EDEXCEL CORE MATHEMATICS C3 (6665) – JANUARY 2011 FINAL MARK SCHEME

Question Number / Scheme / Marks
1.
(a) /
/ / B1
/ or / M1
awrt 1.287 / A1
Hence,
(3)
(b) / Minimum value = / or / B1ft
(1)
(c) /
/ / M1
/ For applying / M1
So, / either or
/ M1
gives, / awrt 3.84 OR 6.16 / A1
awrt 3.84 AND 6.16 / A1
(5)
[9]
Question Number / Scheme / Marks
2.
(a) /
/ An attempt to form a single fraction / M1
/ Simplifies to give a correct quadratic numerator over a correct quadratic denominator / A1 aef
/ An attempt to factorise a 3 term quadratic numerator / M1
/ A1
(4)
(b) /
/ An attempt to form a single fraction / M1
/ Correct result / A1
(2)
(c) /

/ / M1
A1 aef
/ Either or / A1
(3)
[9]
Question Number / Scheme / Marks
3. /
/ Substitutes either
or
or for / M1
/ Forms a “quadratic in sine” = 0 / M1(*)
/ Applies the quadratic formula
See notes for alternative methods. / M1
/ Any one correct answer / A1
180-their pv / dM1(*)
All four solutions correct. / A1
[6]
Question Number / Scheme / Marks
4.
(a) / (eqn *)
/ Substitutes and into eqn * / M1
/ / A1
(2)
(b) /
/ Substitutes and into eqn * and rearranges eqn * to make e±5k the subject. / M1
/ Takes ‘lns’ and proceeds
to make ‘±5k’ the subject. / dM1
/ Convincing proof that / A1
(3)
(c) /
/ where / M1
/ A1 oe
When
Rate of decrease of (3 dp.) / awrt / A1
(3)
[8]
Question Number / Scheme / Marks
5.
(a)
Crosses x-axis
Either or / Either one of {x}=1 OR x={8} / B1
Coordinates are and / Both and / B1
(2)
(b) / Apply product rule: / / M1
/ Any one term correct / A1
Both terms correct / A1
(3)
(c) /

Sign change (and asis continuous) therefore the x-coordinate of Q lies between 3.5 and 3.6. / Attempts to evaluate both and / M1
both values correct to at least 1 sf, sign change and conclusion / A1
(2)
(d) / At Q, / Setting . / M1
/ Splitting up the numerator
and proceeding to x= / M1
(as required) / For correct proof.
No errors seen in working. / A1
(3)
Question Number / Scheme / Marks
(e) / Iterative formula:
/ An attempt to substitute into the iterative formula.
Can be implied by / M1
/ Both
and / A1
to 3 dp. / all stated correctly to 3 dp / A1
(3)
[13]
Question Number / Scheme / Marks
6.
(a) / / Attempt to make x
(or swapped y) the subject / M1
/ Collect x terms together and factorise. / M1
/ / A1 oe
(3)
(b) / Range of g is -9≤ g(x)≤ 4 or -9≤ y ≤ 4 / Correct Range / B1
(1)
(c) / Deduces that is 0.
Seen or implied. / M1
g g(2)= g (0) , from sketch. / -6 / A1
(2)
(d) / / Correct order g followed by f / M1
/ 5 / A1
(2)
(e)(i) / / Correct shape
/ B1
, / B1
Question Number / Scheme / Marks
(e)(ii) / / Correct shape / B1
Graph goes throughand which are marked. / B1
(4)
(f) / Domain of is -9≤ x ≤ 4 / Either correct answer or a follow through from part (b) answer / B1
(1)
[13]
Question Number / Scheme / Marks
7
(a) /
Apply quotient rule:
/ Applying / M1
Any one term correct on the numerator / A1
Fully correct (unsimplified). / A1
/ For correct proof with an understanding that
No errors seen in working.
(as required) / A1*
(4)
(b) / When , / / B1
At / / B1
Either T:
or and
; / with ‘their TANGENT gradient’ and their y1;
or uses with ‘their TANGENT gradient’; / M1
T: / / A1
(4)
[8]
Question Number / Scheme / Marks
8.
(a) /
/ Writes as and gives / M1
or / A1
/ Convincing proof.
Must see both / A1 AG
(3)
(b) /
/ / M1
/ A1
(2)
(c) / / Applies / M1
/ Substitutes for / M1
/ Attempts to use the identity
/ M1
So
/ / A1
(4)
[9]

1