This booklet covers all of AS(first year) pure mathematics for A level, with simplified notes and questions ranging from easy medium or hard for each pure topic. Essential if you want to smash your summer math paper as pure makes up the majority of the papers and this booklet will help build a sol...
AS PURE MATHS REVISION BOOKLEI
Quadratics
tb-42
To sove a" +hr+c =0, factorise, use the formula r =- or cakculalor.
2a
real roots depanding whether
The discriminant is
-4ac There will
be wo distinct roots, equal roots
the discnminant is positive, zero or negative.
or no on
Completing the square -replace x+bx with ( x + - ( orx -bx with ( x - ) - G
Graph of y = ax
+bx +cis aparabolawith U shape when a>0, and n shape when
The graph crosses the y axis when x = 0, s0y C
a < 0.
The graph crosses the x axis wheny = 0, so sove ar + bx t e =0.
When In completed square form, the vertex of y=a(x+ p) +q has coordinates (-p.9), the equation of the
ine of symmetry isX-p
For intervals of the number ine, (a, b) ={xe R:a<1< b}; la,b])= {xe R:asIsb
o sove linear inegualites, treal as equations, but if you divide or multiply by a negalive, reverSe the sign.
» 0 when the graph is above the xaxis,
Tosolve aquadraticinequality, sketchthe graphofthequadralic -if's
R s O W n e nthe graph is below the x axis.
Epress r-8+ 1gin the form (-pY* 9. herep and q are inlegers.
)ence. or otherMse. show that the equation - 8r + 19 - 0 h a s no real so ulions
Skech te graph ofy--8 19. stating the coordinates af the mnimumpoint and the
point where the graph erosses the - a s
Find the values of t for which the ecuation -2E-1)x-2t*-7-0 has equal roots
3)Factorse9-8r-
b) Show that
25-r 4r
can be w t e n as
9-8
-
9 -8r-*
A Gune has equatonr
Wrte down the equation of ts hne of symmmelry.
F i n d the coordinales of fs vertex
) Sketch the curve, indicating the vaues of the interoeps on the x-axs and the -ads.
ind the values for x which saisty both inequalites r 2 r > 8 and 3{2r-1)s15.
Surds,Indices and Equations
To simplity surds, use yab vavb and
To ratonalise the denominator of e. g multiply top & bottom byy2.
To rationalise the denorminator of e. 9 mutiply top & bottom by 4 + v3.
Rules of Indices:
a" " =am*n; am; («"m)" = am*; a - 1; «" = a = Va; a = (Va)".
For simultaneous equations, ane linear & one quadratic, rearrange the linear one to make x or y the subject.
substtute into the
quadralic, simplily & solve.
intersection of curves, solve If line is a tangent to a curve, the equalion
To find the points of simultaneously. a
oained by solving them simullaneously wil have egual rools.
Disguised quadratics: . g. 2"" + 32-0;et y so 2y43y-2-0.
Solve to Iind y & henca x.
, Exampless
a) Wrde dow the valuesofp.q and r. gven tha
54
b F n d the value
of *r for which
Show that i s an inleger and find ts value
3
A rectangte has length 19-53jcmand ares (15-73 |cm
Shew
tha: the width of the rectangie is (-n3).where m and n are integers
t a s i n g the siubsituton 3. show that the equation
9-3''-2 0
can be witen as
T-1-2)-0
br Hence showthat the aqualion
-3 "2-0
has a solution
and. by using logarithms, find the other soluion grving your answer to four decma
places.
A arde wth centre Chas equation
--10y 20-0
A line has equation
a) Show that the -coordinate of any point of mtersecion at the ime and the arce satisfes
the equationx-Ar-4-0
0 e n e o w that the ine is a tangent to tne circle and find the coorcinates of the point of
The curve Chas equalion
tr-3)
e r e ksa constant
The ine L has equation
-2r+2
how hat the x-cnordienates of any points of ntersection of the curve Cwih the ine L satisly
the equaton
k - 2 r 3k-2-D
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