General Certificate of Education Advanced Subsidiary Examination January 2011
Mathematics
MPC1
Unit Pure Core 1 Monday 10 January 2011
9.00 am to 10.30 am
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For this paper you must have: * the blue AQA booklet of formulae and statistical tables. You must not use a calculator.
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Time allowed * 1 hour 30 minutes
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Instructions * Use black ink or black ball-point pen. Pencil should only be used for drawing. * Fill in the boxes at the top of this page. * Answer all questions. * Write the question part reference (eg (a), (b)(i) etc) in the left-hand margin. * You must answer the questions in the spaces provided. Do not write outside the box around each page. * Show all necessary working; otherwise marks for method may be lost. * Do all rough work in this book. Cross through any work that you do not want to be marked. * The use of calculators is not permitted.
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Information * The marks for questions are shown in brackets. * The maximum mark for this paper is 75. Advice * Unless stated otherwise, you may quote formulae, without proof, from the booklet.
P36488/Jan11/MPC1 6/6/
MPC1
2
The curve with equation y ¼ 13 þ 18x þ 3x 2 4x 3 passes through the point P where x ¼ 1 .
1
dy . dx
(3 marks)
(a)
Find
(b)
Show that the point P is a stationary point of the curve and find the other value of x where the curve has a stationary point. (3 marks)
(c) (i)
d2 y Find the value of at the point P. dx 2
(3 marks)
(ii) Hence, or otherwise, determine whether P is a maximum point or a minimum point.
(1 mark)
pffiffiffi 2 Simplify 3 3 .
2 (a)
(1 mark)
pffiffiffiffiffi pffiffiffi pffiffiffi 4 3þ3 7 m þ 21 , where m and n are integers. Express pffiffiffi pffiffiffi in the form n 3 3þ 7
(b)
(4 marks)
The line AB has equation 3x þ 2y ¼ 7 . The point C has coordinates ð2, 7Þ .
3 (a) (i)
Find the gradient of AB .
(2 marks)
(ii) The line which passes through C and which is parallel to AB crosses the y-axis at the
point D. Find the y-coordinate of D.
(3 marks)
(b)
The line with equation y ¼ 1 4x intersects the line AB at the point A. Find the coordinates of A. (3 marks)
(c)
The point E has coordinates ð5, kÞ . Given that CE has length 5, find the two possible values of the constant k. (3 marks)
P36488/Jan11/MPC1
3
The curve sketched below passes through the point Að2, 0Þ .
4
y Pð1, 12Þ
A 2
O
1
x
The curve has equation y ¼ 14 x x 4 and the point Pð1, 12Þ lies on the curve. (a) (i)
Find the gradient of the curve at the point P.
(3 marks)
(ii) Hence find the equation of the tangent to the curve at the point P, giving your
answer in the form y ¼ mx þ c . ð1 (b) (i)
Find
(2 marks)
ð14 x x 4 Þ dx .
(5 marks)
2
(ii) Hence find the area of the shaded region bounded by the curve y ¼ 14 x x 4 and
5 (a) (i)
the line AP.
(2 marks)
Sketch the curve with equation y ¼ xðx 2Þ2 .
(3 marks)
(ii) Show that the equation xðx 2Þ2 ¼ 3 can be expressed as
x 3 4x 2 þ 4x 3 ¼ 0
(1 mark)
The polynomial pðxÞ is given by pðxÞ ¼ x 3 4x 2 þ 4x 3 .
(b) (i)
Find the remainder when pðxÞ is divided by x þ 1 .
(2 marks)
(ii) Use the Factor Theorem to show that x 3 is a factor of pðxÞ .
(2 marks)
(iii) Express pðxÞ in the form ðx 3Þðx 2 þ bx þ cÞ , where b and c are integers.
(2 marks) (c)
Hence show that the equation xðx 2Þ2 ¼ 3 has only one real root and state the value of this root. (3 marks)
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Turn over
P36488/Jan11/MPC1
4
A circle has centre Cð3, 1Þ and radius
6 (a) (i)
pffiffiffiffiffi 13 .
Express the equation of the circle in the form ðx aÞ2 þ ðy bÞ2 ¼ k
(2 marks)
(ii) Hence find the equation of the circle in the form
x 2 þ y 2 þ mx þ ny þ p ¼ 0 where m, n and p are integers.
(3 marks)
(b)
The circle cuts the y-axis at the points A and B. Find the distance AB.
(3 marks)
(c) (i)
Verify that the point Dð5, 2Þ lies on the circle.
(1 mark)
(ii) Find the gradient of CD.
(2 marks)
(iii) Hence find an equation of the tangent to the circle at the point D.
(2 marks)
7 (a) (i)
Express 4 10x x 2 in the form p ðx þ qÞ2 .
(2 marks)
(ii) Hence write down the equation of the line of symmetry of the curve with equation
y ¼ 4 10x x 2 .
(1 mark)
The curve C has equation y ¼ 4 10x x 2 and the line L has equation y ¼ kð4x 13Þ , where k is a constant.
(b)
(i)
Show that the x-coordinates of any points of intersection of the curve C with the line L satisfy the equation x 2 þ 2ð2k þ 5Þx ð13k þ 4Þ ¼ 0
(1 mark)
(ii) Given that the curve C and the line L intersect in two distinct points, show that
4k 2 þ 33k þ 29 > 0 (iii) Solve the inequality 4k 2 þ 33k þ 29 > 0 .
(3 marks) (4 marks)
Copyright ª 2011 AQA and its licensors. All rights reserved.
P36488/Jan11/MPC1
Mark Scheme – General Certificate of Education (A-level) Mathematics – Pure Core 1 – January 2011
Key to mark scheme abbreviations M m or dM A B E or ft or F CAO CSO AWFW AWRT ACF AG SC OE A2,1 –x EE NMS PI SCA c sf dp
mark is for method mark is dependent on one or more M marks and is for method mark is dependent on M or m marks and is for accuracy mark is independent of M or m marks and is for method and accuracy mark is for explanation follow through from previous incorrect result correct answer only correct solution only anything which falls within anything which rounds to any correct form answer given special case or equivalent 2 or 1 (or 0) accuracy marks deduct x marks for each error no method shown possibly implied substantially correct approach candidate significant figure(s) decimal place(s)
No Method Shown Where the question specifically requires a particular method to be used, we must usually see evidence of use of this method for any marks to be awarded. Where the answer can be reasonably obtained without showing working and it is very unlikely that the correct answer can be obtained by using an incorrect method, we must award full marks. However, the obvious penalty to candidates showing no working is that incorrect answers, however close, earn no marks. Where a question asks the candidate to state or write down a result, no method need be shown for full marks. Where the permitted calculator has functions which reasonably allow the solution of the question directly, the correct answer without working earns full marks, unless it is given to less than the degree of accuracy accepted in the mark scheme, when it gains no marks. Otherwise we require evidence of a correct method for any marks to be awarded.
3
Mark Scheme – General Certificate of Education (A-level) Mathematics – Pure Core 1 – January 2011
MPC1 Q
Solution 1(a)
(b)
Marks M1 A1 A1
dy = 18 + 6 x – 12 x 2 dx
18 + 6 x – 12 x 2 = 0 6 ( 3 – 2 x )( x + 1)
x = –1, x =
3 2
Total 3
dy = 0 , PI by attempt to dx solve or factorise
putting their
M1
(= 0)
m1
OE
A1
Comments one of these terms correct another term correct all correct (no + c etc) (penalise + c once only in question)
attempt at factors of their quadratic or use of quadratic equation formula 3
must see both values unless x = –1 is verified separately If M1 not scored, award SC B1 for dy = 0 and verifying that x = –1 leads to dx 3 a further SC B2 for finding x = as other 2 value
(c)(i)
d2 y = 6 – 24 x dx 2 When x = – 1,
B1 d2 y = 6 − (24 × −1) dx 2
M1
d2 y = 30 dx 2 (ii)
dy d2 y but must be correct if 3 d x2 dx marks earned in part (a) d2 y Sub x = –1 into ‘their’ d x2
FT their
Minimum point
A1cso E1
3 1
must have a value in (c)(i) FT “maximum” if their value of
Total
10
4
d2 y 0 2
Discriminant = 4 ( 2k + 5 ) + 4 (13k + 4 )
(
1
stated or used (must be > 0)
M1
condone one slip (may be within formula)
)
or 16k 2 + 132k + 116 > 0
⇒ 4k 2 + 33k + 29 > 0
A1
( 4k + 29 )( k + 1) k=–
AG all correct working and = 0
B1
4 4k 2 + 20k + 25 + 13k + 4 > 0
(iii)
Comments
3
correct factors or correct unsimplified quadratic equation formula
M1
29 , k = –1 4
AG > 0 must appear before final line
−33 ± 332 − 4 × 4 × 29 8 58 condone k = – , –7.25 etc but not left 8 with square roots etc as above
A1 y
–
M1
29 4
sketch or sign diagram including values +
x
–1 O
– –29/4
29 , k > –1 4 Take their final line as their answer k< –
A1
Total TOTAL
4 11 75
9
+ –1
condone use of OR but not AND
Scaled mark unit grade boundaries - January 2011 exams A-level Max. Scaled Mark
Scaled Mark Grade Boundaries and A* Conversion Points A* A B C D E
Code
Title
MD01 MFP1
GCE MATHEMATICS UNIT D01 GCE MATHEMATICS UNIT FP1
75 75
-
61 63
43 43
37 37
MM1A MM1B MPC1
GCE MATHEMATICS UNIT M1A GCE MATHEMATICS UNIT M1B GCE MATHEMATICS UNIT PC1
100 75 75
-
no candidates were entered for this unit 61 53 46 39 56 49 42 36
32 30
GCE MATHEMATICS UNIT S1A GCE MATHEMATICS UNIT S1A - WRITTEN GCE MATHEMATICS UNIT S1A - COURSEWORK
100 75 25
-
84 64 20
74
64
54
44 34 10
MS1B MD02 MFP2 MM2B MPC2 MS2B XMCA2 MFP3 MPC3 MFP4 MPC4
GCE MATHEMATICS UNIT S1B GCE MATHEMATICS UNIT D02 GCE MATHEMATICS UNIT FP2 GCE MATHEMATICS UNIT M2B GCE MATHEMATICS UNIT PC2 GCE MATHEMATICS UNIT S2B GCE MATHEMATICS UNIT XMCA2 GCE MATHEMATICS UNIT FP3 GCE MATHEMATICS UNIT PC3 GCE MATHEMATICS UNIT FP4 GCE MATHEMATICS UNIT PC4
75 75 75 75 75 75 125 75 75 75 75
69 67 63 66 105 66 66 63 68
61 63 60 55 61 59 93 59 59 55 61
53 56 51 47 54 52 81 52 52 47 54
46 50 42 40 47 45 70 45 45 40 47
39 44 34 33 40 38 59 38 38 33 41
32 38 26 26 33 31 48 31 31 26 35
MEST1 MEST2 MEST3 MEST4
GCE MEDIA STUDIES UNIT 1 GCE MEDIA STUDIES UNIT 2 GCE MEDIA STUDIES UNIT 3 GCE MEDIA STUDIES UNIT 4
80 80 80 80
69 72
56 63 58 65
49 54 47 53
42 45 37 42
35 36 27 31
28 28 17 20
PHIL1 PHIL2
GCE PHILOSOPHY UNIT 1 GCE PHILOSOPHY UNIT 2
90 90
-
54 62
48 56
42 50
37 44
32 38
MS1A MS/SS1A/W MS/SS1A/C
55 56
49 49
