Question 8 - A-Level Maths

Edexcel C12 2014 January — Question 8 7 marks

Exam Board	Edexcel
Module	C12 (Core Mathematics 1 & 2)
Year	2014
Session	January
Marks	7
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Discriminant and conditions for roots
Type	Find range for no real roots
Difficulty	Moderate -0.3 This is a straightforward discriminant problem requiring students to set b²-4ac < 0 and solve a quadratic inequality. While it involves multiple algebraic steps and careful inequality handling, it's a standard textbook exercise with no novel insight required, making it slightly easier than average.
Spec	1.02d Quadratic functions: graphs and discriminant conditions

8. Find the range of values of $k$ for which the quadratic equation $$k x ^ { 2 } + 8 x + 2 ( k + 7 ) = 0$$ has no real roots.

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Question 8:

Answer	Marks	Guidance
$kx^2 + 8x + 2(k+7) = 0$; uses $b^2 - 4ac$ with $a=k$, $b=8$, $c=2(k+7)$	M1	May omit bracket or make sign slip or lose the 2
$b^2 - 4ac = 64 - 56k - 8k^2$ or $64 = 56k + 8k^2$	A1	Correct three-term quadratic expression for $b^2 - 4ac$
Attempts to solve $k^2 + 7k - 8 = 0$; critical values $k = 1, -8$	dM1, A1cso	Uses factorisation, formula, or completing the square to find two values of $k$
Uses $b^2 - 4ac < 0$ or $b^2 < 4ac$ or $4ac - b^2 > 0$	M1	Stated anywhere
$k^2 + 7k - 8 > 0$ gives $k > 1$ or $k < -8$	M1 A1	Allow $(-\infty, -8)$ or $(1, \infty)$; $k>1, k<-8$ is A1 but $k>1$ and $k\leq -8$ is A0

## Question 8:

| $kx^2 + 8x + 2(k+7) = 0$; uses $b^2 - 4ac$ with $a=k$, $b=8$, $c=2(k+7)$ | M1 | May omit bracket or make sign slip or lose the 2 |
| $b^2 - 4ac = 64 - 56k - 8k^2$ or $64 = 56k + 8k^2$ | A1 | Correct three-term quadratic expression for $b^2 - 4ac$ |
| Attempts to solve $k^2 + 7k - 8 = 0$; critical values $k = 1, -8$ | dM1, A1cso | Uses factorisation, formula, or completing the square to find two values of $k$ |
| Uses $b^2 - 4ac < 0$ or $b^2 < 4ac$ or $4ac - b^2 > 0$ | M1 | Stated anywhere |
| $k^2 + 7k - 8 > 0$ gives $k > 1$ or $k < -8$ | M1 A1 | Allow $(-\infty, -8)$ or $(1, \infty)$; $k>1, k<-8$ is A1 but $k>1$ and $k\leq -8$ is A0 |

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8. Find the range of values of $k$ for which the quadratic equation

$$k x ^ { 2 } + 8 x + 2 ( k + 7 ) = 0$$

has no real roots.\\

\hfill \mbox{\textit{Edexcel C12 2014 Q8 [7]}}

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Q1 4 Q2 7 Q3 7 Q4 7 Q5 7 Q6 6 Q7 5 Q8 7 Q9 7 Q10 9 Q11 8 Q12 11 Q13 14 Q14 12 Q15 14

Answer	Marks	Guidance
\(kx^2 + 8x + 2(k+7) = 0\); uses \(b^2 - 4ac\) with \(a=k\), \(b=8\), \(c=2(k+7)\)	M1	May omit bracket or make sign slip or lose the 2
\(b^2 - 4ac = 64 - 56k - 8k^2\) or \(64 = 56k + 8k^2\)	A1	Correct three-term quadratic expression for \(b^2 - 4ac\)
Attempts to solve \(k^2 + 7k - 8 = 0\); critical values \(k = 1, -8\)	dM1, A1cso	Uses factorisation, formula, or completing the square to find two values of \(k\)
Uses \(b^2 - 4ac < 0\) or \(b^2 < 4ac\) or \(4ac - b^2 > 0\)	M1	Stated anywhere
\(k^2 + 7k - 8 > 0\) gives \(k > 1\) or \(k < -8\)	M1 A1	Allow \((-\infty, -8)\) or \((1, \infty)\); \(k>1, k<-8\) is A1 but \(k>1\) and \(k\leq -8\) is A0