Higher June 2023 Paper 2 Q17
17
(a) Show that the equation \(x^4 - x^2 - 5 = 0\) can be written in the form \(x = \sqrt[4]{x^2 + 5}\) (1)
(b) Starting with \(x_0 = 1.5\)
use the iteration formula \(x_{n+1} = \sqrt[4]{x_n^2 + 5}\) three times to find an estimate for a solution of \(x^4 - x^2 - 5 = 0\) (3)
use the iteration formula \(x_{n+1} = \sqrt[4]{x_n^2 + 5}\) three times to find an estimate for a solution of \(x^4 - x^2 - 5 = 0\) (3)
| Answer | Mark | Mark scheme |
|---|---|---|
| Shows re- arrangement | C1 | for showing rearrangement, must see \(x^4 = x^2 + 5\) leading to \(x = \sqrt[4]{x^2 + 5}\) |
Additional guidance
Can work backwards
| Answer | Mark | Mark scheme |
|---|---|---|
| 1.669763088 | M1 | for substitution to find \(x_1\) eg \(\sqrt[4]{1.5^2 + 5}\ (= 1.64\ldots)\) |
| M1 | for substitution to find \(x_2\) eg \(\sqrt[4]{{\text{``}1.64\ldots\text{''}}^2 + 5}\ (= 1.66\ldots)\) | |
| A1 | for answer in the range 1.6697 to 1.6698 or 1.67(0) |
Additional guidance
If a correct value is given and then rounded or rounded incorrectly award full marks