OCR C1 2016 June — Question 9 7 marks

Exam BoardOCR
ModuleC1 (Core Mathematics 1)
Year2016
SessionJune
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicInequalities
TypeQuadratic equation real roots
DifficultyStandard +0.3 This is a standard discriminant problem requiring rearrangement to standard form, then applying b²-4ac > 0 for two distinct real roots. It's slightly above average difficulty due to the parameter k and requiring careful algebraic manipulation, but follows a well-practiced technique taught in C1.
Spec1.02d Quadratic functions: graphs and discriminant conditions1.02f Solve quadratic equations: including in a function of unknown
9 Find the set of values of \(k\) for which the equation \(x ^ { 2 } + 2 x + 11 = k ( 2 x - 1 )\) has two distinct real roots.
Question 9:
AnswerMarks Guidance
AnswerMarks Guidance
\(x^2 + (2-2k)x + 11 + k = 0\)M1* Attempt to rearrange to a three-term quadratic. Each M depends on previous M
\((2-2k)^2 - 4(11+k)\)M1dep* Uses \(b^2 - 4ac\), involving \(k\) and not involving \(x\)
\(4k^2 - 12k - 40 > 0\), \(k^2 - 3k - 10 > 0\)A1 Correct simplified inequality obtained www
\((k-5)(k+2)\)M1dep* Correct method to find roots of 3-term quadratic; 5 and \(-2\) seen as roots
A1\(b^2 - 4ac > 0\) and chooses "outside region". \(-2 > k > 5\) scores M1A0
\(k < -2,\ k > 5\)M1dep*, A1 Fully correct, strict inequalities. Allow "\(k < -2\) or \(k > 5\)" for A1. Do not allow "\(k < -2\) and \(k > 5\)" 
## Question 9:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 + (2-2k)x + 11 + k = 0$ | M1* | Attempt to rearrange to a three-term quadratic. Each M depends on previous M |
| $(2-2k)^2 - 4(11+k)$ | M1dep* | Uses $b^2 - 4ac$, involving $k$ and not involving $x$ |
| $4k^2 - 12k - 40 > 0$, $k^2 - 3k - 10 > 0$ | A1 | Correct simplified inequality obtained www |
| $(k-5)(k+2)$ | M1dep* | Correct method to find roots of 3-term quadratic; 5 and $-2$ seen as roots |
| | A1 | $b^2 - 4ac > 0$ and chooses "outside region". $-2 > k > 5$ scores M1A0 |
| $k < -2,\ k > 5$ | M1dep*, A1 | Fully correct, strict inequalities. Allow "$k < -2$ or $k > 5$" for A1. Do not allow "$k < -2$ and $k > 5$" |

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9 Find the set of values of $k$ for which the equation $x ^ { 2 } + 2 x + 11 = k ( 2 x - 1 )$ has two distinct real roots.

\hfill \mbox{\textit{OCR C1 2016 Q9 [7]}}
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