Moderate -0.5 This is a straightforward application of the discriminant condition b²-4ac=0 for equal roots. Students need to substitute into the formula, solve a simple linear equation (9p²-4p=0), and factor out p. While it requires knowing the discriminant condition, it's a standard C1 exercise with minimal algebraic manipulation, making it slightly easier than average.
Attempt to substitute into \(b^2-4ac\) or \(b^2=4ac\) with \(b\) or \(c\) correct; condone \(x\)'s in one term only
\((3p)^2 - 4p = 0\)
A1
Any correct equation: \((3p)^2 - 4\times1\times p = 0\) or better
Attempt to solve for \(p\), e.g. \(p(9p-4)=0\), must potentially lead to \(p=k\), \(k\neq0\)
M1
Attempt to factorize or solve; method must lead to \(p=\frac{4}{9}\)
\(p = \frac{4}{9}\) (ignore \(p=0\) if seen)
A1cso
ALT: Comparing coefficients: M1 for \((x+\alpha)^2 = x^2+\alpha^2+2\alpha x\), A1 for \(3p=2\sqrt{p}\), M1 for \(\sqrt{p}=\frac{2}{3}\)
# Question 6:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $b^2 - 4ac$ attempted in terms of $p$ | M1 | Attempt to substitute into $b^2-4ac$ or $b^2=4ac$ with $b$ or $c$ correct; condone $x$'s in one term only |
| $(3p)^2 - 4p = 0$ | A1 | Any correct equation: $(3p)^2 - 4\times1\times p = 0$ or better |
| Attempt to solve for $p$, e.g. $p(9p-4)=0$, must potentially lead to $p=k$, $k\neq0$ | M1 | Attempt to factorize or solve; method must lead to $p=\frac{4}{9}$ |
| $p = \frac{4}{9}$ (ignore $p=0$ if seen) | A1cso | ALT: Comparing coefficients: M1 for $(x+\alpha)^2 = x^2+\alpha^2+2\alpha x$, A1 for $3p=2\sqrt{p}$, M1 for $\sqrt{p}=\frac{2}{3}$ |
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6. The equation $x ^ { 2 } + 3 p x + p = 0$, where $p$ is a non-zero constant, has equal roots.
Find the value of $p$.\\
\hfill \mbox{\textit{Edexcel C1 2009 Q6 [4]}}