## Miscellaneous Exam-Style Questions.## Problems adapted from questions set for previous Mathematics exams. The questions shown here or their solutions contain the text 'Planet'. |

## 1. | 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.

## 2. | 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} |

## 3. | IB Analysis and Approaches |

NASA’s Transiting Exoplanet Survey Satellite (TESS) has discovered a planet between the sizes of Mars and Earth orbiting a bright, cool, nearby star. The planet, called L 98-59b, marks the smallest found by TESS yet. The radius of L 98-59b is \(5.42\times 10^6\) m.

(a) Write down the diameter of the planet in kilometers.

The volume of the planet in cubic kilometers can be expressed in the form \(n\times 10^k\), where \( 1 \le n\lt 10,k\in \mathbb Z\)

(b) Find the value of \(n\) and the value of \(k\).

## 4. | IB Analysis and Approaches |

The heights, H metres, of flowers called Xylothorn Blooms growing in the dense forests of Verdantem on the luminous planet Aurorion can be modelled by a normal distribution with mean 14.3 metres and standard deviation 3.9 metres.

(a) One of the flowers is selected at random. Find the probability that its height more than 15.5 metres.

According to this model, 40% of the flowers have a height between \(x\) metres and 15.5 metres.

(b) Find the probability that a randomly selected flower has a height less than \(x\) metres.

(c) Find the value of \(x\).

(d) Ten flowers are selected at random.

Find the probability that no more than two of the flowers has a height less than \(x\) metres.

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