By Hugh Neill, Douglas Quadling, Julian Gilbey
Written to compare the contents of the Cambridge syllabus. natural arithmetic 1 corresponds to unit P1. It covers quadratics, services, coordinate geometry, round degree, trigonometry, vectors, sequence, differentiation and integration.
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Extra resources for Advanced Level Mathematics: Pure Mathematics 1
If a and b have the same sign the axis of symmetry is to the left of the y-axis; if a and b have opposite signs the axis of symmetry is to the right of the y -axis. If a is positive the vertex is at the lowest point of the graph; if a is negative the vertex is at the highest point. The larger the size of Ia I the m:ore the graph is elongated, that is, lengthened in they-direction. 7 The point of intersection of two graphs The principle for finding the point of intersection of two curves is the same as that for finding the point of intersection of two graphs which are straight lines.
L l - Fig. 12 2x - 6 and y = 12 + x - 2x 2 . Solving these equations simultaneously gives x 2 - 2x - 6. = 12 + x - 2x 2 , whicli is 3x 2 - 3x -18 = 0. Dividing by 3 gives x 2 - x - 6 = 0, which factorises as (x + 2)(x - 3) = 0, giving x = -2 or x = 3. ·' . ~ubstituting these values in either equation to find y gives the points of ··'intersection as (-2,2) and (3,-3). ~v '-);--;·,~T ~~ -.. ~ ~~li'~:~~l"'~';~ry·~, 1 Find the point or points of intersection for the following lines and curves. (a) x=3andy=x 2 +4x-7 (c) y = 8 and y = x 2 +2x (b) y=3andy=x 2 -5x+7 (d) y+3 = 0 and y = 2x 2 +Sx-6 2 Find the points of intersection for the following lines and curves.
Also, as 41 is not a perfect square, the roots are irrational. (b) As a=2, b=-3 and c=-5, b 2 -4ac=(-3) 2 -4x2x(-5)=9+40=49. The discriminant is positive, so the equation 2x 2 - 3x - 5 = 0 has two roots. Also, as 49 is a perfect square, the roots are rational. (c) b 2 -4ac = (~4) 2 -4x2x5=16-40 =-24. As the discriminant is negative, the equation 2x 2 - 4x + 5 = 0 has no roots. 57 (r-~ 58 PuRE MATHEMATICS 1 2 (d) b 2 - 4ac = (-4) - 4x2x2=16-16 = 0. As the discriminant is zero, the equation 2x 2 - 4x + 2 = 0 has only one (repeated) root.