| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Show discriminant inequality, then solve |
| Difficulty | Moderate -0.3 This is a standard discriminant question requiring students to apply b²-4ac ≥ 0 for real roots, expand and simplify to obtain the given inequality, then solve a quadratic inequality. While it involves algebraic manipulation across multiple steps, it follows a completely routine procedure taught explicitly in C1 with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \((k - 2)^2 - 4 \times (2k - 7)(k - 3)\) | M1 | discriminant − condone one slip −condone omission of brackets |
| \(k^2 - 4k + 4 - 4(2k^2 - 6k - 7k + 21)\) | A1 | |
| "their" \(-7k^2 + 48k - 80 \geq 0\) | B1 | real roots condition; f(k) ⩾ 0 must appear before final line |
| \(7k^2 - 48k + 80 \leq 0\) | A1cso | AG (all working correct with no missing brackets etc); 4 marks total |
| (b) \(7k^2 - 48k + 80 = (7k - 20)(k - 4)\) | M1 | correct factors |
| A1 | or roots unsimplified \(\frac{48 \pm \sqrt{64}}{14}\); accept \(\frac{56}{14}\), \(\frac{40}{14}\) etc here | |
| critical values are 4 and \(\frac{20}{7}\) | ||
| M1 | sketch or sign diagram including values | |
| \(\begin{array}{ccccc} + & - & + \\ \frac{2 \cdot 0'}{7} & & 4 \end{array}\) | ||
| \(\frac{20}{7} \leq k \leq 4\) | A1cao | fractions must be simplified here; 4 marks total |
**(a)** $(k - 2)^2 - 4 \times (2k - 7)(k - 3)$ | M1 | discriminant − condone one slip −condone omission of brackets
$k^2 - 4k + 4 - 4(2k^2 - 6k - 7k + 21)$ | A1 |
"their" $-7k^2 + 48k - 80 \geq 0$ | B1 | real roots condition; f(k) ⩾ 0 must appear before final line
$7k^2 - 48k + 80 \leq 0$ | A1cso | AG (all working correct with no missing brackets etc); 4 marks total
**(b)** $7k^2 - 48k + 80 = (7k - 20)(k - 4)$ | M1 | correct factors
| | | A1 | or roots unsimplified $\frac{48 \pm \sqrt{64}}{14}$; accept $\frac{56}{14}$, $\frac{40}{14}$ etc here
critical values are 4 and $\frac{20}{7}$ | |
| | | M1 | sketch or sign diagram including values
| | | | $\begin{array}{ccccc} + & - & + \\ \frac{2 \cdot 0'}{7} & & 4 \end{array}$
$\frac{20}{7} \leq k \leq 4$ | A1cao | fractions must be simplified here; 4 marks total
**Take their final line as their answer**
**Total: 8 marks**
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## GRAND TOTAL: 75 marks7 The quadratic equation
$$( 2 k - 7 ) x ^ { 2 } - ( k - 2 ) x + ( k - 3 ) = 0$$
has real roots.
\begin{enumerate}[label=(\alph*)]
\item Show that $7 k ^ { 2 } - 48 k + 80 \leqslant 0$.
\item Find the possible values of $k$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2013 Q7 [8]}}