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\(\int \frac{\mathrm{dx}}{\sqrt{2 \mathrm{ax}-\mathrm{x}^{2}}}=\mathrm{a}^{\mathrm{n}} \sin ^{-1}\left[\frac{\mathrm{x}}{\mathrm{a}}-1\right]\)

The value of \(\mathrm{n}\) is:

1. \(0\)

2. \(-1\)

3. \(1\)

4. none of these

Subtopic: Â Dimensions |

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The dimensions of \(\left [ML^{-1} T^{-2} \right ]\) may correspond to:

a. | work done by a force |

b. | linear momentum |

c. | pressure |

d. | energy per unit volume |

Choose the correct option:

1. | (a) and (b) |

2. | (b) and (c) |

3. | (c) and (d) |

4. | none of the above |

Subtopic: Â Dimensions |

Â 81%

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The radius of a circle is stated as \(2.12\) cm. Its area should be written as:

1. | \(14\mathrm{~cm^2}\) | 2. | \(14.1\mathrm{~cm^2}\) |

3. | \(14.11\mathrm{~cm^2}\) | 4. | \(14.1124\mathrm{~cm^2}\) |

Subtopic: Â Significant Figures |

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A dimensionless quantity,

1. | never has a unit |

2. | always has a unit |

3. | may have a unit |

4. | does not exist |

Subtopic: Â Dimensions |

Â 79%

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A physical quantity is measured and the result is expressed as \(nu\) where \(u\) is the unit used and \(n\) is the numerical value. If the result is expressed in various units then:

1. \(n\propto \mathrm{size~of}~u\)

2. \(n\propto u^2\)

3. \(n\propto \sqrt u\)

4. \(n\propto \frac{1}{u}\)

Subtopic: Â Dimensions |

Â 77%

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Which of the following equations is dimensionally correct?

\((I)~~ v=\sqrt{\frac{P}{\rho}}~~~~~~(II)~~v=\sqrt{\frac{mgl}{I}}~~~~~~(III)~~v=\frac{Pr^2}{2\eta l}\)

(where \(v=\) speed, \(P=\) pressure; \(r,\) *\(l\)* are lengths; \(\rho=\) density, \(m=\) mass, \(g=\) acceleration due to gravity, \(I=\) moment of inertia, and \(\eta=\) coefficient of viscosity)

1. | \(I~ and~II\) |

2. | \(I~ and~III\) |

3. | \(II~ and~III\) |

4. | \(I,~II~and~III\) |

Subtopic: Â Dimensions |

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The acceleration due to gravity on the surface of the earth is \(g=10\) m/s^{2}. The value in km/(minute)^{2} is:

1. \(36\)

2. \(0.6\)

3. \(\frac{10}{6}\)

4. \(3.6\)

Subtopic: Â Dimensions |

Â 62%

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Darcy's law describes the volume flow rate \((Q)\) of a viscous fluid (viscosity : \(\mu\)) through a porous medium, under the action of a pressure difference. \(\frac{Q}{A}=\frac{k}{\mu}\left(\frac{{\large p}_{1}-{\large p}_{2}}{L}\right) \)

Here, \(k\) is the permeability of the medium. The unit of \(k\) is (SI units)

1. m

2. m^{2}

3. m^{2}/s

4. m/s^{2}

Here, \(k\) is the permeability of the medium. The unit of \(k\) is (SI units)

1. m

2. m

3. m

4. m/s

Subtopic: Â Dimensions |

Â 52%

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Plane angle and solid angle have:

1. | both units and dimensions |

2. | units but no dimensions |

3. | dimensions but no units |

4. | no units and no dimensions |

Subtopic: Â Dimensions |

Â 78%

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The area of a rectangular field (in \(\text{m}^2\)) of length \(55.3\) m and breadth \(25\) m after rounding off the value, for correct significant digits is:

1. \(14\times10^{2}\)

2. \(138\times10^{1}\)

3. \(1382\)

4. \(1382.5\)

1. \(14\times10^{2}\)

2. \(138\times10^{1}\)

3. \(1382\)

4. \(1382.5\)

Subtopic: Â Significant Figures |

Â 59%

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