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Date	November 2018	Marks available	4	Reference code	18N.1.SL.TZ0.S_10
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 0
Command term	Find and Hence	Question number	S_10	Adapted from	N/A

Question

Let $f\left( x \right) = {x^3} - 2{x^2} + ax + 6$ . Part of the graph of $f$ is shown in the following diagram.

The graph of $f$ crosses the $y$ -axis at the point P. The line L is tangent to the graph of $f$ at P.

Find the coordinates of P.

[2]

Find $f'\left( x \right)$ .

[2]

b.i.

Hence, find the equation of L in terms of $a$ .

[4]

b.ii.

The graph of $f$ has a local minimum at the point Q. The line L passes through Q.

Find the value of $a$ .

[8]

Markscheme

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

valid approach (M1)

eg $f\left( 0 \right)$ , ${0^3} - 2{\left( 0 \right)^2} + a\left( 0 \right) + 6$ , $f\left( 0 \right) = 6$ , $\left( {0,\,\,y} \right)$

(0, 6) (accept $x$ = 0 and $y$ = 6) A1 N2

[2 marks]

$f' = 3{x^2} - 4x + a$ A2 N2

[2 marks]

b.i.

valid approach (M1)

eg $f'\left( 0 \right)$

correct working (A1)

eg $3{\left( 0 \right)^2} - 4\left( 0 \right) + a$ , slope = $a$ , $f'\left( 0 \right) = a$

attempt to substitute gradient and coordinates into linear equation (M1)

eg $y - 6 = a\left( {x - 0} \right)$ , $y - 0 = a\left( {x - 6} \right)$ , $6 = a\left( 0 \right) + c$ , L $= ax + 6$

correct equation A1 N3

eg $y = ax + 6$ , $y - 6 = ax$ , $y - 6 = a\left( {x - 0} \right)$

[4 marks]

b.ii.

valid approach to find intersection (M1)

eg $f\left( x \right) = L$

correct equation (A1)

eg ${x^3} - 2{x^2} + ax + 6 = ax + 6$

correct working (A1)

eg ${x^3} - 2{x^2} = 0$ , ${x^2}(x - 2) = 0$

$x = 2$ at Q (A1)

valid approach to find minimum (M1)

eg $f'\left( x \right) = 0$

correct equation (A1)

eg $3{x^2} - 4x + a = 0$

substitution of their value of $x$ at Q into their $f'\left( x \right) = 0$ equation (M1)

eg $3{\left( 2 \right)^2} - 4\left( 2 \right) + a = 0$ , $12 - 8 + a = 0$

$a$ = −4 A1 N0

[8 marks]

Examiners report

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

Syllabus sections

Topic 4—Statistics and probability » SL 4.1—Concepts, reliability and sampling techniques

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