Q. No.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. \( {[{L}],[{LT}] \text { and }\left[{T}^2\right]}\)
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. \( {[{L}],[{LT}] \text { and }\left[{T}^2\right]}\)
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. \(-10\%\)
If the error in the measurement of the radius of a sphere is 2%, then the error in the determination of the volume of the sphere will be:
1. 4%
2. 6%
3. 8%
4. 2%
| 1. | pressure if \(a=1,\) \(b=-1,\) \(c=-2\) |
| 2. | velocity if \(a=1,\) \(b=0,\) \(c=-1\) |
| 3. | acceleration if \(a=1,\) \(b=1,\) \(c=-2\) |
| 4. | force if \(a=0,\) \(b=-1,\) \(c=-2\) |
\(P=\dfrac{a^3b^2}{cd}\)
Percentage error in \(P\) is:
1. \(10\%\)
2. \(7\%\)
3. \(4\%\)
4. \(14\%\)
If dimensions of critical velocity \({v_c}\) of a liquid flowing through a tube are expressed as \(\eta^{x}\rho^yr^{z}\), where \(\eta, \rho~\text{and}~r\) are the coefficient of viscosity of the liquid, the density of the liquid, and the radius of the tube respectively, then the values of \({x},\) \({y},\) and \({z},\) respectively, will be:
1. \(1,-1,-1\)
2. \(-1,-1,1\)
3. \(-1,-1,-1\)
4. \(1,1,1\)
| 1. | \(0.521\) cm | 2. | \(0.525\) cm |
| 3. | \(0.053\) cm | 4. | \(0.529\) cm |
Planck's constant (\(h\)), speed of light in the vacuum (\(c\)), and Newton's gravitational constant (\(G\)) are the three fundamental constants. Which of the following combinations of these has the dimension of length?
| 1. | \(\dfrac{\sqrt{hG}}{c^{3/2}}\) | 2. | \(\dfrac{\sqrt{hG}}{c^{5/2}}\) |
| 3. | \(\dfrac{\sqrt{hG}}{G}\) | 4. | \(\dfrac{\sqrt{Gc}}{h^{3/2}}\) |
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
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:
1. \(0.25~\text{mm}\)
2. \(0.5~\text{mm}\)
3. \(1.0~\text{mm}\)
4. \(0.01~\text{mm}\)
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