Higher January 2022 Paper 2R Q20
20 \(y\) is inversely proportional to \(\sqrt{x}\)
\(x\) is directly proportional to \(T^3\)
Given that \(y = 8\) when \(T = 25\)
find the exact value of \(T\) when \(y = 27\)
(4)
| Scheme | Marks |
|---|---|
\(y = \dfrac{k}{\sqrt{x}}\) or \(ky = \dfrac{1}{\sqrt{x}}\) or \(x = pT^3\) or \(y = \dfrac{k}{\sqrt{pT^3}}\) or \(y = \dfrac{c}{\sqrt{T^3}}\) oe or Alternative \(y^2T^3 = n\) oe | M1 |
\(c = 8 \times \sqrt{25^3}\ (= 1000)\) oe or \(n = 8^2 \times 25^3\ (= 1000000)\) oe | M1 |
\(27 = \dfrac{\text{‘}{1000}\text{’}}{\sqrt{T^3}}\) and \(T^3 = \left(\dfrac{\text{‘}{1000}\text{’}}{27}\right)^2\) oe \(27 = \dfrac{\text{‘}{1000}\text{’}}{\sqrt{T^3}}\) and \(T^{\frac{1}{2}} = \left(\dfrac{\text{‘}{1000}\text{’}}{27}\right)^{\frac{1}{3}}\) oe or \(T^3 = \dfrac{\text{‘}{1000000}\text{’}}{27^2}\) oe | M1 |
| \(\dfrac{100}{9}\) | A1 |
| (4) | |
| (4 marks) |
Notes
M1: Constant of proportionality must be a symbol such as \(k\) or \(p\) or \(c\) or \(n\)
\(k \neq 1\), \(p \neq 1\) and \(c \neq 1\) and \(n \neq 1\)
M1: dep M1 for rearranging for \(c\) or \(n\) with \((y =)\ 8\) and \((T =)\ 25\) substituted correctly into their equation
M1: for substitution of \(y\) and a correct rearrangement for \(T^3\) or \(T^{\frac{1}{2}}\) or \(T\).
A1: oe eg \(11\dfrac{1}{9}\) or \(11.\dot{1}\) or 11.111(…)