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Date May 2021 Marks available 1 Reference code 21M.1.SL.TZ1.12
Level Standard Level Paper Paper 1 Time zone Time zone 1
Command term Hence or otherwise and Find Question number 12 Adapted from N/A

Question

Ellis designs a gift box. The top of the gift box is in the shape of a right-angled triangle GIK.

A rectangular section HIJL is inscribed inside this triangle. The lengths of GH, JK, HL, and LJ are pcm, qcm, 8cm and 6cm respectively.

The area of the top of the gift box is Acm2.

Ellis wishes to find the value of q that will minimize the area of the top of the gift box.

Find A in terms of p and q.

[1]
a.i.

Show that A=192q+3q+48.

[3]
a.ii.

Find dAdq.

[2]
b.

Write down an equation Ellis could solve to find this value of q.

[1]
c.i.

Hence, or otherwise, find this value of q.

[1]
c.ii.

Markscheme

A=12×6×q+12×8×p+48  OR  A=12p+6q+8  OR  A=3q+4p+48            A1

 

[1 mark]

a.i.

valid attempt to link p and q, using tangents, similar triangles or other method        (M1)

eg.  tanθ=8p and tanθ=q6  OR  tanθ=p8 and tanθ=6q  OR  8p=q6

correct equation linking p and q            A1

eg.  pq=48  OR  p=48q  OR  q=48p

substitute p=48q into a correct area expression        M1

eg.  A=12×6×q+12×8×48q+48  OR  A=1248q+6q+8

A=3q+192q+48         AG

 

Note: The AG line must be seen with no incorrect, intermediate working, for the final M1 to be awarded.

 

[3 marks]

a.ii.

-192q2+3            A1A1


Note:
Award A1 for -192q2, A1 for 3. Award A1A0 if extra terms are seen.


[2 marks]

b.

-192q2+3=0            A1


[1 mark]

c.i.

q=8cm            A1


[1 mark]

c.ii.

Examiners report

[N/A]
a.i.
[N/A]
a.ii.
[N/A]
b.
[N/A]
c.i.
[N/A]
c.ii.

Syllabus sections

Topic 5—Calculus » AHL 5.17—Phase portrait
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