Date | May Specimen paper | Marks available | 7 | Reference code | SPM.2.AHL.TZ0.12 |
Level | Additional Higher Level | Paper | Paper 2 | Time zone | Time zone 0 |
Command term | Verify | Question number | 12 | Adapted from | N/A |
Question
Show that .
Verify that and satisfy the equation .
Hence, or otherwise, show that the exact value of .
Using the results from parts (b) and (c) find the exact value of .
Give your answer in the form where , .
Markscheme
stating the relationship between and and stating the identity for M1
and
⇒ AG
[1 mark]
METHOD 1
attempting to substitute for and using the result from (a) M1
LHS = A1
(= RHS) A1
so satisfies the equation AG
attempting to substitute for and using the result from (a) M1
LHS = A1
A1
(= RHS) A1
so satisfies the equation AG
METHOD 2
let and
attempting to find the sum of roots M1
A1
(from part (a)) A1
attempting to find the product of roots M1
A1
= −1 A1
the coefficient of and the constant term in the quadratic are and −1 respectively R1
hence the two roots are and AG
[7 marks]
METHOD 1
and are roots of R1
Note: Award R1 if only is stated as a root of .
A1
attempting to solve their quadratic equation M1
A1
() R1
so AG
METHOD 2
attempting to substitute into the identity for M1
A1
attempting to solve their quadratic equation M1
A1
R1
so AG
[5 marks]
is the sum of the roots of R1
A1
A1
attempting to rationalise their denominator (M1)
A1A1
[6 marks]