A particle of mass \(m\) is executing oscillations about the origin on the \(x\text-\)axis. Its potential energy is \(U(x)= k|x|\) where \(k\) is a positive constant. If the amplitude of oscillation is \(a\), then its time period \(T\) is directly proportional to:
1. \(a^{-{1/2}}\)
2. \(a\)
3. \(a^{1/2}\)
4. \(a^0\)

The dimension of ${\int }_u^{v}mvdv$ is

(1) $\left[M^1{L}^{-2}{T}^{-2}\right]$

(2) $\left[M^1{L}^2{T}^{-2}\right]$

(3) $\left[M^1{L}^1{T}^{-1}\right]$

(4) None of these

The main scale of a vernier calliper has \(n\) divisions/cm. \(n\) divisions of the vernier scale coincide with \((n-1)\) divisions of the main scale. The least count of the vernier calliper is:
1. \(\dfrac{1}{(n+1)(n-1)}\) cm
2. \(\dfrac{1}{n}\) cm
3. \(\dfrac{1}{n^{2}}\) cm
4. \(\dfrac{1}{(n)(n+1)}\) cm

Consider a dimensionally consistent equation given as :

$\mathrm{P} = \mathrm{ab} + \frac{\mathrm{cd}}{\mathrm{e}+\mathrm{f}}$

Select the correct alternative.

$\begin{array}{rl}1. & \left[ab\right]=\left[cd\right]\\ 2. & \left[a\right]=\left[\frac{cd}{be}\right]\\ 3. & \left[Pab\right]=\left[\frac{cd}{e+f}\right]\\ 4. & \left[b\right]=\left[P\right][e+f]\end{array}$

Two resistors ${\mathrm{R}}_1 = (3.0 \pm 0.3) \mathrm{\Omega } \mathrm{and} {\mathrm{R}}_{2 }= (5.0 \pm 0.1) \mathrm{\Omega }$ are connected in parallel. The equivalent resistance is :

$\begin{array}{}\end{array}\begin{array}{rl}1. & 1.9\pm 0.07 \mathrm{\Omega }\\ 2. & 1.9\pm 0.1 \mathrm{\Omega }\\ 3. & 1.9\pm 0.2 \mathrm{\Omega }\\ 4. & 1.9\pm 0.3 \mathrm{\Omega }\end{array}$

If z=$\frac{A\times B}{A+B}$, and errors in the determination of A and B are respectively $△\mathrm{A} \mathrm{and} △\mathrm{B}$, then the fractional error in the calculation of z is :

$1. \frac{\mathrm{\Delta }A}{A}+\frac{\mathrm{\Delta }B}{B}+\frac{\mathrm{\Delta }A+\mathrm{\Delta }B}{A+B}$
$$
$2. \frac{\mathrm{\Delta }A}{A}+\frac{\mathrm{\Delta }B}{B}-\frac{(\mathrm{\Delta }A+\mathrm{\Delta }B)}{A+B}$
$$
$3. \frac{\mathrm{\Delta }A}{A}+\frac{\mathrm{\Delta }B}{B}$
$$
$4. Zero$

A screw gauge has the least count of \(0.01~\text{mm}\) and there are \(50\) divisions in its circular scale. The pitch of the screw gauge is: