## Exam-Style Questions on Standard Form## Problems on Standard Form adapted from questions set in previous Mathematics exams. |

## 1. | GCSE Higher |

The number 1.25 × 10^{n} is a cube number.

Find the smallest possible integer value of n.

## 2. | GCSE Higher |

The following table shows the distance from the sun and mass of the eight planets.

Planet | Distance from the Sun (km) | Mass (tonnes) |
---|---|---|

Mercury | 5.79×10^{7} |
3.3×10^{20} |

Venus | 1.08×10^{8} |
4.87×10^{21} |

Earth | 1.5×10^{8} |
5.98×10^{21} |

Mars | 2.28×10^{8} |
6.42×10^{20} |

Jupiter | 7.78×10^{8} |
1.9×10^{24} |

Saturn | 1.43×10^{9} |
5.69×10^{23} |

Uranus | 2.87×10^{9} |
8.68×10^{22} |

Neptune | 4.5×10^{9} |
1.02×10^{23} |

(a) Which planet has the second smallest mass?

(b) Find the difference between the mass of Saturn and the mass of Jupiter.

(c) It is claimed that Saturn is about ten times further from the sun than the Earth. By showing your working explain why you either agree or disagree with this statement.

## 3. | IB Studies |

The Moon appears to orbit around the Earth with an average radius of 385 thousand kilometres.

(a) Write down this radius, in kilometres, in the form \(a\times 10^k\), where \( 1\le a\lt 10,k\in \mathbb Z\)

(b) Calculate the distance, in kilometres, the Moon travels around the Earth in one orbit, assuming that the orbit is a circle.

It takes approximately 27.3 days for the Moon to complete one orbit.

(c) Calculate the total distance the Moon appears to travel around the earth during a leap year.

## 4. | GCSE Higher |

The volume of the Moon is \(2.2 \times 10^{10}\) km^{3}

The volume of the Saturn is \(8.3 \times 10^{14}\) km^{3}

How many times larger is the radius of Saturn than the radius of the Moon?

Assume that Saturn and the Moon are both spheres and that the volume \(v\) of a sphere with radius \(r\) is given by the following formula.

$$v=\frac43\pi r^3$$Give your answer to 2 significant figures.

## 5. | GCSE Higher |

The surface gravity, \(g\), of a planet is the gravitational acceleration experienced at its surface. The following formula can be used to find how a planet's gravity compares to Earth's.

$$g=\frac{6.67 \times 10^{-11} \times m}{r^2}$$Where \(m\) is the mass of the planet and \(r\) is the radius.

Find the ratio of the surface gravity on Jupiter and the surface gravity Mars. Give your answer in the form \(n:1\).

Mass (kg) | Radius (m) | |

Jupiter | 1.90 × 10^{27} | 6.68 × 10^{7} |

Mars | 6.39 × 10^{23} | 3.39 × 10^{6} |

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