Date | November 2017 | Marks available | 7 | Reference code | 17N.1.sl.TZ0.7 |
Level | SL only | Paper | 1 | Time zone | TZ0 |
Command term | Find | Question number | 7 | Adapted from | N/A |
Question
Consider f(x)=logk(6x−3x2), for 0<x<2, where k>0.
The equation f(x)=2 has exactly one solution. Find the value of k.
Markscheme
METHOD 1 – using discriminant
correct equation without logs (A1)
eg6x−3x2=k2
valid approach (M1)
eg−3x2+6x−k2=0, 3x2−6x+k2=0
recognizing discriminant must be zero (seen anywhere) M1
egΔ=0
correct discriminant (A1)
eg62−4(−3)(−k2), 36−12k2=0
correct working (A1)
eg12k2=36, k2=3
k=√3 A2 N2
METHOD 2 – completing the square
correct equation without logs (A1)
eg6x−3x2=k2
valid approach to complete the square (M1)
eg3(x2−2x+1)=−k2+3, x2−2x+1−1+k23=0
correct working (A1)
eg3(x−1)2=−k2+3, (x−1)2−1+k23=0
recognizing conditions for one solution M1
eg(x−1)2=0, −1+k23=0
correct working (A1)
egk23=1, k2=3
k=√3 A2 N2
[7 marks]