Moderate -0.3 This is a straightforward discriminant problem requiring students to apply b²-4ac < 0 for no real roots, then solve a simple linear inequality. It's slightly easier than average because it's a direct application of a standard technique with minimal algebraic manipulation required.
- 1st M: Could also be, for example, comparing or equating \(b^2\) and \(4ac\). Must be considering the given quadratic equation. There must not be x terms in the expression, but there must be a k term.
- 1st A: Correct expression (need not be simplified) and correct inequality sign. Allow also \(-3^2 - 4 \times 2 \times -(k+1) < 0\).
- 2nd M: Condone sign or bracketing mistakes in manipulation. Not dependent on 1st M, but should not be given for irrelevant work. M0 M1 could be scored.
- Special cases:
1. Where there are x terms in the discriminant expression, but then division by \(x^2\) gives an inequality/equation in k: (This could score M0 A0 M1 A1)
2. Use of ≤ instead of < loses one A mark only, at first occurrence, so an otherwise correct solution leading to \(k \leq -\frac{17}{8}\) scores M1 A0 M1 A1.
- N.B. Use of \(b = 3\) instead of \(b = -3\) implies no A marks.
Use of $b^2 - 4ac$, perhaps implicit (e.g. in quadratic formula) | M1 |
$(-3)^2 - 4 \times 2 \times -(k+1) < 0$ | $(9 + 8(k+1) < 0)$ | A1 |
$8k < -17$ (Manipulate to get $pk < q$, or $pk > q$, or $pk = q$) | M1 |
$k < -\frac{17}{8}$ (Or equiv.: $k < -2\frac{1}{8}$ or $k < -2.125$) | A1cso | (4 marks)
**Total: 4 marks**
**Guidance:**
- 1st M: Could also be, for example, comparing or equating $b^2$ and $4ac$. Must be considering the given quadratic equation. There must not be x terms in the expression, but there must be a k term.
- 1st A: Correct expression (need not be simplified) and correct inequality sign. Allow also $-3^2 - 4 \times 2 \times -(k+1) < 0$.
- 2nd M: Condone sign or bracketing mistakes in manipulation. Not dependent on 1st M, but should not be given for irrelevant work. M0 M1 could be scored.
- Special cases:
1. Where there are x terms in the discriminant expression, but then division by $x^2$ gives an inequality/equation in k: (This could score M0 A0 M1 A1)
2. Use of ≤ instead of < loses one A mark only, at first occurrence, so an otherwise correct solution leading to $k \leq -\frac{17}{8}$ scores M1 A0 M1 A1.
- **N.B.** Use of $b = 3$ instead of $b = -3$ implies no A marks.
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5. The equation $2 x ^ { 2 } - 3 x - ( k + 1 ) = 0$, where $k$ is a constant, has no real roots.
Find the set of possible values of $k$.\\
\hfill \mbox{\textit{Edexcel C1 2007 Q5 [4]}}