Oxford Cambridge and RSA
Wednesday 16 May 2018 – Morning
AS GCE MATHEMATICS (MEI)
4751/01 Introduction to Advanced Mathematics (C1)
QUESTION PAPER
Candidates answer on the Printed Answer Book.
* 7 3 0 9 4 0 4 4 8 3 *
OCR supplied materials: Duration: 1 hour 30 minutes
• Printed Answer Book 4751/01
• MEI Examination Formulae and Tables (MF2)
Other materials required:
None
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the Printed Answer Book and the Question Paper.
• The Question Paper will be found inside the Printed Answer Book.
• Write your name, centre number and candidate number in the spaces provided on the
Printed Answer Book. Please write clearly and in capital letters.
• Write your answer to each question in the space provided in the Printed Answer Book.
If additional space is required, you should use the lined page(s) at the end of this booklet.
The question number(s) must be clearly shown.
• Use black ink. HB pencil may be used for graphs and diagrams only.
• Read each question carefully. Make sure you know what you have to do before starting your
answer.
• Answer all the questions.
• Do not write in the barcodes.
• You are not permitted to use a calculator in this paper.
• Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
This information is the same on the Printed Answer Book and the Question Paper.
• The number of marks is given in brackets [ ] at the end of each question or part question on
the Question Paper.
• You are advised that an answer may receive no marks unless you show sufficient detail of
the working to indicate that a correct method is being used.
• The total number of marks for this paper is 72.
• The Printed Answer Book consists of 12 pages. The Question Paper consists of 4 pages.
Any blank pages are indicated.
INSTRUCTION TO EXAMS OFFICER / INVIGILATOR
• Do not send this Question Paper for marking; it should be retained in the centre or recycled.
Please contact OCR Copyright should you wish to re-use this document.
No calculator can
be used for this
paper
© OCR 2018 [H/102/2647] OCR is an exempt Charity
DC (SC/SW) 152199/2 Turn over
, 2
Section A (36 marks)
Simplify ^5a 2 ch # 2a 4 c -5 .
3
1 [2]
2 Find the equation of the line joining the points (-1, 9) and (2, -3), giving your answer in the form
y = mx + c. State the coordinates of the points where this line intersects the axes. [5]
3 Find the value of
(i) a2 14k ,
-2
[2]
(ii) ^8000h3 .
2
[2]
4 For the following equation, express x in terms of y.
x 2x + 1
= [4]
3y y+2
5 Find the coordinates of the point of intersection of the lines y = 4x + 3 and 3x + 2y = 9. [4]
Find the term that is independent of x in the binomial expansion of a - 3xk .
1 6
6 [3]
x
7 (i) Express 28 + 3 175 in the form a b, where a and b are integers and b is as small as possible. [2]
6 3 2 a+b 2
(ii) Simplify - , giving your answer in the form , where a, b and c are integers. [3]
5- 2 5+ 2 c
8 For each of the following pairs of sentences A and B, give a reason why the statement A + B is false and
write either ‘A & B ’ or ‘A % B ’ to show the correct relationship.
(i) A: n is positive.
B: n2 + 6 is positive. [2]
(ii) A: The diagonals of a quadrilateral bisect each other but not at right angles.
B: The quadrilateral is a rectangle but not a square. [2]
9 You are given that f(x) = ax3 + cx and that f(-1) = 3. You are also given that when f(x) is divided by (x - 4),
the remainder is 108. Find the values of a and c. [5]
© OCR 2018 4751/01 Jun18
, 3
Section B (36 marks)
10 (i) Express 3x2 - 9x + 5 in the form a(x + b)2 + c. Hence state the equation of the line of symmetry and
the y-coordinate of the minimum point of the curve with equation y = 3x2 - 9x + 5. [6]
(ii) Find the coordinates of the points where the graph of y = 3x2 - 9x + 5 intersects the axes. Give your
answers in an exact form. Hence state the solution of the inequality 3x2 - 9x + 5 < 0. [4]
11 You are given that f(x) = (2x + 5)(x2 - 5x + 4).
(i) Sketch the graph of y = f(x). [4]
(ii) You are given that g(x) = 2x3 - 5x2 - 17x + 48. Show that x = -3 is a root of g(x) = 0 and that it is the
only real root. [6]
(iii) Show that y = g(x) is a translation of y = f(x) by c m , finding the value of k.
0
[3]
k
12
y
A (7, 4)
x
O
Fig. 12
Fig. 12 shows a sketch of the circle with equation (x - 2)2 + (y + 1)2 = 50. You are given that the point
A (7, 4) lies on the circle.
(i) Write down the radius of this circle and the coordinates of its centre. [2]
(ii) The line L has equation y = 2x - 10 and passes through the point A (7, 4). Use algebra to find the
coordinates of the point B where the line L meets the circle again. Hence show that the perpendicular
distance from the centre of the circle to the line L is 5. [6]
(iii) Show that, when the line y = 2x + k is a tangent to the circle, k satisfies the equation
k2 + 10k - 225 = 0. [5]
END OF QUESTION PAPER
© OCR 2018 4751/01 Jun18
, 4
Oxford Cambridge and RSA
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2018 4751/01 Jun18
Wednesday 16 May 2018 – Morning
AS GCE MATHEMATICS (MEI)
4751/01 Introduction to Advanced Mathematics (C1)
QUESTION PAPER
Candidates answer on the Printed Answer Book.
* 7 3 0 9 4 0 4 4 8 3 *
OCR supplied materials: Duration: 1 hour 30 minutes
• Printed Answer Book 4751/01
• MEI Examination Formulae and Tables (MF2)
Other materials required:
None
INSTRUCTIONS TO CANDIDATES
These instructions are the same on the Printed Answer Book and the Question Paper.
• The Question Paper will be found inside the Printed Answer Book.
• Write your name, centre number and candidate number in the spaces provided on the
Printed Answer Book. Please write clearly and in capital letters.
• Write your answer to each question in the space provided in the Printed Answer Book.
If additional space is required, you should use the lined page(s) at the end of this booklet.
The question number(s) must be clearly shown.
• Use black ink. HB pencil may be used for graphs and diagrams only.
• Read each question carefully. Make sure you know what you have to do before starting your
answer.
• Answer all the questions.
• Do not write in the barcodes.
• You are not permitted to use a calculator in this paper.
• Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
This information is the same on the Printed Answer Book and the Question Paper.
• The number of marks is given in brackets [ ] at the end of each question or part question on
the Question Paper.
• You are advised that an answer may receive no marks unless you show sufficient detail of
the working to indicate that a correct method is being used.
• The total number of marks for this paper is 72.
• The Printed Answer Book consists of 12 pages. The Question Paper consists of 4 pages.
Any blank pages are indicated.
INSTRUCTION TO EXAMS OFFICER / INVIGILATOR
• Do not send this Question Paper for marking; it should be retained in the centre or recycled.
Please contact OCR Copyright should you wish to re-use this document.
No calculator can
be used for this
paper
© OCR 2018 [H/102/2647] OCR is an exempt Charity
DC (SC/SW) 152199/2 Turn over
, 2
Section A (36 marks)
Simplify ^5a 2 ch # 2a 4 c -5 .
3
1 [2]
2 Find the equation of the line joining the points (-1, 9) and (2, -3), giving your answer in the form
y = mx + c. State the coordinates of the points where this line intersects the axes. [5]
3 Find the value of
(i) a2 14k ,
-2
[2]
(ii) ^8000h3 .
2
[2]
4 For the following equation, express x in terms of y.
x 2x + 1
= [4]
3y y+2
5 Find the coordinates of the point of intersection of the lines y = 4x + 3 and 3x + 2y = 9. [4]
Find the term that is independent of x in the binomial expansion of a - 3xk .
1 6
6 [3]
x
7 (i) Express 28 + 3 175 in the form a b, where a and b are integers and b is as small as possible. [2]
6 3 2 a+b 2
(ii) Simplify - , giving your answer in the form , where a, b and c are integers. [3]
5- 2 5+ 2 c
8 For each of the following pairs of sentences A and B, give a reason why the statement A + B is false and
write either ‘A & B ’ or ‘A % B ’ to show the correct relationship.
(i) A: n is positive.
B: n2 + 6 is positive. [2]
(ii) A: The diagonals of a quadrilateral bisect each other but not at right angles.
B: The quadrilateral is a rectangle but not a square. [2]
9 You are given that f(x) = ax3 + cx and that f(-1) = 3. You are also given that when f(x) is divided by (x - 4),
the remainder is 108. Find the values of a and c. [5]
© OCR 2018 4751/01 Jun18
, 3
Section B (36 marks)
10 (i) Express 3x2 - 9x + 5 in the form a(x + b)2 + c. Hence state the equation of the line of symmetry and
the y-coordinate of the minimum point of the curve with equation y = 3x2 - 9x + 5. [6]
(ii) Find the coordinates of the points where the graph of y = 3x2 - 9x + 5 intersects the axes. Give your
answers in an exact form. Hence state the solution of the inequality 3x2 - 9x + 5 < 0. [4]
11 You are given that f(x) = (2x + 5)(x2 - 5x + 4).
(i) Sketch the graph of y = f(x). [4]
(ii) You are given that g(x) = 2x3 - 5x2 - 17x + 48. Show that x = -3 is a root of g(x) = 0 and that it is the
only real root. [6]
(iii) Show that y = g(x) is a translation of y = f(x) by c m , finding the value of k.
0
[3]
k
12
y
A (7, 4)
x
O
Fig. 12
Fig. 12 shows a sketch of the circle with equation (x - 2)2 + (y + 1)2 = 50. You are given that the point
A (7, 4) lies on the circle.
(i) Write down the radius of this circle and the coordinates of its centre. [2]
(ii) The line L has equation y = 2x - 10 and passes through the point A (7, 4). Use algebra to find the
coordinates of the point B where the line L meets the circle again. Hence show that the perpendicular
distance from the centre of the circle to the line L is 5. [6]
(iii) Show that, when the line y = 2x + k is a tangent to the circle, k satisfies the equation
k2 + 10k - 225 = 0. [5]
END OF QUESTION PAPER
© OCR 2018 4751/01 Jun18
, 4
Oxford Cambridge and RSA
Copyright Information
OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders
whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright
Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (www.ocr.org.uk) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible
opportunity.
For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a
department of the University of Cambridge.
© OCR 2018 4751/01 Jun18
