June 2023 Paper 1 Q7
7
(a) Given that \(n\) is a positive integer, express\[\frac{7}{3 + 5\sqrt{n}} - \frac{7}{5\sqrt{n} - 3}\]as a single fraction not involving surds. [3 marks]
(b) Hence, deduce that\[\frac{7}{3 + 5\sqrt{n}} - \frac{7}{5\sqrt{n} - 3}\]is a rational number for all positive integer values of \(n\) [1 mark]
| Scheme | Marks | AO |
|---|---|---|
| Multiplies both the numerator and denominator of at least one of the given fractions by an appropriate conjugate. Or Obtains a common denominator with numerators which simplify to \(35\sqrt{n} - 21 - \left(21 + 35\sqrt{n}\right)\) or a single fraction with numerator \(35\sqrt{n} - 21 - \left(21 + 35\sqrt{n}\right)\) | M1 | 1.1a |
| Obtains a correct single unsimplified fraction. | A1 | 1.1b |
| Obtains correct simplified fraction \(-\dfrac{42}{25n - 9}\) OE | A1 | 2.1 |
| (3) |
Typical solution
\[\frac{7}{3 + 5\sqrt{n}} - \frac{7}{5\sqrt{n} - 3}\]\[= \frac{7\left(5\sqrt{n} - 3\right)}{\left(3 + 5\sqrt{n}\right)\left(5\sqrt{n} - 3\right)} - \frac{7\left(3 + 5\sqrt{n}\right)}{\left(3 + 5\sqrt{n}\right)\left(5\sqrt{n} - 3\right)}\]\[= \frac{35\sqrt{n} - 21 - \left(21 + 35\sqrt{n}\right)}{\left(3 + 5\sqrt{n}\right)\left(5\sqrt{n} - 3\right)}\]\[= -\frac{42}{25n - 9}\]| Scheme | Marks | AO |
|---|---|---|
| Explains that the numerator and denominator are both integers, or rational, and concludes it is rational | E1F | 2.4 |
| (1) | ||
| (4 marks) |
Typical solution
Since 42 and \(25n\)-9 are both integers the expression is rational