Moderate -0.3 This is a straightforward discriminant question requiring students to set b²-4ac < 0 and solve a quadratic inequality. While it involves multiple steps (forming inequality, rearranging, factorising/solving), it's a standard textbook exercise testing routine application of the discriminant condition with no novel problem-solving required. Slightly easier than average due to its predictable structure.
Attempts \(b^2 - 4ac\) with \(a=2, b=5p, c=p\). Condone \(5p^2 - 8p\) for this mark. Also allow attempt at \(b^2...4ac\) with any equality or inequality
\(= 25p^2 - 8p\)
A1
Correct and simplified \(b^2 - 4ac = 25p^2 - 8p\), without brackets
Sets \(b^2 - 4ac...0\) with their values, proceeds to find value(s) for \(p\)
Chooses inside region for their two values
M1
Chooses inside region for their two roots. Can be awarded for \(0 \leqslant p \leqslant \frac{8}{25}\)
\(0 < p < \frac{8}{25}\)
A1
Or equivalent: \((0, 0.32)\), \(p \in (0, 0.32)\), \(\left]0, \frac{8}{25}\right[\), \(p > 0 \text{ ... } p < \frac{8}{25}\) where ... is comma, and, or nothing
## Question 7:
| Working/Answer | Mark | Guidance |
|---|---|---|
| Attempts $b^2 - 4ac = (5p)^2 - 8p$ | M1 | Attempts $b^2 - 4ac$ with $a=2, b=5p, c=p$. Condone $5p^2 - 8p$ for this mark. Also allow attempt at $b^2...4ac$ with any equality or inequality |
| $= 25p^2 - 8p$ | A1 | Correct and simplified $b^2 - 4ac = 25p^2 - 8p$, without brackets |
| Sets $b^2 - 4ac...0 \Rightarrow 25p^2 - 8p...0 \Rightarrow$ values for $p$ | M1 | Sets $b^2 - 4ac...0$ with their values, proceeds to find value(s) for $p$ |
| Chooses inside region for their two values | M1 | Chooses inside region for their two roots. Can be awarded for $0 \leqslant p \leqslant \frac{8}{25}$ |
| $0 < p < \frac{8}{25}$ | A1 | Or equivalent: $(0, 0.32)$, $p \in (0, 0.32)$, $\left]0, \frac{8}{25}\right[$, $p > 0 \text{ ... } p < \frac{8}{25}$ where ... is comma, and, or nothing |
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7. The equation $2 x ^ { 2 } + 5 p x + p = 0$, where $p$ is a constant, has no real roots.
Find the set of possible values for $p$.\\
\hfill \mbox{\textit{Edexcel C12 2019 Q7 [5]}}