Standard +0.3 This is a standard discriminant problem requiring students to set b²-4ac < 0 and solve the resulting quadratic inequality. While it involves multiple steps (forming the discriminant, simplifying, factorizing, and determining the inequality solution), these are all routine C1 techniques with no novel insight required, making it slightly above average difficulty.
Attempts \(b^2 - 4ac\) or an equation or inequality involving \(b^2\) and \(4ac\). Must involve \(k^2\) in first term (but no \(x\) anywhere). If \(b^2 - 4ac\) not stated, must be clear attempt.
Must be working with the discriminant explicitly and not only as part of the quadratic formula. Allow \(\sqrt{b^2 - 4ac}\) for first M1 A1
A1
Correct discriminant, simplified to 3 terms
M1
States discriminant \(< 0\) or \(b^2 < 4ac\).
Can be awarded at any stage. Doesn't need first M1. No square root here.
DM1
Correct method to find roots of a three term quadratic
A1
Both values of \(k\) correct
M1
Chooses "inside region" of inequality
Allow correct region for their inequality. Do not allow "\(k < -\frac{1}{9}\) or \(k > -1\)"; Do not allow "\(k < -\frac{1}{9}\) or \(k > -1\)".
A1
Allow "\(k < -\frac{1}{9}\) and \(k > -1\)" etc. must be strict inequalities for A mark
[7]
$(3k - 1)^2 - 4 \times k \times k - 4 = 9k^2 + 10k + 1$, $9k^2 + 10k + 1 < 0$, $(9k + 1)(k + 1) < 0$, $-1, -\frac{1}{9}$, $-1 < k < -\frac{1}{9}$
| *M1 | Attempts $b^2 - 4ac$ or an equation or inequality involving $b^2$ and $4ac$. Must involve $k^2$ in first term (but no $x$ anywhere). If $b^2 - 4ac$ not stated, must be clear attempt. | Must be working with the discriminant explicitly and not only as part of the quadratic formula. Allow $\sqrt{b^2 - 4ac}$ for first M1 A1 |
| A1 | Correct discriminant, simplified to 3 terms | |
| M1 | States discriminant $< 0$ or $b^2 < 4ac$. | Can be awarded at any stage. Doesn't need first M1. No square root here. |
| DM1 | Correct method to find roots of a three term quadratic | |
| A1 | Both values of $k$ correct | |
| M1 | Chooses "inside region" of inequality | Allow correct region for their inequality. Do not allow "$k < -\frac{1}{9}$ or $k > -1$"; Do not allow "$k < -\frac{1}{9}$ or $k > -1$". |
| A1 | Allow "$k < -\frac{1}{9}$ and $k > -1$" etc. must be strict inequalities for A mark | |
| [7] | | |
The quadratic equation $kx^2 + (3k - 1)x - 4 = 0$ has no real roots. Find the set of possible values of $k$. [7]
\hfill \mbox{\textit{OCR C1 2013 Q8 [7]}}