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.
Find u.
Find the acute angle between y=xy=x and LL.
Find (f∘f)(x)(f∘f)(x).
Hence, write down f−1(x)f−1(x).
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.
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′ , −16x−2
−0.25 (exact) A1 N2
[2 marks]
u =(4−1) or any scalar multiple A2 N2
[2 marks]
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 4−1√12+12√42+(−1)2, −3√2√17, 2.1112, 120.96°
1.03037 , 59.0362°
angle = 1.03 , 59.0° A1 N4
[5 marks]
attempt to form composite (f∘f)(x) (M1)
eg f(f(x)) , f(16x) , 16f(x)
correct working (A1)
eg 1616x , 16×x16
(f∘f)(x)=x A1 N2
[3 marks]
f−1(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 f−1(x)=16x by interchanging x and y.
[1 mark]
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), (1−4)
substitution of their values into correct formula (must be from vectors) (M1)
eg −4−4√12+42√42+(−1)2, 8√17√17
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]