A student performs an experiment to determine Young's modulus of a wire, exactly 1 m long by Searle's method. In a particular reading, the student measures extensions in the length of the wire to be 4.00 mm with an uncertainty of ± 0.01 mm at a load of exactly 1 kg. The student also measures the radius of wire to be 1.00 mm with an uncertainty of ± 0.005 mm. The Young's modulus obtained from the reading is (Take g = 10 $\mathrm{m}/{\mathrm{s}}^2$) :

 

$1. (7.96 \pm 0.10) \times {10}^8 N/m^2$
$$
$2. (7.96\pm 0.16)\times {10}^8\mathrm{N}/{\mathrm{m}}^2$
$$
$3. (8.0\pm 0.2)\times {10}^8\mathrm{N}/{\mathrm{m}}^2$
$$
$4. (7.04\pm 0.32)\times {10}^8\mathrm{N}/{\mathrm{m}}^2\begin{array}{}\end{array}$

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 :

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

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

In an experiment, the percentage errors that occurred in the measurement of physical quantities \(A,\) \(B,\) \(C,\) and \(D\) are \(1\%\), \(2\%\), \(3\%\), and \(4\%\) respectively. Then, the maximum percentage of error in the measurement of \(X,\) where \(X=\frac{A^2 B^{\frac{1}{2}}}{C^{\frac{1}{3}} D^3}\), will be:
1. \(10\%\)
2. \(\frac{3}{13}\%\)
3. \(16\%\)
4. \(-10\%\)

The velocity \(v\) of a particle at time \(t\) is given by \(v=at+\dfrac{b}{t+c}\), where \(a,\) \(b\) and \(c\) are constants. The dimensions of \(a,\) \(b\) and \(c\) are respectively:
1. \(\left[{LT}^{-2}\right],[{L}] \text { and }[{T}]\)
2. \( {\left[{L}^2\right],[{T}] \text { and }\left[{LT}^2\right]}  \)
3. \( {\left[{LT}^2\right],[{LT}] \text { and }[{L}]}  \)
4. \( {[{L}],[{LT}] \text { and }\left[{T}^2\right]}\)