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Date May 2019 Marks available 5 Reference code 19M.2.SL.TZ1.S_9
Level Standard Level Paper Paper 2 Time zone Time zone 1
Command term Find Question number S_9 Adapted from N/A

Question

Let f(x)=16xf(x)=16x. The line LL is tangent to the graph of ff at x=8x=8.

LL can be expressed in the form r =(82)+t=(82)+tu.

The direction vector of y=xy=x is (11)(11).

Find the gradient of LL.

[2]
a.

Find u.

[2]
b.

Find the acute angle between y=xy=x and LL.

[5]
c.

Find (ff)(x)(ff)(x).

[3]
d.i.

Hence, write down f1(x)f1(x).

[1]
d.ii.

Hence or otherwise, find the obtuse angle formed by the tangent line to ff at x=8x=8 and the tangent line to ff at x=2x=2.

[3]
d.iii.

Markscheme

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

attempt to find f(8)     (M1)

eg   f(x) ,  y ,  16x2

−0.25 (exact)     A1 N2

[2 marks]

a.

u =(41)  or any scalar multiple    A2 N2

[2 marks]

b.

correct scalar product and magnitudes           (A1)(A1)(A1)

scalar product =1×4+1×1(=3)

magnitudes =12+12,  42+(1)2  (=2,17)

substitution of their values into correct formula           (M1)

eg  4112+1242+(1)23217,  2.1112,  120.96° 

1.03037 ,  59.0362°

angle = 1.03 ,  59.0°    A1 N4

[5 marks]

c.

attempt to form composite (ff)(x)     (M1)

eg   f(f(x)) ,  f(16x) ,  16f(x)

correct working     (A1)

eg  1616x ,  16×x16

(ff)(x)=x     A1 N2

[3 marks]

d.i.

f1(x)=16x  (accept y=16x, 16x)    A1 N1

Note: Award A0 in part (ii) if part (i) is incorrect.
Award A0 in part (ii) if the candidate has found f1(x)=16x by interchanging x and y.

[1 mark]

d.ii.

METHOD 1

recognition of symmetry about y=x    (M1)

eg   (2, 8) ⇔ (8, 2) 

evidence of doubling their angle        (M1)

eg   2×1.03,  2×59.0

2.06075, 118.072°

2.06 (radians)  (118 degrees)     A1  N2

 

METHOD 2

finding direction vector for tangent line at x=2      (A1)

eg   (14),  (14)

substitution of their values into correct formula (must be from vectors)      (M1)

eg   4412+4242+(1)2,  81717

2.06075, 118.072°

2.06 (radians)  (118 degrees)     A1  N2

 

METHOD 3

using trigonometry to find an angle with the horizontal      (M1)

eg   tanθ=14,  tanθ=4

finding both angles of rotation      (A1)

eg   θ1=0.244978,  14.0362,  θ1=1.81577,  104.036

2.06075, 118.072°

2.06 (radians)  (118 degrees)     A1  N2

[3 marks]

d.iii.

Examiners report

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

Syllabus sections

Topic 4—Statistics and probability » SL 4.1—Concepts, reliability and sampling techniques
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Topic 2—Functions
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