Q. No.If \(u_1\) and \(u_2\) are the units selected in two systems of measurement and \(n_1\) and \(n_2\) are their numerical values, then:
| 1. | \(n_1u_1=n_2u_2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \) |
| 2. | \(n_1u_1+n_2u_2=0\) |
| 3. | \(n_1n_2=u_1u_2\) |
| 4. | \((n_1+u_1)=(n_2+u_2)\) |
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The velocity \(v\) of a particle at time \(t\) is given by \({v}={at}+\frac{{b}}{{t}+{c}}.\) The dimensions of \({a}\), \({b}\), and \({c}\) are respectively:
1. \( {\left[{LT}^{-2}\right],[{L}],[{T}]} \)
2. \( {\left[{L}^2\right],[{T}] \text { and }\left[{LT}^2\right]} \)
3. \( {\left[{LT}^2\right],[{LT}] \text { and }[{L}]} \)
4. \( {[{L}],[{LT}], \text { and }\left[{T}^2\right]}\)
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| 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 \(a=0, ~b= -1,~c=-2\) |
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| 1. | \(\left[MLT^{-2}\right]\) | 2. | \(\left[ML^{-1}T^{2}\right]\) |
| 3. | \(\left[ML^{-1}T^{-2}\right]\) | 4. | \(\left[MLT^{2}\right]\) |
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| 1. | \([MLT^{-2}]\) | 2. | \([MLT^{-1}]\) |
| 3. | \([ML^2T^{-1}]\) | 4. | \([M^2LT^{-1}]\) |
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In the relation, \(y=a \cos (\omega t-k x)\), the dimensional formula for \(k\) will be:
1. \( {\left[M^0 L^{-1} T^{-1}\right]} \)
2. \({\left[M^0 L T^{-1}\right]} \)
3. \( {\left[M^0 L^{-1} T^0\right]} \)
4. \({\left[M^0 L T\right]}\)
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When units of mass, length, and time are taken as \(10~\text{kg}, 60~\text{m}~\text{and}~60~\text{s}\) respectively, the new unit of energy becomes \(x\) times the initial SI unit of energy. The value of \(x\) will be:
1. \(10\)
2. \(20\)
3. \(60\)
4. \(120\)
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1. \(16\) times
2. \(8\) times
3. \(2\) times
4. \(4\) times
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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. \(\left[L^{{-1/2}}T^{{-1/2}}\right]\)
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For the expression, \(10^{(at+3)}\), the dimensions of \(a\) will be:
1. \(\left[M^0L^0T^{0}\right]\)
2. \(\left[M^0L^0T^{1}\right]\)
3. \(\left[M^0L^0T^{-1}\right]\)
4. None of these
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