OCR ADVANCED SUBSIDIARY GCE IN MATHEMATICS (3840, 3841, 3842, 3843 and 3844) OCR ADVANCED GCE IN MATHEMATICS (7840, 7842 and 7844) Specimen Question Papers and Mark Schemes These specimen question papers and mark schemes are intended to accompany the OCR Advanced Subsidiary GCE and Advanced GCE specifications in Mathematics for teaching from September 2000. Centres are permitted to copy material from this booklet for their own internal use. The GCE awarding bodies have prepared new specifications to incorporate the range of features required by new GCE and and subject criteria. The specimen assessment assessment material accompanying accompanying the new specifications is provided to give centres a reasonable idea of the general shape and character of the planned question papers in advance of the first operational examination.
Specimen Materials - Mathematics
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© OCR 2000
CONTENTS
Advanced Subsidiary GCE Unit 2631: Pure Mathematics 1
P1
Question Paper Mark Scheme Unit 2637: Mechanics 1
Page Page
53 57
Page Page
85 89
S1
Question Paper Mark Scheme Unit 2645: Discrete Mathematics 1
5 9
M1
Question Paper Mark Scheme Unit 2641: Probability and Statistics 1
Page Page
D1
Question Paper Mark Scheme
Page 117 Page 121
A2 Unit 2632: Pure Mathematics 2
P2
Question Paper Mark Scheme Unit 2633: Pure Mathematics 3
Page Page
29 33
Page Page
37 41
Page Page
45 49
P6
Question Paper Mark Scheme
Specimen Materials - Mathematics
21 25
P5
Question Paper Mark Scheme Unit 2636: Pure Mathematics 6
Page Page P4
Question Paper Mark Scheme Unit 2635: Pure Mathematics 5
13 17
P3
Question Paper Mark Scheme Unit 2634: Pure Mathematics 4
Page Page
2
© OCR 2000
CONTENTS
Advanced Subsidiary GCE Unit 2631: Pure Mathematics 1
P1
Question Paper Mark Scheme Unit 2637: Mechanics 1
Page Page
53 57
Page Page
85 89
S1
Question Paper Mark Scheme Unit 2645: Discrete Mathematics 1
5 9
M1
Question Paper Mark Scheme Unit 2641: Probability and Statistics 1
Page Page
D1
Question Paper Mark Scheme
Page 117 Page 121
A2 Unit 2632: Pure Mathematics 2
P2
Question Paper Mark Scheme Unit 2633: Pure Mathematics 3
Page Page
29 33
Page Page
37 41
Page Page
45 49
P6
Question Paper Mark Scheme
Specimen Materials - Mathematics
21 25
P5
Question Paper Mark Scheme Unit 2636: Pure Mathematics 6
Page Page P4
Question Paper Mark Scheme Unit 2635: Pure Mathematics 5
13 17
P3
Question Paper Mark Scheme Unit 2634: Pure Mathematics 4
Page Page
2
© OCR 2000
A2 continued Unit 2638: Mechanics 2
M2
Question Paper Mark Scheme Unit 2639: Mechanics 3
77 81
Page Page
93 97
Page 101 Page 105 S4
Question Paper Mark Scheme Unit 2646: Discrete Mathematics 2
Page Page
S3
Question Paper Mark Scheme Unit 2644: Probability and Statistics 4
69 73
S2
Question Paper Mark Scheme Unit 2643: Probability and Statistics 3
Page Page M4
Question Paper Mark Scheme Unit 2642: Probability and Statistics 2
61 65
M3
Question Paper Mark Scheme Unit 2640: Mechanics 4
Page Page
Page 109 Page 113 D2
Question Paper Insert Mark Scheme
Page 125 Page 133 Page 137
Marking Instructions
Page 143
Specimen Materials - Mathematics
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© OCR 2000
Specimen Materials Mathematics
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© OCR 2000 Oxford, Cambridge and RSA Examinations
General Certificate of Education Advanced Subsidiary (AS) and Advanced Level
P1
MATHEMATICS Pure Mathematics 1
Additional materials: Answer paper Graph paper List of Formulae
TIME
1 hour 20 minutes
INSTRUCTIONS TO CANDIDATES Write your name, Centre number and candidate number in the spaces provided on the answer paper. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures, unless a different degree of accuracy is specified in the question or is clearly appropriate. You are permitted to use only a scientific calculator in this paper. INFORMATION FOR CANDIDATES The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 60. Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper. You are reminded of the need for clear presentation in your answers.
This question paper consists of 3 printed pages and 1 blank page. Specimen Materials - Mathematics
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© OCR 2000
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(i) Write down the exact value of 7 − 2 . (ii) Simplify
2
[1]
( x x )3 . 2 x4
[2]
(i) Solve the simultaneous equations y
= x 2 − 3x + 2,
y
= 3x − 7.
[4]
(ii) Interpret your solution to part (i) geometrically.
3
[1]
The point A has coordinates (7, 4) . The straight lines with equations x + 3 y + 1 = 0 and 2x + 5 y intersect at the point B. Show that one of these two lines is perpendicular to AB.
4
[6]
Show that the equation 15cos 2 θ
= 13 + sin θ
may be written as a quadratic equation in sin θ .
[2]
Hence solve the equation, giving all values of θ such that 0 ° ≤ θ
5
=0
Sketch the graph of y
= cos x ° , for values of x from 0 to 360.
Sketch, on the same diagram, the graph of y
= cos(x − 60)° .
≤ 360° .
[5]
[1] [2]
Use your diagram to solve the equation cos x° = cos(x − 60)°
for values of x between 0 and 360. Indicate clearly on your diagram how the solutions relate to the graphs. [3] State how many values of x satisfying the equation cos(10x)° = cos(10 x − 60)°
lie between 0 and 360. (You should explain your reasoning briefly, but no further detailed working or sketching is necessary.) [2]
Specimen Materials - Mathematics
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© OCR 2000
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(i) Evaluate
4
∫ x(4 − x) dx .
[3]
0
(ii)
= x(4 − x) , together with a straight line. the origin O and at the point P with x-coordinate k , where 0 < k < 4 . The diagram shows the curve y
This line cuts the curve at
(a) Show that the area of the shaded region, bounded by the line and the curve, is 16 k 3 .
[4]
(b) Find, correct to 3 decimal places, the value of k for which the area of the shaded region is half of [2] the total area under the curve between x = 0 and x = 4 .
7
A quadratic function is defined by f( x) = x 2 + kx + 9 ,
where k is a constant. It is given that the equation f(x ) = 0 has two distinct real roots. Find the set of [3] values that k can take. For the case where k = −4 3 , (i) express f( x) in the form ( x + a )2 + b , stating the values of a and b , and hence write down the least value taken by f( x) , [4] (ii) solve the equation f(x)
8
= 0 , expressing your answer in terms of surds, simplified as far as possible.
The equation of a curve is y
= 6x2 − x3 .
[3]
Find the coordinates of the two stationary points on the curve,
and determine the nature of each of these stationary points.
[6]
State the set of values of x for which 6 x 2 − x 3 is a decreasing function of x.
[2]
The gradient at the point M on the curve is 12. Find the equation of the tangent to the curve at M .
[4]
Specimen Materials - Mathematics
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© OCR 2000
BLANK PAGE
Specimen Materials - Mathematics
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© OCR 2000
General Certificate of Education Advanced Subsidiary (AS) and Advanced Level
P1
MATHEMATICS Pure Mathematics 1
MARK SCHEME
MAXIMUM MARK
60
For live examinations, each Mark Scheme includes the General Instructions for Marking set out on pages 143 to 145.
This mark scheme consists of 4 printed pages. Specimen Materials - Mathematics
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© OCR 2000
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1 B1 (i) 49 1 Correct value stated as final answer --------------------------------------------------------------------------------------------------------------------------------------------------------
(ii)
( x x )3 2x 4 1 2
2
41
=
x 2 2 x4
M1
1 2
x
A1
3x − 7 = x 2 − 3x + 2 x 2 − 6x + 9 = 0 x = 3 only y = 2 only
M1
Eliminate y to obtain an equation in x only
A1 A1 A1
Correct 3-te rm equation in x Obtained by any correct solution method If two values of x are found both y-values
1
x 2 or
(i) EITHER :
Power 3 × 112 or 3 + 1 12 in numerator 2 Or any equally simple equivalent
must follow correctly 2
y+7 y+7 = 3 − 3 3 + 2 2 y − 4y + 4 = 0 y = 2 only x = 3 only y
OR :
M1
Eliminate x to obtain an equation in y only
A1
Correct 3-term equation in y
A1
Obtained by any correct solution method
4 If two values of y are found both x-values must follow correctly -------------------------------------------------------------------------------------------------------------------------------------------------------(ii) The line y = 3x − 7 is the tangent to the curve y
3
= x 2 − 3x + 2
at the point (3, 2)
Solve x + 3y + 1 = 0 and 2x + 5y
x
= 5,
y
= −2
= 0 simultaneously
at B
A1
B1
1 For identifying tangency
M1
Attempt soln and obtain at least one answer
A1
Identify correct coordinates with B, either explicitly or implicitly
Gradient of AB is
4 − ( −2) 7 −5
A1
For simplified follow-through value
− 25 Perpendicular lines require m1m2 = −1 AB is perpendicular to x + 3y + 1 = 0
B1
For either gradient correctly stated or used
M1
Any statement or use of the correct relation
15(1 − sin 2 θ ) = 13 + sin θ
M1
Gradients of the lines are
4
=3
− 13
and
A1
6 Correct use of 3 × − 13
= − 1 , or equivalent
Attempted relevant use of sin 2 θ
+ cos 2 θ = 1
15sin 2 θ + sinθ − 2 = 0 2 Any correct 3-term form A1 -------------------------------------------------------------------------------------------------------------------------------------------------------(5 sin θ + 2)(3sin θ − 1) = 0 M1 Any recognisable solution method attempted
= − 25 or 13 = 19.5°, 160 . 5°, 203 .6°, 336 .4°
sin θ
A1
Both correct values
θ
A1
For any one correct value
A1 A1
Specimen Materials - Mathematics
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For a second correct value 5 For both remaining values, and no others
© OCR 2000
