FPM 20130213 [1] | Scoins.net | DJS

## FPM 20130213 [1]

At BSB we have a class studying Edexcel Further Pure Maths Paper from in Year 11 as an accelerated class. I converted the FPM papers to Word form and took out all the blank pages from the .pdf that one can download from the internet. For copyright reasons I’m not sharing those - I made a page and then changed my mind about the propriety of doing that.

However, having made the files, you could ask me (nicely) for a copy, couldn’t you?

More pertinently, I then started writing questions in the style of Edexcel. I expect that I will re-order these into paper-sized groups; meanwhile, you are welcome to some practice. Do tell me how I’m able to make tabs work inside iWeb - I’m using spaces, which is why the formatting doesn’t line up.

The scope of the course syllabus makes C1 and C2 look slow and ponderous. The work goes so quickly (I’m happy, but the kids are not) that students for whom maths has always been easy find themselves coming apart and adrift - a state of affairs that sets them up very well for Sixth Form work, but it will provide a GCSE grade that will be lower than they’d expect. I must persuade them to make sure effort goes to other subjects (that will be dropped at the end of Y11) first.

Suffice it to say that if you’re looking for C1 and C2 revision, these will be more than adequate.

DJS 20130213

20130228 adjusted Q2. Q12 needed adjustment (10² to 12²), though it worked.

Coordinate Geometry:

1  Show that 7x + 3y = 13 and 6x – 14y = k are perpendicular.                                    [4]

These lines meet when x=1. Find k.                                                                                 [4]

2    Two lines, with equations 3x + 2y = 5 and 12x – ay = k meet at P, for constants a & k.

a Express the angle between these two lines as a difference of tangents.                   [2]

b If the lines meet at right angles, state the value of a.                                                  [2]

c For the case where the lines are perpendicular, find the value of k that places the intersection of the two lines nearest the origin                                                                 [6]

3   The points A and B have coordinates (1,9) and (13,3) respectively.

a  Find an equation of AB, giving your answer in the implicit form ax + by = c, where a,b,c are rational numbers.                                                                                                        [3]

P is a point on AB and between them such that AP:PB = 2:1. D is the point on the x-axis such that APD is a right angle. C is on DP produced so that P bisects CD.

b   Find the equation of DP in its implicit form. Find Point C[4,2]

c   Express the length of AB and CD exactly.                                                                  [4]

d   Identify ACBD as a quadrilateral, and give its area.                                                   [2]

4    The points A and B have coordinates (1,4) and (13,8) respectively.

a   Find an equation of AB, giving your answer in the form y = ax + b, where a,bR.     [3]

The line l is the perpendicular bisector of AB.

b   Find the equation of l in the form ax+by=c where a,b,c are rationals.                         [4]

c   The point C has coordinates (5,q). Given that C lies on l find the value of q.             [2]

The line l meets the x-axis at the point D. (d) Find the exact area of the kite ACBD.      [4]

e   Identify the best description of ACBD as a quadrilateral, showing why.                      [2]

Co-ordinate geometry and calculus

5 Line L has equation  3x+y = 5

a Write an expression for the distance of a point from the origin.                                 [1]

b By eliminating one variable from your expression and differentiating (or otherwise)

find the closest point, P, that L comes to the origin                                                        [5]

c Write the equation of the line OP and show that it is perpendicular to L.                  [4]

6   A curve C, with equation y2 = 8x and the line l intersect at the point A with coordinates (a, 2a), a ≠ 0. Sketch curve C and line l in the positive quadrant. The line l has gradient −2 and intersects the x-axis at the point B.

a    Find the non-zero value of a.                                                                                    [2]
b    Find the x-coordinate of B. State angle OAB                                                          [2,1]
The region AOB is rotated through 360° about the x-axis.

c    Find the volume of the solid generated, in the form k2nπ (k∈Q, n∈N).       [5]

Curves and calculus

7  a Differentiate x³-4x²-3x+18 with respect to x.                                                            [2]

The curve with equation y = x³-4x²-3x+18  has stationary points at C and D.

b   Find the position values of both C and D  and identify point D as the minimum          [6]

c   Use position D to factorise x³-4x²-3x+18  completely.                                                 [4]

d   The curve meets the x and y axes at A and B respectively. Show that A is (-2,0)      [1]

e   Sketch y= x³-4x²-3x+18 on axes -5<x<5, showing the positions of A,B,C,D              [2]

f    By integration find exactly the area ABCD, a little over 50 square units                      [4]

8  A curve has equation     y = x³ - 6x² - x + m, where m is a positive integer. The curve crosses the x-axis at the point (-2,0).

a   Show that m = 30                                                                                                       [2]

b   Factorise y = y = x³ - 6x² - x + 30 completely                                                             [3]

c   The curve also crosses the x-axis at two other points. Write down the x-coordinate of each of these two points                                                                                                 [1]

d   Sketch the curve.                                                                                                       [3]

Series

9    The third term of an arithmetic series is 50 and the sum of the first 9 terms of the series is 720

a  Calculate the common difference of the series.                                                         [4]

The sum of the first n terms of the series is Sn.         Given that 1000< Sn <1500
(b) find the set of possible values of
n.                                                                           [6]

10  The sum of the first and third terms of a geometric series G is 70

The sum of the second and third terms of G is 60
a   Find the possible vales for the common ratio of G                                                 [4]

b   Find the corresponding values for the infinite sum.                                                [2]

c   Find the least value of n for which Sn > S                                                               [6]

11  The sum of the second and fourth terms of a geometric series G is 181

The sum of the second and third terms of G is 10
a   Find the common ratio of G, given that G is convergent                                           [4]

b   Find the corresponding values for the infinite sum.                                                   [2]

c   There is another series, H, for which the same conditions apply, the sum of the second and fourth terms add to 181 and the sum of the second and third terms add to 10.

d   Write down the common ratio, find the first term                                                    [1,2]
e   Find the least value of n for which S
n > 106                                                              [3]

* There is a general form for this question; this uses the tenth member (181,10) of the set I found.

12.The sum of the 15th and 17th terms of a geometric series is 122. The sum of the 15th and 16th terms is 12
a   Find the common ratio of G, given that G is convergent                                            [4]

b   Find the corresponding values for the infinite sum.                                                   [2]

c   There is another series, H, for which the same conditions apply, that the sum of the 15th and 17th terms is 122 and the sum of the 15th and 16th terms is 12.

d   Write down the common ratio, find the first term exactly                                        [1,2]
e   Find the value of n for which S
n is nearest two million.                                              [3]

* There is a general form for this question; this uses the sixth member (61,6) of the set. More in Series at A-level.

2   a=18, closest is (15/16. 10/16) => k=0

Q3 CD is 2x-y+13=0, C is (11.5, 10); 75 sq units

Q4 C is at (5,12)

Q5 min at x=3/2.

Q6  y=8-2x;   Vol in two parts, (16+32/3)π

Q7  (-2,), (-1/3, 18 14/27), (0, 18), (3,0) 52 1/12

Q8 (x+2)(x-3)(x-5);   turning points at 0- and 4+

Q9 n=12±1, (a,d)=(20,15)

Q10  one series is 43.86, 33.86, 26.14 ...  and the other starts 1.14, -8.86, 68.86... only th efirst has an infinite sum, close to 57.

Q11  r = 0.9 [a=1000/171] converges to about 58.5 or r=19 a=10/551 and n>7.035

Q12   r=5/6  [a=772.88], S=4637.26;   or r=11, a=11-14, n=21, S21=1.948x106.

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