Higher November 2022 Unit 1 Q11
11.
(a) Given that \(y\) is directly proportional to \(x^3\) and that \(y = 108\) when \(x = 3\),
(i) find an expression for \(y\) in terms of \(x\). [3]
(ii) Use the expression you found in part (i) to complete the following table. [2]
| \(x\) | 3 | 5 | |
| \(y\) | 108 | 4000 |
(b) It is known that \(e\) is inversely proportional to \(f\).
Describe what happens to \(e\) when \(f\) is doubled. [1]
Describe what happens to \(e\) when \(f\) is doubled. [1]
(a)(i)
| Scheme | Marks | Comments |
|---|---|---|
| \(y \;\alpha\; x^3\) OR \(y = kx^3\) | B1 | Allow \(y \;\alpha\; kx^3\) |
| \(108 = k \times 3^3\) OR \(k = 4\) | M1 | M1 implies B1. F.T. from \(y \;\alpha\; x^n\) with \(n > 1\) or n = –3 Use of n = –3 leads to k = 2916 Use of n = 2 leads to k = 12 |
| \((y =)\; 4x^3\) | A1 | May be seen in part (ii) |
(a)(ii)
| Scheme | Marks | Comments | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| B2 | B1 for each correct value. Check working space if table is empty. F.T. from ‘their k’, provided M1 awarded (accept answer left as a root) (No FT for \(y = (1)x^3\)) F.T. from \(y \;\alpha\; x^n\) with \(n > 1\) or n = –3 Use of n = –3 leads to answers of 23.328 and 0.9 Use of n = 2 leads to answers of 300 and \(\sqrt{(1000/3)}\) |
| Scheme | Marks | Comments |
|---|---|---|
| Valid statement e.g. \(e\) is halved; \(e\) is divided by 2 | E1 |