Higher June 2025 Paper 2 Q26
26 A and B are two similar solids.
The height of solid A is 31 cm
The height of solid B is 18.6 cm
Given that
volume of solid A – volume of solid B = 735 cm3
find the volume of solid A
(4)
| Scheme | Marks |
|---|---|
A : B = 31 : 18.6 (= 5 : 3) oe or A3 : B3 = 313 : 18.63 (= 53 : 33) oe or \(\dfrac{31}{18.6}\left(= \dfrac{5}{3}\right)\) oe or \(\dfrac{18.6}{31}\left(= \dfrac{3}{5}\right)\) oe or \(\left(\dfrac{18.6}{31}\right)^3\) oe or \(\dfrac{27}{125}\) oe or \(\left(\dfrac{31}{18.6}\right)^3\) oe or \(\dfrac{125}{27}\) oe | M1 |
\(V_A - \left(\dfrac{3}{5}\right)^3 V_A\;(= 735)\) oe or \(V_A - \dfrac{27}{125}V_A\;(= 735)\) oe or \(\dfrac{98}{125}V_A\;(= 735)\) oe or \(\dfrac{V_A}{V_A - 735} = \dfrac{31^3}{18.6^3}\) oe or \(\left(\dfrac{5}{3}\right)^3 V_B - V_B\;(= 735)\) oe or \(\dfrac{125}{27}V_B - V_B\;(= 735)\) oe or \(\dfrac{98}{27}V_B\;(= 735)\) oe or \(\dfrac{V_B + 735}{V_B} = \dfrac{31^3}{18.6^3}\) oe or \(1 - \left(\dfrac{3}{5}\right)^3\left(= \dfrac{98}{125} = 0.784\right)\) oe or \(\left(\dfrac{5}{3}\right)^3 - 1\left(= \dfrac{98}{27} = 3.62(962\ldots)\right)\) oe or 53 – 33 (= 125 – 27 = 98) | M1 |
\((V_A =)735 \times \dfrac{125}{98}\) oe or \((V_A =)735 \div \dfrac{98}{125}\) oe or \((V_A =)\dfrac{31^3 \times 735}{31^3 - 18.6^3}\) oe or \((V_B =)735 \times \dfrac{27}{98}(= 202.5)\) oe or \((V_B =)735 \div \dfrac{98}{27}(= 202.5)\) or \((V_B =)\dfrac{18.6^3 \times 735}{31^3 - 18.6^3}(= 202.5)\) oe or 735 ÷ 98 × 53 oe or 7.5 × 125 oe or 735 ÷ 98 × 33 (= 202.5) oe or 7.5 × 27 (= 202.5) oe | M1 |
| Working not required, so correct answer scores full marks (unless from obvious incorrect working) Answer: 937.5 | A1 |
| (4) | |
| (4 marks) |
Notes
M1: for correct linear SF or volume SF either as a fraction or ratio. Allow \(\dfrac{5}{3} = 1.6(6\ldots.)\) truncated or rounded
M1: Note: 735 is given in the equation
Allow any letter for \(V_A\) or for \(V_B\)
\(V_A - \left(\dfrac{3}{5}\right)^3 V_A\;(= 735)\) can be written as \(V_A - \dfrac{V_A}{\left(\frac{5}{3}\right)^3}\;(= 735)\) or
\(\left(\dfrac{5}{3}\right)^3 V_B - V_B\;(= 735)\) can be written as \(\dfrac{V_B}{\left(\frac{3}{5}\right)^3} - V_B\;(= 735)\)
M1: for a correct method to find \(V_A\) or \(V_B\)
A1: oe allow 938 from correct working