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Date	November 2020	Marks available	6	Reference code	20N.1.SL.TZ0.S_5
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Find	Question number	S_5	Adapted from	N/A

Question

Let $f (x) = - x^{2} + 4 x + 5$ and $g (x) = - f (x) + k$ .

Find the values of $k$ so that $g (x) = 0$ has no real roots.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

METHOD 1 – (discriminant)

correct expression for $g$ (A1)

eg $- (- x^{2} + 4 x + 5) + k, x^{2} - 4 x - 5 + k = 0$

evidence of discriminant (M1)

eg $b^{2} - 4 a c, Δ$

correct substitution into discriminant of $g$ (A1)

eg ${(- 4)}^{2} - 4 (1) (- 5 + k), 16 - 4 (k - 5)$

recognizing discriminant is negative (M1)

eg $Δ < 0, {(- 4)}^{2} - 4 (1) (- 5 + k) < 0, 16 < 4 (k - 5), 16 - 4 (- 1) (5) < 0$

correct working (must be correct inequality) (A1)

eg $- 4 k < - 36, k - 5 > 4, 16 + 20 - 4 k < 0$

$k > 9$ A1 N3

METHOD 2 – (transformation of vertex of $f$ )

valid approach for finding $f (x)$ vertex (M1)

eg $- \frac{b}{2 a} = 2, f' (x) = 0$

correct vertex of $f (x)$ (A1)

eg $(2, 9)$

correct vertex of $- f (x)$ (A1)

eg $(2, - 9)$

correct vertex of $g (x)$ (A1)

eg $(\begin{matrix} 2 \\ - 9 \end{matrix}) + (\begin{matrix} 0 \\ k \end{matrix}), (2, - 9 + k)$

recognizing when vertex is above $x$ -axis (M1)

eg $- 9 + k > 0$ , sketch

$k > 9$ A1 N3

METHOD 3 – (transformation of $f$ )

recognizing vertical reflection of $f (x)$ (M1)

eg $- f (x), x^{2} - 4 x - 5$ , sketch

correct expression for $g (x)$ (A1)

eg $x^{2} - 4 x - 5 + k$

valid approach for finding vertex of $g (x)$ (M1)

eg $- \frac{b}{2 a} = 2, g' (x) = 0$

correct $y$ coordinate of vertex of $g (x)$ (A1)

eg $y = - 9 + k, (2, - 9 + k)$

recognizing when vertex is above $x$ -axis (M1)

eg $- 9 + k > 0$ , sketch

$k > 9$ A1 N3

[6 marks]

Examiners report

[N/A]

Syllabus sections

Topic 2—Functions » SL 2.7—Solutions of quadratic equations and inequalities, discriminant and nature of roots

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