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Date	May 2022	Marks available	3	Reference code	22M.1.SL.TZ2.11
Level	Standard Level	Paper	Paper 1	Time zone	Time zone 2
Command term	Justify and Determine	Question number	11	Adapted from	N/A

Question

Consider the function $f (x) = x^{2} - \frac{3}{x}, x \neq 0$ .

Line $L$ is a tangent to $f (x)$ at the point $(1, - 2)$ .

Find $f' (x)$ .

[2]

Use your answer to part (a) to find the gradient of $L$ .

[2]

Determine the number of lines parallel to $L$ that are tangent to $f (x)$ . Justify your answer.

[3]

Markscheme

$(f' (x) =) 2 x + \frac{3}{x^{2}}$ A1A1

Note: Award A1 for $2 x$ , A1 for $+ \frac{3}{x^{2}}$ OR $= 3 x^{- 2}$

[2 marks]

attempt to substitute $1$ into their part (a) (M1)

$(f' (1) =) 2 (1) + \frac{3}{1^{2}}$

$5$ A1

[2 marks]

EITHER

$5 = 2 x + \frac{3}{x^{2}}$ M1

$x = - 0.686, 1, 2.19 (- 0.686140 \dots, 1, 2.18614 \dots)$ A1

sketch of $y = f' (x)$ with line $y = 5$ M1

three points of intersection marked on this graph A1

(and it can be assumed no further intersections occur outside of this window)

THEN

there are two other tangent lines to $f (x)$ that are parallel to $L$ A1

Note: The final A1 can be awarded provided two solutions other than $x = 1$ are shown OR three points of intersection are marked on the graph.

Award M1A1A1 for an answer of “3 lines” where $L$ is considered to be parallel with itself (given guide definition of parallel lines), but only if working is shown.

[3 marks]

Examiners report

Was reasonably well done, with the stronger candidates able to handle a negative exponent appropriately when finding the derivative. There were a few who confused the notation for derivative with the notation for inverse.

Most knew to substitute $x = 1$ into the derivative to find the gradient at that point, but some also tried to substitute the y-coordinate for $f' (x)$ .

There was a lot of difficulty understanding what approach would help them determine the number of tangents to $f (x)$ that are parallel to L. Several wrote just an answer, which is not adequate when justification is required.

Syllabus sections

Topic 2—Functions » SL 2.1—Equations of straight lines, parallel and perpendicular

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