A screw gauge gives the following readings when used to measure the diameter of a wire:
Main scale reading: \(0\) mm
Circular scale reading: \(52\) divisions 
Given that \(1\) mm on the main scale corresponds to \(100\) divisions on the circular scale, the diameter of the wire that can be inferred from the given data is:
1. \(0.26\) cm 
2. \(0.052\) cm 
3. \(0.52\) cm 
4. \(0.026\) cm 

We measure the period of oscillation of a simple pendulum. In successive measurements, the readings turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71 s, and 2.80 s. The average absolute error and percentage error, respectively, are:

1.  0.22 s and 4%

2.  0.11 s and 4%

3.  4 s and 0.11%

4.  5 s and 0.22%

If \(y = a\sin(bt-cx)\), where \(y\) and \(x\) represent length and \(t\) represents time, then which of the following has the same dimensions as that of \(\frac{ab^2}{c}\)$\frac{\mathrm{}}{}$?
1. \((\text{speed})^2\)
2. \(\text{momentum}\)
3. \(\text{angle}\)
4. \(\text{acceleration}\)

The dimensions of \((\mu_0\varepsilon_0)^{\frac{-1}{2}}\) are:
1. \(\left[L^{-1}T\right]\)
2. \(\left[LT^{-1}\right]\)
3. \(\left[L^{{-1/2}}T^{{1/2}}\right]\)
4. \(\left[L^{{-1/2}}T^{{-1/2}}\right]\)

An object is moving through a liquid. The viscous damping force acting on it is proportional to the velocity. Then the dimensions of the constant of proportionality are:
1. \(\left[ML^{-1}T^{-1}\right]\)
2. \(\left[MLT^{-1}\right]\)
3. \(\left[M^0LT^{-1}\right]\)
4. \(\left[ML^{0}T^{-1}\right]\)

The density of a material in a CGS system of units is \(4~\text{g/cm}^3\). In a system of units in which the unit of length is \(10~\text{cm}\) and the unit of mass is \(100~\text{g}\), the value of the density of the material will be:
1. \( 0.04 \)
2. \( 0.4 \)
3. \( 40 \)
4. \(400\)

On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is/are not correct.
a. \(y = a\sin \left(2\pi t / T\right)\)
b. \(y = a\sin(vt)\)
c. \(y = \left({a \over T}\right) \sin \left({t \over a}\right)\)
d. \(y = a \sqrt 2 \left(\sin \left({2 \pi t \over T}\right) - \cos \left({2 \pi t \over T}\right)\right)\)
(Symbols have their usual meanings.)
Choose the correct option:
1. (a), (c)
2. (a), (b)
3. (b), (c)
4. (a), (d)

The mean length of an object is \(5~\text{cm}\). Which of the following measurements is the most accurate?
1. \(4.9~\text{cm}\)
2. \(4.805~\text{cm}\)
3. \(5.25~\text{cm}\)
4. \(5.4~\text{cm}\)

The length and breadth of a rectangular sheet are 16.2 cm and 10.1 cm, respectively. The area of the sheet in appropriate significant figures and error would be, respectively,
1. 164 $\pm$ 3 ${\mathrm{cm}}^2$
2. 163.62 $\pm$ 2.6 ${\mathrm{cm}}^2$ 
3. 163.6 $\pm$ 2.6 ${\mathrm{cm}}^2$ 
4. 163.62 $\pm$ 3 ${\mathrm{cm}}^2$

The numbers \(2.745\) and \(2.735\) on rounding off to \(3\) significant figures will give respectively,