H Graphs by simeonsrage

VIEWS: 39 PAGES: 33

									Q1.         Here are the equations of two straight lines.

                      y = 7x – 6

                      y = mx + c

      (a)    Write down the value of m if the lines are perpendicular.

             .........................................................................................................................

                                                 Answer .......................................................................
                                                                                                                                                      (1)

      (b)    Write down the values of m and c if the lines are reflections of each other in the y axis.

             .........................................................................................................................

                                                  Answer m = ...............................................................

                                                                c = ...............................................................
                                                                                                                                                      (2)
                                                                                                                                         (Total 3 marks)




                                                                                                                                                            Page 1 of 33
Q2.         On the grid below, indicate clearly the region defined by the three inequalities

                                                     x ≥ 1
                                                     y ≥ x – 1
                                                     x + y ≤ 7
      Mark the region with an R.

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................




                                                                                                                                           (Total 3 marks)




                                                                                                                                                             Page 2 of 33
Q3.        Find the equation of the straight line passing through the point (0, 5) which is perpendicular
      to the line
                                                              y=        x+3

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

                                                                          Answer .................................................
                                                                                                                                           (Total 2 marks)




Q4.       A is the point (2, 9)
      B is the point (8, 7)
      M is the midpoint of AB
      C is the point (8, 18)




                                                        Not drawn accurately




      Is MC perpendicular to AB?
      You must justify your answer.
      Do not use graph paper to answer this question.

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................
                                                                                                                                           (Total 4 marks)




                                                                                                                                                             Page 3 of 33
Q5.         Here are the equations of four straight lines.

             Line 1:          y=x+4

             Line 2:          y = 3x

             Line 3:          y = 3x + 5

             Line 4:          y = –x + 5

      (a)    Which two lines are parallel?

             .........................................................................................................................

                                                    Answer .............................. and ..............................
                                                                                                                                                      (1)


      (b)    Which two lines intersect the y axis at the same point?

             .........................................................................................................................

                                                    Answer .............................. and ..............................
                                                                                                                                                      (1)
                                                                                                                                         (Total 2 marks)




Q6.         The line PQ is shown on the grid.




                                                                                                                                                            Page 4 of 33
      (a)     Find the gradient of a line which is perpendicular to PQ.

              ..........................................................................................................................

              ..........................................................................................................................

              ..........................................................................................................................

                                                                        Answer .................................................
                                                                                                                                                        (3)


      (b)     Hence find the equation of the perpendicular bisector of the line PQ.

              ..........................................................................................................................

              ..........................................................................................................................

              ..........................................................................................................................

                                                                        Answer .................................................
                                                                                                                                                        (2)
                                                                                                                                           (Total 5 marks)




Q7.         Two gas supply companies have different ways of charging for the gas they supply.

                                          Alpha gasCO
            Fixed Charge                       £9.60
            Price per kilowatt hour of gas    First 5 kilowatt hours free then
                                              £1.30 for every kilowatt hour over 5.


                                           Beta gasCO
            Fixed Charge                       No fixed charge
            Price per kilowatt hour of gas     £1.50 for every kilowatt hour.




                                                                                                                                                              Page 5 of 33
Find the number of kilowatt hours after which Alpha gasCo becomes cheaper than Beta gasCo.
You might want to use some graph paper.
You must show your method clearly.

..................................................................................................................................

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                                                          Answer ................................... kilowatt hours
                                                                                                                                     (Total 4 marks)




                                                                                                                                                       Page 6 of 33
Q8.       A is the point (1, –2).
      B is the point (5, 4).




      Find the equation of the line perpendicular to AB, passing through the mid-point of AB.

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

      ..................................................................................................................................

                                                                          Answer .................................................
                                                                                                                                           (Total 4 marks)




                                                                                                                                                             Page 7 of 33
Q9.          Find the equation of the line through (0, –2) and (4, 18).

       ..................................................................................................................................

       ..................................................................................................................................

       ..................................................................................................................................

       ..................................................................................................................................

       ..................................................................................................................................

       ..................................................................................................................................

       ..................................................................................................................................

                                                                           Answer .................................................
                                                                                                                                            (Total 3 marks)




Q10.           (a)       A straight line has gradient 2 and passes through the point (0, 8)

                 Write down the equation of the line.

                                                                          Answer ..................................................
                                                                                                                                                         (1)

       (b)      Write down the gradient of a line that is perpendicular to the line                                           y = 4x

                                                                          Answer ..................................................
                                                                                                                                                         (1)


       (c)      Write down the equation of a line that is perpendicular to the line                                           y = 4x

                                                                          Answer ..................................................
                                                                                                                                                         (1)
                                                                                                                                            (Total 3 marks)




                                                                                                                                                               Page 8 of 33
Q11.         The line l on the graph passes through the points A (0, 3) and B (–4, 11).




       (a)   Calculate the gradient of the line l.

             ..........................................................................................................................

             ..........................................................................................................................

                                                                  Answer ......................................................
                                                                                                                                                       (2)


       (b)   Write down the equation of the line l.

             ..........................................................................................................................

                                                                  Answer ......................................................
                                                                                                                                                       (1)

       (c)   Write down the equation of the line which also passes through the point (0, 3) but is
             perpendicular to line l.

             ..........................................................................................................................

             ..........................................................................................................................

                                                                  Answer ......................................................
                                                                                                                                                       (2)
                                                                                                                                          (Total 5 marks)




                                                                                                                                                             Page 9 of 33
Q12.         (a)     Find the equation of the line AB.




             .........................................................................................................................

             .........................................................................................................................

             .........................................................................................................................

                                                                       Answer .................................................
                                                                                                                                                      (3)


       (b)   Give the y-coordinate of the point on the line with an x-coordinate of 6.

             .........................................................................................................................

             .........................................................................................................................

                                                                       Answer .................................................
                                                                                                                                                      (2)

       (c)   Write down the gradient of a line perpendicular to AB.

                                                                       Answer .................................................
                                                                                                                                                      (1)
                                                                                                                                         (Total 6 marks)




                                                                                                                                                        Page 10 of 33
Q13.        A sketch of the line 2y – x = 4 is shown.
       The line crosses the axes at A and B.




       (a)   Calculate the coordinates of A and B.

             ..........................................................................................................................

             ..........................................................................................................................

             ..........................................................................................................................

             ..........................................................................................................................

                                                      Answer A(..........., .............), B (............., ............)
                                                                                                                                                       (2)

       (b)   Calculate the gradient of the line AB.

             ..........................................................................................................................

             ..........................................................................................................................

             ..........................................................................................................................

             ..........................................................................................................................

                                                                       Answer .................................................
                                                                                                                                                       (2)
                                                                                                                                          (Total 4 marks)




                                                                                                                                                         Page 11 of 33
M1.   (a)
            –0.14(…)
                       B1




                            Page 12 of 33
(b)   –7
           B1

      –6
           B1
                [3]




                  Page 13 of 33
M2.   Correct region indicated
                   Award marks dependent upon number of lines drawn correctly and
                   extent of shading
                                                                                    B3
                                                                                         [3]




                                                                                           Page 14 of 33
M3.     Sight of –1   or –1.5 or –3/2
                      accept –1 / (     ) or –1 / 0.66 ... for M1 only
                                                                         M1

      y=–   x+5
                      oe eg. 2y = –3x + 10
                                                                         A1
                                                                              [2]




                                                                                Page 15 of 33
M4.       Mid point (5, 8)
                                                                     B1

      Gradient AB

                        Accept any indication
                        eg, 6 across, 2 down
                                                                     B1

      Attempt to find gradient MC or ‘stepping’ from M to C
                        M1 for using ‘Their gradient’
                                                                     M1

      Valid conclusion with justification.
      eg, No because gradient MC not 3
                       Accept any indication
                       eg, (5, 8) plus (3, 9) = (8, 17), mm’ ≠ – 1
                                                                     A1




                                                                          Page 16 of 33
Alt

Mid point (5, 8)
                                               B1

Use of Pythagoras
                                               M1

Three correct lengths


                                               A1

Correct conclusion at least 2 correct values
                                               A1
                                                    [4]




                                                      Page 17 of 33
M5.         (a)   2 and 3
                            oe
                                 B1

      (b)    3 and 4
                            oe
                                 B1
                                      [2]




                                        Page 18 of 33
M6.   (a)   Gradient of PQ =
                                                                                        M1

        Perp. grad. =


                   (=   oe)
                   Drawing method:
                   Perpendicular line drawn and attempt at finding its gradient M2
                                                                                     M1 dep




                   oe
                                                                                        A1




                                                                                              Page 19 of 33
(b)   y = (their –     )x+c
                                                          M1

      y=–       x+
                     oe
                     Accept 1.4 to 1.6 for   from graph
                                                          A1
                                                               [5]




                                                                 Page 20 of 33
M7.      9.60 + (x – 5) × 1.30
                       Alt: M1 for graph of Alpha parcels
                                                            M1

      = 1.50x
                      M1 for graph of Beta
                                                            M1

      3.10 = 0.20x
                      A1 accuracy
                                                            A1

      x = 15.5
                      A1 answer. Accept 16 but not 15.
                      T&I gets M1 iff taken as far as 15.
                      A1 for both schemes at 15
                      A1 for both schemes at 16
                      A1 conclusion
                                                            A1
                                                                 [4]




                                                                   Page 21 of 33
M8.       Attempt to find gradient of perpendicular line
                        Must be negative reciprocal of their gradient for AB
                                                                                M1

      (Gradient =) –
                       oe   eg –0.66, –0.67
                                                                                 A1

      Use of midpoint (3, 1)
                       Must be used either on the diagram with an attempt
                       at a perpendicular or in y = mx + c to find c.
                                                                                M1

      y=–    x +3
                       ft their gradient if first M1 awarded
                       Accept equivalents eg 3y + 2x = 9
                                                                               A1 ft
                                                                                       [4]




                                                                                         Page 22 of 33
M9.       Identifying –2 as constant term in equation y = mx + c
                                                                   B1

      Gradient =
                       Attempt to find gradient
                                                                   M1

      y = 5x – 2
                       oe
                                                                   A1
                                                                        [3]




                                                                          Page 23 of 33
M10.         (a)   y = 2x + 8
                                    B1



       (b)   –
                         oe –0.25
                                    B1




                                         Page 24 of 33
(c)   y=–   (+ c) or x = –4y (+ c)
                                     B1 ft
                                             [3]




                                               Page 25 of 33
M11.   (a) Any correct attempt at
        (y-step) ÷ (x-step)
                    Might be marked on diagram
                                                 M1

       –2
                                                 A1




                                                      Page 26 of 33
(b)   y = -2x + 3
                    ft. their gradient
                                         B1




                                              Page 27 of 33
(c)   Gradient =
                   Attempt at gradient of perpendicular line, ft. from their gradient in
                   part (a) using (m1 × m2 = –1) as long as there is no contradiction
                   between parts (a) and (b)
                                                                                            M1

      y = ½x + 3
                   or equivalent
                                                                                           A1 ft
                                                                                                   [5]




                                                                                                     Page 28 of 33
M12.   (a)   Intercept = 9
                   i.e. identifying that 9 is the constant
                   term in the equation.
                                                             B1



       Gradient = –      =–3
                   Any attempt at gradient for M1.
                   i.e ±9/ ±3
                                                             M1

       y = – 3x + 9
                   Accept equivalent forms.
                   NB y = 3x + 9 is B1, Ml, A0
                                                             A1




                                                                  Page 29 of 33
(b)   Substitute x = 6 into their equation
                  Or recognise that y-step from 0 to 3
                  is the same as 3 to 6. eg sight of 9.
                  M1 can be implied by answer only.
                                                          M1

      –9
                                                          A1




                                                               Page 30 of 33
(c)
      ft on their gradient in (a), Allow an
      'embedded' answer in an equation,

      e.g. y =    x+9
                                              B1 ft
                                                      [6]




                                                        Page 31 of 33
M13.   (a)   A(–4, 0)
                   or 2y = 4 and – x = 4 seen
                                                B1

       B(0,2)
                   SC1 reversed answers
                                                B1




                                                     Page 32 of 33
(b)   (their difference in y’s) ÷ (± their difference in x’s)
                    or attempt to rearrange 2y – x = 4 to y = 0.5x + 2
                    and 2y = 4 + x or y – 0.5x = 2 seen
                    ft condone A(0, ?) and/or B(?, 0) from (a)
                                                                          M1



      0.5 or
                  ft for 0 < gradient < 1 only
                                                                         A1 ft
                                                                                 [4]




                                                                                   Page 33 of 33

								
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