| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2014 |
| Session | June |
| Marks | 3 |
| Topic | Discriminant and conditions for roots |
| Type | Find range for two distinct roots |
| Difficulty | Moderate -0.8 This is a straightforward discriminant question requiring only recall of the formula Δ = b² - 4ac and the condition Δ > 0 for two distinct roots. It involves simple algebraic manipulation (k² - 16 > 0) and solving a basic quadratic inequality, making it easier than average with no problem-solving insight required. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions |
| Answer | Marks | Guidance |
|---|---|---|
| A1 [3] Allow BOD on 'and' not 'or'. \( | k | > 4\) gets A1A1 |
(i) $\Delta = b^2 - 4ac$
$= k^2 - 16$
M1
A1 [2] Simplify to this
(ii) $k^2 - 16 > 0$
M1 Must be $>$ seen, or implied by answer. Allow incorrect answer from (i), as long as $b^2 - 4ac$ attempted
$k > 4$
$k < -4$
A1 A1A0 for $-4 > k > 4$ or $k > \pm 4$
A1 [3] Allow BOD on 'and' not 'or'. $|k| > 4$ gets A1A1
Attempting to solve $f'(x) > 0$ can get M1A1A1 as above2 Let $\mathrm { f } ( x ) = x ^ { 2 } + k x + 4$, where $k$ is a constant.\\
(i) Find an expression for the discriminant of f in terms of $k$.\\
(ii) Hence find the range of values of $k$ for which the equation $\mathrm { f } ( x ) = 0$ has two distinct real roots.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2014 Q2 [3]}}