Date | November 2015 | Marks available | 6 | Reference code | 15N.1.sl.TZ0.7 |
Level | SL only | Paper | 1 | Time zone | TZ0 |
Command term | Find | Question number | 7 | Adapted from | N/A |
Question
An arithmetic sequence has the first term lnalna and a common difference ln3ln3.
The 13th term in the sequence is 8ln98ln9. Find the value of aa.
Markscheme
Note: There are many approaches to this question, and the steps may be done in any order. There are 3 relationships they may need to apply at some stage, for the 3rd, 4th and 5th marks. These are
equating bases eg recognising 9 is 3232
log rules: lnb+lnc=ln(bc), lnb−lnc=ln(bc)lnb+lnc=ln(bc), lnb−lnc=ln(bc),
exponent rule: lnbn=nlnblnbn=nlnb.
The exception to the FT rule applies here, so that if they demonstrate correct application of the 3 relationships, they may be awarded the A marks, even if they have made a previous error. However all applications of a relationship need to be correct. Once an error has been made, do not award A1FT for their final answer, even if it follows from their working.
Please check working and award marks in line with the markscheme.
correct substitution into u13u13 formula (A1)
eglna+(13−1)ln3lna+(13−1)ln3
set up equation for u13u13 in any form (seen anywhere) (M1)
eglna+12ln3=8ln9lna+12ln3=8ln9
correct application of relationships (A1)(A1)(A1)
a=81a=81 A1 N3
[6 marks]
Examples of application of relationships
Example 1
correct application of exponent rule for logs (A1)
eglna+ln312=ln98lna+ln312=ln98
correct application of addition rule for logs (A1)
egln(a312)=ln98ln(a312)=ln98
substituting for 9 or 3 in ln expression in equation (A1)
egln(a312)=ln316, ln(a96)=ln98ln(a312)=ln316, ln(a96)=ln98
Example 2
recognising 9=329=32 (A1)
eglna+12ln3=8ln32, lna+12ln912=8ln9lna+12ln3=8ln32, lna+12ln912=8ln9
one correct application of exponent rule for logs relating ln9ln9 to ln3ln3 (A1)
eglna+12ln3=16ln3, lna+6ln9=8ln9lna+12ln3=16ln3, lna+6ln9=8ln9
another correct application of exponent rule for logs (A1)
eglna=ln34, lna=ln92lna=ln34, lna=ln92