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UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS
Cambridge International Level 3 Pre-U Certificate
Principal Subject
Mathematics
Paper 1
9794/01
For Examination from 2012
MARK SCHEME
Maximum Mark : 80
IMPORTANT NOTICE
Mark Schemes have been issued on the basis of one copy per Assistant examiner and two
copies per Team Leader.
Centre (–5, 3)
Radius 6
B1
B1
(i)
Show f(2)=0
B1
(ii)
Method shown e.g. division to get quadratic
Obtain two factors or roots
M1
A1
1
2
x  22 x  3x  3
x  2,
3
(i)
3
, 3
2
Attempt S 40 
A1
(follow through their factors)
40
2  7  40  1d 
2
Obtain correct unsimplified expression
Equate attempt at S 40 to 4960 and attempt to find d.
Obtain d = 6
(ii)
Attempt use of S  
a
1 r
(i)
Form the equations 2r  rθ  40
5
(i)
(ii)
M1
A1
B1
1
and r 2 θ  100
2
(ii)
M1
A1
M1
A1
Obtain 20
4
B1 ft
B1
200
, or equivalent, to eliminate θ
r2
Obtain r 2  20r  100  0 Answer given
Use θ 
Solve quadratic for r
Obtain correct value r = 10
Substitute and obtain correct value θ = 2
M1
A1
M1
A1
A1
Attempt integration to obtain at least one ln term
M1
Obtain ln(x – 2) – ln(2x + 3)
x2
Obtain ln
2x  3
+c
A1
A1
A1
dv
 x2
dx
x3
du 1

v
dx x
3
Obtain and expression of the form f x   g( x)dx
u  ln x

3

M1
M1
3
x ln x
x
1

 dx
3
3 x
x 3 ln x x 3

 c 
Obtain
3
9
n.b. Mark for + c may be awarded in this part if withheld in (i).
Obtain
M1
A1
A1
6
(i)
(ii)
Obtain 1 – 18x
Attempt binomial expansion of at least one more term with each term product of
binomial coefficient and power of –2x
Obtain 144x 2
Obtain  672x 3
Multiply together two relevant pairs of terms
Obtain 144  18a  66
37
Obtain a 
3
Attempt use of correct Newton-Raphson formula with appropriate f(x)
7
2
Use e.g. f ( x)  1 
x  13
8
(i)
(ii)
M1
A1
A1
M1
A1ft
A1
M1
Use x 0  2 and continue until at least 2 iterates agree.
Obtain final answer 1.879
B1
M1
A1
Attempt to differentiate
Obtain 6 x 2  10 x  4
M1
A1
Setting their
dy
=0
dx
Solving quadratic to obtain x  2
(iii)
B1
M1
x
1
3
d2 y
dy
Looks at sign of
, derived correctly from their
, or other correct method
2
dx
dx
A1
M1
2
d y
> 0 therefore minimum
dx 2
1 d2 y
When x   ,
<0 therefore maximum
3 dx 2
When x = 2,
A1
A1
9
(i)
(ii)
Parabola correct
Line correct
B1
B1
Equating and attempting to solve equation
Obtain x = –1 and x = 2
M1
A1
EITHER:
M1
A1
M1
Attempt subtraction f(x) – g(x) in the correct order
Obtain 2  x  x 2
Attempt integration of their difference
Obtain 2 x 
OR:
1 2 1 3
x  x
2
3
Use limits correctly
A1
M1
Obtain 4
1
2
A1
Attempt
 3  2 x  x dx
Obtain 3 x  x 2 
2
M1
1 3
x
3
A1
M1
A1
Use limits correctly
Obtain 9
Calculate area of triangle as
1
1
 3 3  4
2
2
Subtract to obtain area between curve and line as 4
10
(i)
Find a – b or b – a
Use correct method to find the magnitude of any vector
154 or equivalent
(ii)
Using ( AO or OA ) and ( AB or BA )
scalar product of any two vectors
cos θ 
product of their moduli

32.8 or better, or 0.572 rad or better
M1
1
2
A1
M1
M1
A1
B1
M1
A1
11
Separate variables prior to integration
1
1
 sec y dy   x
2
dx
A1
1
(+ c)
x
π
3
Substitute in y  and x = 4 to get c 
6
4
M1
3 1
y  sin 1    o.e.
4 x
A1
sin y  
12
M1
A1
A1
Attempt expression of cos θ  2 sin θ in any of the forms R cosθ  α  or
R sin θ  α 
Obtain e.g. R cos α  1
And R sin α  2
Solve to obtain R  3
And e.g. α  54.7  or 0.955 rad
Attempt to link at least one critical value with a value of θ
3 corresponds to θ  54.7  or 0.955 rad
State that  3 corresponds to θ  234.7  or 4.097 rad
1
1
Identify maximum as
and/or minimum as
2R
2 R
1
State maximum as
, o.e., and 234.7  o.e.
2 3
1
State minimum as
, o.e., and 54.7  o.e.
2 3
State that
M1
A1
A1
A1
A1
M1
A1
A1
M1
A1
A1