(\vec{F}=\) force; \(t=\) time)
Choose the correct answer from the options given below :

A particle of mass \(m\) with an initial velocity \(u\hat i\) collides perfectly elastically with a mass \(3m\) at rest. It moves with a velocity \(v\hat j\) after collision, then, \(v\) is given by

A body of mass \({M}\) moving at speed \({V}_{0}\) collides elastically with a mass \('m'\) at rest. After the collision, the two masses move at angles \(\theta_{1}\) and \(\theta_{2}\) with respect to the initial direction of motion of the body of mass \({M}\). The largest possible value of the ratio \({M} / {m}\), for which the angles \(\theta_{1}\) and \(\theta_{2}\) will be equal, is :

An ideal fluid flows (laminar flow) through a pipe of non-uniform diameter. The maximum and minimum diameters of the pipes are \(6.4 \;\mathrm{cm}\) and \(4.8 \;\mathrm{cm},\) respectively. The ratio of the minimum and the maximum velocities of fluid in this pipe is:

Two particles \(A, B\) are moving on two concentric circles of radii \(R_1\) and \(R_2\) with equal angular speed \(\omega \). At \(t = 0,\) their positions and direction of motion are shown in the figure The relative velocity \(\overrightarrow {{V_A}}  - \overrightarrow {{V_B}} \) at \(t = \frac{\pi }{{2\omega }}\) is given by

The difference of temperature in a material can convert heat energy into electrical energy. To harvest the heat energy, the material should have _________.

If in a \(G.P.\) of \(64\) terms, the sum of all the terms is \(7\) times the sum of the odd terms of the \(G.P,\) then the common ratio of the \(G.P\). is equal to

Let \(f(x)=\frac{x-1}{x+1}, x \in R-\{0,-1,1)\). If \(f^{a+1}(x)=f\left(f^{n}(x)\right)\) for all \(n \in N\), then \(f^{\prime}(6)+f(7)\) is equal to

The value of \(\int_0^1\left(2 x^3-3 x^2-x+1\right)^{\frac{1}{3}} d x\) is equal to :

Consider the following three statements for the function \( f:(0,\infty)\rightarrow\mathbb{R} \) defined by
\( f(x)=|log_{e}x|-|x-1| \):
(I) f is differentiable at all \( x>0 \).
(II) f is increasing in (0, 1).
(III) f is decreasing in (1, ∞).
Then:

A balloon has mass of \(10\,g\) in air. The air escapes from the balloon at a uniform rate with velocity \(4.5\,cm / s\). If the balloon shrinks in \(5\,s\) completely. Then, the average force acting on that balloon will be (in dyne).

Let \(f(x)=\begin{cases} e^{x-1}, & x<0 \\ x^2-5x+6, & x \geq 0 \end{cases}\) and \(g(x)=f(|x|)+|f(x)|\). If the number of points where \(g\) is not continuous and is not differentiable are \(\alpha\) and \(\beta\) respectively, then \(\alpha+\beta\) is equal to ______