AS Revision 4

This is intended to be (and expected to be) easy. You are expected to have done the sheets previous to this one, completely. That should have meant asking questions or extended work to establish complete understanding. I keep telling you; do the work I set and you will get an A grade.

1   Write the equation of the line through (3, -4) and (-4,10) in the ax+by=c form.         [3]

2   Write the equation of the line perpendicular to 2x-5y=16 that passes through (3,-4).  [3]

3   Show that x² + 18x + y² – 6y + 108 = 0 is a circle by finding its centre and radius.  [4]

4   Identify and classify the stationary values of y = x+ 2x³ – 2x² – 6x +79.                 [6]

5   Differentiate the following with respect to x (so find d/dx of what I have written)

a)   15x⁻² + 12x-1/2                                                                                                    [3]
b)   y
³ + t                                                                                                                    [3]
c)   (x
² + 5)                                                                                                               [4]
d)   (
x² + 1)    / (3x-2~)                                                                                               [4]

6   The line y = 3x -7 does not pass through the origin.
Write an expression for the distance from a general point on the line to the origin.
Show that this distance is a minimum when x = 7/2                                             [6]

Total [36]

If you have finished and others are far from finished, here is some extension work. Explore...

7   Investigate shapes whose perimeter is numerically equal to their area.              [8 plus]

8   A farmer has a wall to which he wishes to attach a fence. The length of fencing available is fixed. What rectangle will maximise the area enclosed?  What more general shape will enclose even more area?   Can you prove this (rather than just show it?).            [around 10]

9  The curve y= x² - 3x + 4 does not pass through (1,0). Just how close does it get? Do you get a similar answer for (p, p-1)?                                                                           [about 14]

10  I suggest that the number N = 4n+1 for integer n is either prime or is the sum of at least two pairs of squares. A counter-example would disprove my conjecture and a proof would confirm it.                                                                                                          [lots of marks]
For example n=15 makes N=61 which is prime, but n=16, N=65 and 65 = 8
²+1² = 7²+4².

© David Scoins 2017