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  • Core 1

    Jan 2012

    June 2012

    Jan 2013

    June 2013

  • Centre Number Candidate Number

    Surname

    Other Names

    Candidate Signature

    General Certificate of EducationAdvanced Subsidiary ExaminationJanuary 2012

    Mathematics MPC1

    Unit Pure Core 1

    Friday 13 January 2012 9.00 am to 10.30 am

    For this paper you must have:* the blue AQA booklet of formulae and statistical tables.

    You must not use a calculator.

    Time allowed* 1 hour 30 minutes

    Instructions* Use black ink or black ball-point pen. Pencil should only be used fordrawing.

    * 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-handmargin.

    * You must answer the questions in the spaces provided. Do not writeoutside the box around each page.

    * Show all necessary working; otherwise marks for method may belost.

    * Do all rough work in this book. Cross through any work that you donot want to be marked.

    * The use of calculators is not permitted.

    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.

    * You do not necessarily need to use all the space provided.

    For Examiners Use

    Examiners Initials

    Question Mark

    1

    2

    3

    4

    5

    6

    7

    TOTAL

    P45841/Jan12/MPC1 6/6/6/ MPC1(JAN12MPC101)

  • 2Answer all questions in the spaces provided.

    1 The point A has coordinates 6, 4 and the point B has coordinates 2, 7 .

    (a) Given that the point O has coordinates 0, 0 , show that the length of OA is lessthan the length of OB. (3 marks)

    (b) (i) Find the gradient of AB. (2 marks)

    (ii) Find an equation of the line AB in the form px qy r , where p, q and r areintegers. (3 marks)

    (c) The point C has coordinates k, 0 . The line AC is perpendicular to the line AB.Find the value of the constant k. (3 marks)

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    QUESTION

    PART

    REFERENCE

  • 42 (a) Factorise x2 4x 12 . (1 mark)

    (b) Sketch the graph with equation y x2 4x 12 , stating the values where the curvecrosses the coordinate axes. (4 marks)

    (c) (i) Express x2 4x 12 in the form x p2 q , where p and q are positive integers.(2 marks)

    (ii) Hence find the minimum value of x2 4x 12 . (1 mark)

    (d) The curve with equation y x2 4x 12 is translated by the vector 32

    .

    Find an equation of the new curve. You need not simplify your answer. (2 marks)

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  • 63 (a) (i) Simplify 32

    p 2. (1 mark)

    (ii) Show that 32

    p 1 2 3 2p 2 is an integer and find its value. (4 marks)

    (b) Express4

    5

    p 7 2p2

    5

    p 2p in the form mn

    p, where m and n are integers. (4 marks)

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  • 84 The curve with equation y x5 3x2 x 5 is sketched below. The point O is atthe origin and the curve passes through the points A1, 0 and B1, 4 .

    (a) Given that y x5 3x2 x 5 , find:

    (i)dy

    dx; (3 marks)

    (ii)d2y

    dx2. (1 mark)

    (b) Find an equation of the tangent to the curve at the point A1, 0 . (2 marks)

    (c) Verify that the point B, where x 1 , is a minimum point of the curve. (3 marks)

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    y

    A

    1 O xB1, 4

    (08)

  • 10

    4 (d) The curve with equation y x5 3x2 x 5 is sketched below. The point O is atthe origin and the curve passes through the points A1, 0 and B1, 4 .

    (i) Find

    11x5 3x2 x 5 dx . (5 marks)

    (ii) Hence find the area of the shaded region bounded by the curve between A and B and

    the line segments AO and OB. (2 marks)

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    A

    1 O xB1, 4

    (10)

  • 12

    5 The polynomial px is given by px x3 cx2 dx 12 , where c and d areconstants.

    (a) When px is divided by x 2 , the remainder is 150 .

    Show that 2c d 65 0 . (3 marks)

    (b) Given that x 3 is a factor of px, find another equation involving c and d.(2 marks)

    (c) By solving these two equations, find the value of c and the value of d. (3 marks)

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  • 14

    6 A rectangular garden is to have width x metres and length x 4 metres.

    (a) The perimeter of the garden needs to be greater than 30 metres.

    Show that 2x > 11 . (1 mark)

    (b) The area of the garden needs to be less than 96 square metres.

    Show that x2 4x 96 < 0 . (1 mark)

    (c) Solve the inequality x2 4x 96 < 0 . (4 marks)

    (d) Hence determine the possible values of the width of the garden. (1 mark)

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  • 16

    7 A circle with centre C has equation x2 y2 14x 10y 49 0 .

    (a) Express this equation in the form

    x a2 y b2 r2 (3 marks)

    (b) Write down:

    (i) the coordinates of C ;

    (ii) the radius of the circle. (2 marks)

    (c) Sketch the circle. (2 marks)

    (d) A line has equation y kx 6 , where k is a constant.

    (i) Show that the x-coordinates of any points of intersection of the line and the circle

    satisfy the equation k 2 1x2 2k 7x 25 0 . (2 marks)

    (ii) The equation k 2 1x2 2k 7x 25 0 has equal roots. Show that

    12k 2 7k 12 0 (3 marks)

    (iii) Hence find the values of k for which the line is a tangent to the circle. (2 marks)

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    QUESTION

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    REFERENCE

    Turn over

    s

  • Centre Number Candidate Number

    Surname

    Other Names

    Candidate Signature

    General Certificate of EducationAdvanced Subsidiary ExaminationJune 2012

    Mathematics MPC1

    Unit Pure Core 1

    Wednesday 16 May 2012 9.00 am to 10.30 am

    For this paper you must have:* the blue AQA booklet of formulae and statistical tables.

    You must not use a calculator.

    Time allowed* 1 hour 30 minutes

    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 each question in the space provided for that

    question. If you require extra space, use an AQA supplementaryanswer book; do not use the space provided for a different question.

    * 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.

    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.* You do not necessarily need to use all the space provided.

    For Examiners Use

    Examiners Initials

    Question Mark

    1

    2

    3

    4

    5

    6

    7

    TOTAL

    P50000/Jun12/MPC1 6/6/6/ MPC1(JUN12MPC101)

  • 2Answer all questions.

    Answer each question in the space provided for that question.

    1 Express5

    3

    p 62

    3

    p 3 in the form m n3

    p, where m and n are integers. (4 marks)

    Answer space for question 1

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    QUESTION

    PART

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    (02)

  • 42 The line AB has equation 4x 3y 7 .

    (a) (i) Find the gradient of AB. (2 marks)

    (ii) Find an equation of the straight line that is parallel to AB and which passes through

    the point C3, 5, giving your answer in the form px qy r , where p, q and rare integers. (3 marks)

    (b) The line AB intersects the line with equation 3x 2y 4 at the point D. Find thecoordinates of D. (3 marks)

    (c) The point E with coordinates k 2, 2k 3 lies on the line AB. Find the value ofthe constant k. (2 marks)

    Answer space for question 2

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  • 63 The polynomial px is given by

    px x3 2x2 5x 6

    (a) (i) Use the Factor Theorem to show that x 1 is a factor of px . (2 marks)

    (ii) Express px as the product of three linear factors. (3 marks)

    (b) Verify that p0 > p1 . (2 marks)

    (c) Sketch the curve with equation y x3 2x2 5x 6 , indicating the values wherethe curve crosses the x-axis. (3 marks)

    Answer space for question 3

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    (06)

  • 7Answer space for question 3

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    QUESTION

    PART

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  • 84 The diagram shows a solid cuboid with sides of lengths x cm, 3x cm and y cm.

    The total surface area of the cuboid is 32 cm2.

    (a) (i) Show that 3x2 4xy 16 . (2 marks)

    (ii) Hence show that the volume, V cm3 , of the cuboid is given by

    V 12x 9x3

    4(2 marks)

    (b) FinddV

    dx. (2 marks)

    (c) (i) Verify that a stationary value of V occurs when x 43. (2 marks)

    (ii) Findd2V

    dx2and hence determine whether V has a maximum value or a minimum

    value when x 43. (2 marks)

    Answer space for question 4

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    y

    3x

    x

    (08)

  • 9Answer space for question 4

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  • 11

    Answer space for question 4

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    (11)

  • 12

    5 (a) (i) Express x2 3x 5 in the form x p2 q . (2 marks)

    (ii) Hence write down the equation of the line of symmetry of the curve with equation

    y x2 3x 5 . (1 mark)

    (b) The curve C with equation y x2 3x 5 and the straight line y x 5 intersectat the point A0, 5 and at the point B, as shown in the diagram below.

    (i) Find the coordinates of the point B. (3 marks)

    (ii) Find

    x2 3x 5 dx . (3 marks)

    (iii) Find the area of the shaded region R bounded by the curve C and the line segment AB.

    (4 marks)

    Answer space for question 5

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    y

    xO

    AR

    B

    (12)

  • 15

    Answer space for question 5

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    (15)

    QUESTION

    PART

    REFERENCE

  • 16

    6 The circle with centre C5, 8 touches the y-axis, as shown in the diagram.

    (a) Express the equation of the circle in the form

    x a2 y b2 k (2 marks)

    (b) (i) Verify that the point A2, 12 lies on the circle. (1 mark)

    (ii) Find an equation of the tangent to the circle at the point A, giving your answer in the

    form sx ty u 0 , where s, t and u are integers. (5 marks)

    (c) The points P and Q lie on the circle, and the mid-point of PQ is M7, 12.

    (i) Show that the length of CM is n5

    p, where n is an integer. (2 marks)

    (ii) Hence find the area of triangle PCQ. (3 marks)

    Answer space for question 6

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  • 18

    7 The gradient,dy

    dx, of a curve C at the point x, y is given by

    dy

    dx 20x 6x2 16

    (a) (i) Show that y is increasing when 3x2 10x 8 < 0 . (2 marks)

    (ii) Solve the inequality 3x2 10x 8 < 0 . (4 marks)

    (b) The curve C passes through the point P2, 3 .

    (i) Verify that the tangent to the curve at P is parallel to the x-axis. (2 marks)

    (ii) The point Q3, 1 also lies on the curve. The normal to the curve at Q and thetangent to the curve at P intersect at the point R. Find the coordinates of R.

    (7 marks)

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    QUESTION

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  • Centre Number Candidate Number

    Surname

    Other Names

    Candidate Signature

    General Certificate of EducationAdvanced Subsidiary ExaminationJanuary 2013

    Mathematics MPC1

    Unit Pure Core 1

    Monday 14 January 2013 9.00 am to 10.30 am

    For this paper you must have:* the blue AQA booklet of formulae and statistical tables.

    You must not use a calculator.

    Time allowed* 1 hour 30 minutes

    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 each question in the space provided for that

    question. If you require extra space, use an AQA supplementaryanswer book; do not use the space provided for a different question.

    * 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.

    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.* You do not necessarily need to use all the space provided.

    For Examiners Use

    Examiners Initials

    Question Mark

    1

    2

    3

    4

    5

    6

    7

    8

    TOTAL

    P56476/Jan13/MPC1 6/6/6/ MPC1(JAN13MPC101)

  • 2Answer all questions.

    Answer each question in the space provided for that question.

    1 The point A has coordinates 3, 2 and the point B has coordinates 7, k .

    The line AB has equation 3x 5y 1 .

    (a) (i) Show that k 4 . (1 mark)

    (ii) Hence find the coordinates of the midpoint of AB. (2 marks)

    (b) Find the gradient of AB. (2 marks)

    (c) A line which passes through the point A is perpendicular to the line AB. Find an

    equation of this line, giving your answer in the form px qy r 0 , where p, qand r are integers. (3 marks)

    (d) The line AB, with equation 3x 5y 1 , intersects the line 5x 8y 4 at thepoint C. Find the coordinates of C. (3 marks)

    Answer space for question 1

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