Edexcel P1 2020 January — Question 8 6 marks

Exam BoardEdexcel
ModuleP1 (Pure Mathematics 1)
Year2020
SessionJanuary
Marks6
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Mark schemeDownload PDF ↗
TopicDiscriminant and conditions for roots
TypeRange of k, line not intersecting curve
DifficultyModerate -0.3 This is a standard discriminant problem requiring students to set the equations equal, rearrange to standard form, and apply b²-4ac < 0 for no intersection. While it involves algebraic manipulation and solving a quadratic inequality in k, it follows a well-practiced procedure with no novel insight required, making it slightly easier than average.
Spec1.02d Quadratic functions: graphs and discriminant conditions1.02f Solve quadratic equations: including in a function of unknown
8. The straight line \(l\) has equation \(y = k ( 2 x - 1 )\), where \(k\) is a constant. The curve \(C\) has equation \(y = x ^ { 2 } + 2 x + 11\) Find the set of values of \(k\) for which \(l\) does not cross or touch \(C\).
(6)
Question 8:
AnswerMarks Guidance
AnswerMarks Guidance
Equates \(y = k(2x-1)\) and \(y = x^2 + 2x + 11 \Rightarrow k(2x-1) = x^2 + 2x + 11\)M1 Attempts to equate the two expressions
\(\Rightarrow x^2 + (2-2k)x + 11 + k \ (= 0)\)A1 Correct quadratic with terms collected. Condone missing "\(= 0\)". May be implied by values of \(a\), \(b\), \(c\) used in later work.
Attempts \(b^2 - 4ac \ldots 0 \Rightarrow (2-2k)^2 - 4(11+k) \ldots 0\) and proceeds to critical valuesM1 Where \(a = \pm 1\), \(b = \pm 2 \pm 2k\), \(c = \pm 11 \pm k\). Condone arithmetical slips. FYI \(b^2 - 4ac = 4k^2 - 12k - 40\)
Critical values \((k =)\ 5, -2\)A1 Must have come from correct working
No roots so \(b^2 - 4ac < 0\), choose inside regionM1 Finds inside region for their critical values. May award for \(-2 \leq k \leq 5\) or \(-2 \leq k < 5\)
\(-2 < k < 5\)A1 oe such as \(k \in (-2, 5)\) cso. Note \(-2 < x < 5\) is A0 
## Question 8:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Equates $y = k(2x-1)$ and $y = x^2 + 2x + 11 \Rightarrow k(2x-1) = x^2 + 2x + 11$ | M1 | Attempts to equate the two expressions |
| $\Rightarrow x^2 + (2-2k)x + 11 + k \ (= 0)$ | A1 | Correct quadratic with terms collected. Condone missing "$= 0$". May be implied by values of $a$, $b$, $c$ used in later work. |
| Attempts $b^2 - 4ac \ldots 0 \Rightarrow (2-2k)^2 - 4(11+k) \ldots 0$ and proceeds to critical values | M1 | Where $a = \pm 1$, $b = \pm 2 \pm 2k$, $c = \pm 11 \pm k$. Condone arithmetical slips. FYI $b^2 - 4ac = 4k^2 - 12k - 40$ |
| Critical values $(k =)\ 5, -2$ | A1 | **Must have come from correct working** |
| No roots so $b^2 - 4ac < 0$, choose inside region | M1 | Finds inside region for their critical values. May award for $-2 \leq k \leq 5$ or $-2 \leq k < 5$ |
| $-2 < k < 5$ | A1 | oe such as $k \in (-2, 5)$ cso. Note $-2 < x < 5$ is A0 |

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8. The straight line $l$ has equation $y = k ( 2 x - 1 )$, where $k$ is a constant.

The curve $C$ has equation $y = x ^ { 2 } + 2 x + 11$\\
Find the set of values of $k$ for which $l$ does not cross or touch $C$.\\
(6)\\

\hfill \mbox{\textit{Edexcel P1 2020 Q8 [6]}}
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