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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 fx=x2-3x, x0.

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

Find f'x.

[2]
a.

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

[2]
b.

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

[3]
c.

Markscheme

f'x= 2x+3x2           A1A1


Note:
Award A1 for 2x, A1 for +3x2  OR  =3x-2
 

[2 marks]

a.

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

f'1= 21+312

5           A1
 

[2 marks]

b.

EITHER

5=2x+3x2          M1

x=-0.686, 1, 2.19   -0.686140, 1, 2.18614          A1


OR

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 fx 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]

c.

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.

a.

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).

b.

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.

c.

Syllabus sections

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