Moderate -0.5 This is a straightforward discriminant question requiring students to set b²-4ac < 0 and solve a simple quadratic inequality. While it tests understanding of the discriminant condition for no real roots, it's a standard textbook exercise with routine algebraic manipulation, making it slightly easier than average.
allow \(=, >, <\) etc instead of \(<\); or M1 for \(b^2 - 4ac\) soi (may be in formula) or for attempt at completing square
allow M2 for \(2k^2 < 20, 2k^2 - 20 = 0\) etc but M1 only for just \(2k^2 - 20\); ignore rest of quadratic formula; ignore \(\sqrt{b^2 - 4ac} < 0\) seen if \(b^2 - 4ac < 0\) then used, otherwise just M1 for \(\sqrt{b^2 - 4ac} < 0\)
\(-\sqrt{5} < k < \sqrt{5}\)
A2
may be two separate inequalities or A1 for one 'end' correct or B1 for 'endpoint' \(= \sqrt{5}\)
[4]
$4k^2 - 4 \times 1 \times 5$ or $k^2 - 5 [< 0]$ oe or $[(x + k)^2 +] 5 - k^2 [> 0]$ oe | M2 | allow $=, >, <$ etc instead of $<$; or M1 for $b^2 - 4ac$ soi (may be in formula) or for attempt at completing square
| allow M2 for $2k^2 < 20, 2k^2 - 20 = 0$ etc but M1 only for just $2k^2 - 20$; ignore rest of quadratic formula; ignore $\sqrt{b^2 - 4ac} < 0$ seen if $b^2 - 4ac < 0$ then used, otherwise just M1 for $\sqrt{b^2 - 4ac} < 0$
$-\sqrt{5} < k < \sqrt{5}$ | A2 | may be two separate inequalities or A1 for one 'end' correct or B1 for 'endpoint' $= \sqrt{5}$ | allow SC1 for $-\sqrt{10} < k < \sqrt{10}$ following at least M1 for $2k^2 - 20$ oe
| [4] |
Find the set of values of $k$ for which the graph of $y = x^2 + 2kx + 5$ does not intersect the $x$-axis. [4]
\hfill \mbox{\textit{OCR MEI C1 2012 Q7 [4]}}