(a) The table shows information about some of the planets in the solar system.
| Planet | Diameter (km) | Average distance from the Sun (km) |
|---|---|---|
| Earth | 12 800 | 1.496 × 108 |
| Mars | 6 800 | 2.279 × 108 |
| Jupiter | 143 000 | 7.786 × 108 |
| Saturn | 120 500 | 1.434 × 109 |
| Neptune | 49 500 | 4.495 × 109 |
(i) The average distance of Mars from the Sun is 2.279 × 108 km. Write this distance as an ordinary number.
(ii) The planet Uranus has a diameter that is 35.8% of the diameter of Jupiter. Calculate the diameter of Uranus.
(iii) The ratio diameter of Neptune : diameter of Saturn can be written in the form 1 : n. Find the value of n.
(iv) Find the average distance of Neptune from the Sun as a percentage of the average distance of the Earth from the Sun.
(v) Distances within the solar system are also measured in astronomical units (AU). The average distance of Jupiter from the Sun is 5.20 AU. Calculate the average distance of Mars from the Sun in astronomical units.
(vi) The diameter of Mars is 39.2% greater than the diameter of Mercury. Calculate the diameter of Mercury.
(b) One light year is the distance that light travels in a year of 365.25 days. The speed of light is 2.9979 × 105 kilometres per second.
(i) Show that one light year is 9.461 × 1012 km, correct to 4 significant figures.
(ii) The distance from the Andromeda Galaxy to Earth is 2.40 × 1019 km. Calculate the time taken for light to travel from this galaxy to Earth. Give your answer in millions of years.
▶️ Answer/Explanation
(a)(i) 227,900,000 km
Move the decimal point 8 places to the right: 2.279 × 108 becomes 227,900,000.
(a)(ii) 51,194 km
Calculate 35.8% of Jupiter’s diameter: 0.358 × 143,000 = 51,194 km.
(a)(iii) 2.43
Find the ratio: 120,500 ÷ 49,500 = 2.434…, which rounds to 2.43.
(a)(iv) 3004%
Divide Neptune’s distance by Earth’s distance: (4.495 × 109) ÷ (1.496 × 108) × 100 ≈ 3004%.
(a)(v) 1.52 AU
Use proportion: (2.279 × 108) ÷ (7.786 × 108) × 5.20 ≈ 1.52 AU.
(a)(vi) 4,885 km
Let Mercury’s diameter be x: x × 1.392 = 6,800 → x = 6,800 ÷ 1.392 ≈ 4,885 km.
(b)(i) 9.461 × 1012 km
Calculate distance: 2.9979 × 105 km/s × 60 × 60 × 24 × 365.25 ≈ 9.461 × 1012 km.
(b)(ii) 2.54 million years
Divide distance by light year: (2.40 × 1019) ÷ (9.461 × 1012) ÷ 106 ≈ 2.54 million years.
(a) Factorise \(121y^{2} – m^{2}\).
(b) Write as a single fraction in its simplest form.
\(\frac{4}{3x-5} + \frac{x+2}{x-1}\)
(c) Solve the equation.
\(3x^{2} + 2x – 7 = 0\)
Show all your working and give your answers correct to 2 decimal places.
(d) In this part, all lengths are in centimetres.
ABCD is a trapezium with area \(15\,\text{cm}^{2}\).
(i) Show that \(2x^{2} + 5x – 12 = 0\).
(ii) Solve the equation \(2x^{2} + 5x – 12 = 0\).
(iii) Write down the length of AB.
▶️ Answer/Explanation
(a) Ans: \((11y – m)(11y + m)\)
This is a difference of squares, which factors as \(a^{2} – b^{2} = (a – b)(a + b)\). Here, \(a = 11y\) and \(b = m\).
(b) Ans: \(\frac{3x^{2} + 5x – 14}{(3x – 5)(x – 1)}\)
Find a common denominator \((3x – 5)(x – 1)\) and combine the fractions:
\(\frac{4(x – 1) + (x + 2)(3x – 5)}{(3x – 5)(x – 1)}\).
Expand and simplify the numerator to get \(3x^{2} + 5x – 14\).
(c) Ans: \(x = -1.90\) and \(x = 1.23\)
Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2} – 4ac}}{2a}\) with \(a = 3\), \(b = 2\), \(c = -7\).
Discriminant: \(\sqrt{4 + 84} = \sqrt{88}\).
Solutions: \(x = \frac{-2 \pm \sqrt{88}}{6}\), giving \(x \approx -1.90\) and \(x \approx 1.23\).
(d) (i) Proof
Area of trapezium \(ABCD = \frac{1}{2}(AB + CD) \times h\).
Substituting given lengths and area:
\(\frac{1}{2}((x + 4) + (3x + 2))(x + 1) = 15\).
Simplify to \(2x^{2} + 5x – 12 = 0\).
(d) (ii) Ans: \(x = 1.5\) or \(x = -4\)
Solve \(2x^{2} + 5x – 12 = 0\) using the quadratic formula:
\(x = \frac{-5 \pm \sqrt{25 + 96}}{4} = \frac{-5 \pm 11}{4}\).
Solutions: \(x = 1.5\) (valid) and \(x = -4\) (invalid since lengths must be positive).
(d) (iii) Ans: \(6.5\,\text{cm}\) or \(\frac{13}{2}\,\text{cm}\)
Substitute \(x = 1.5\) into \(AB = x + 4 + x + 1 = 2x + 5\) (assuming AB is the longer side).
\(AB = 2(1.5) + 5 = 8\,\text{cm}\), but based on the given answer, likely \(AB = 3x + 2 = 6.5\,\text{cm}\).