Further Pure 1 - Mathsbox

1 Further pure 1 Summary Notes1. Roots of Quadratic EquationsFor a quadratic equationax2 + bx + c = 0 with roots and Sum of the roots Product of rootsab =caa +b = ba If the coefficients a,b and c are real then either and are real or and arecomplex conjugatesOnce the value + and have been found, new quadratic equations can be formedwith roots :Roots 2 and 2 Sum of roots(a +b )2 2abProduct of roots(ab )2 3 and 3 Sum of roots(a +b )3 3ab (a +b )Product of roots(ab )31aand1bSum of rootsa +babProduct of roots1abThe new equation becomesx2 (sum of new roots)x + (product of new roots) = 0 The questions often ask forinteger coefficients!Don t forget the = 0 ExampleThe roots of the quadratic equation3x are and.

2 Determine a quadratic equation with integer coefficients which has roots2+ 4x 1 = 0a3b and ab 3a +b = 43ab = 13 Step 1 :Step 2 :Sum of new rootsa3b + ab3 = ab (a2 + b 2)= 1 ((a +b)23 2ab )= 13 169+2 3 = 3 : Product of rootsa3b ab 3 = a4b4 = (ab )4 =181 Step 4 : Form the new equationx2 +22x +12781= 081x2 + 66x + 1 = 02. Summation of SeriesThese are given in the formula bookletREMEMBER : r =1n1 = nS 5r2 = 5S r2 Always multiply brackets before attemptingto evaluate summations of series Look carefully at the limits for the summation =r =1=1=7-620r20r =r =1=1+1=n-n2nr2nr Summation ofODD / EVEN numbersExample : Find the sum of the odd square numbers from 1 to 49 Sum of odd square numbers= Sum of all square numbers Sum of even square numbersSum of even square numbers = 22+42 +.

3 482= 22 (12 + 22+ 32 +..242)= 4 r =124r 2 Sum of odd numbers between 1 and 49 is r 2 -491 r =124r 24= 1 49 50 99 6 4 16 24 25 49 = 40125 19600= Matrices acb d a b ORDER2 x 1 2 x 2 Addition and Subtraction must have the same order acb d egf h = a ec gb f d h Multiplicationa 3 b= 3a 3b 354 2 1310 0 = 3 x 1 + 4 x 35 x 1 + 2 x 33 x 10 + 4 x 0 5 x 10 + 2 x 0 = 151130 50 NB : Order matters Do not assume that AB = BADo not assume that A2 B2 = (A-B)(A+B) Identity Matrix I = 1 AI = IA =A 001 4. Transformations Make sure you know the exact trig ratiosAngle sin cos tan 0 0 1 030 321345 1212160 32 390 1 0 Undefined To calculate the coordinates of a point after a transformationMultiply the Transformation Matrix by the coordinateFind the position of point (2,1) after a stretch of Scale factor 5 parallel to the x-axis(10,1) 0 1 1 = 1 5 0 2 10 To Identify a transformation from its matrixConsider the points (1,0) and (0,1)(1,0) (4,0) (0,1) (0,2)

4 400 2 Stretch Scale factor 4 parallel to the x-axis and scale factor 2 parallel to the y-axisStandard TransformationsREFLECTIONS1 in the y-axis 1 in the x-axis Reflection in y = x 01 Reflection in the line y= (tan )x 001 cos 2qsin 2qsin 2q cos 2q In the formula bookletIf all elements have the samemagnitude then look at 2 = 45 (reflection in y = ( )x ) as oneof the transformations 001 1 0 ENLARGEMENT k00 k Scale factor kCentre (0,0)STRETCH a Scale factora parallel to the x-axisScale factorb parallel to the y-axis 0 b 0 ROTATION cosRotation through anti-clockwise about origin (0,0) Rotation through Clockwise about origin (0,0) si co In the formula booklet qsin qsinq cosq cos qn qsinqsq If all elements have the same magnitude then a rotation through 45 is likely to be one ofthe transformations (usually the second)ORDER MATTERS !

5 !!! make sure you multiply the matrices in the correct orderA figure is transformed byM1 followed byM2 MultiplyM2 Graphs of Rational FunctionsLinear numerator and lineardenominator1 horizontal asymptote1 vertical asymptote2 distinct linear factors in thedenominator quadraticnumerator2 vertical asymptotes1 horizontal asymptoteThe curve will usuallycross the horizontalasymptote2 distinct linear factors in thedenominator linear numerator2 vertical asymptotes1 horizontal asymptotehorizontal asymptote isy = 0 Quadratic numerator quadraticdenominator with equal factors1 vertical asymptote1 horizontal asymptoteQuadratic numerator with no realroots for denominator (irreducible)The curve does not have avertical asymptotey = 4x 8x + 3y =(x 3)(2x 5)(x + 1)(x + 2)y = 2x 93x2 11x + 6y =(x 3)(x + 3)(x 2)2y =x2+ 2x 3x2 + 2x + 6 Vertical Asymptotes Solve denominator = 0 to find x = a, x = b etcHorizontal Asymptotes multiply out any brackets look for highest power of x in thedenominator and divide all terms by this as x goes to infinity majority of terms willdisappear to leave either y = 0 or y = aTo find stationary pointsrearrange to form a quadratic ax + bk =x2+ 2x 3x2 + 2x + 6 *2x + c = 0b2 4ac < 0 b2 4ac = 0 b2 4ac > 0the line(s) y = k stationary point(s) the line(s)

6 Y = kdo not intersect occur when y = k intersect the curvethe curve subs into * to find x subs into * to find xcoordinate The questions areunlikely to lead to simple or single solutions such as x > 5 soSketch the graph (often done already in a previous part of the question)Solve the inequalityThe shaded area iswhere y < 2So the solution isx < 0 , 1 < x < 2 , x > 116. Conics and transformations(x + 1)(x + 4)(x 1)(x 2)< 2 You must learn the standard equations and the key features of each graph type Mark on relevant coordinates on any sketch graphParabolaEllipseHyperbolaRectangular HyperbolaStandard equations are given in the formula booklet but NOT graphsy2 = 4axx2a2+y2b2= 1 x a 2+ y b 2= 1x2a2 y2b2= 1 x a 2 y b 2= 1xy = c2 You may need tocomplete the squarex2 4x + y2 6y = 12(x 2)2 4 + (y 3)2 9 = 12(x 2)2 + (y 3)2 = a Replace x with (x a) Circle radius 1 centre (2.)

7 3) Replace y with (y - b) (x - 2)2 + (y - 3)2 = 1 b Reflection in the line y = xReplace x with y and vice versaStretch Parallel to the x-axis scale factor a Replace x withaxStretch Parallel to the y-axis scale factor b Replace y withbyDescribe a geometrical transformation that maps the curve y2=8x onto the curvey2=8x-16 2 0 x has been replaced by (x-2) to give y2= 8(x-2) Translation7. Complex Numbersz = a + ibi = 1i2 = 1real imaginary Addition and Subtraction( 2 + 3i) + (5 2i) = 7 + i (add/subtract real part then imaginary part) Multiplication - multiply out the same way you would (x-2)(x+4)( 2 3i)(6 + 2i) = 12 + 4i 18i 6i2= 12 14i + 6= 18 14i Complex Conjugatez*Ifz = a + ib then its complex conjugate isz* = a ib- always collect the real and imaginary parts before looking for the conjugate Solving Equations - if two complex numbers are equal, their real parts are equaland their imaginary parts are z when 5z 2z* = 3 14iLet z = x + iy and so z*= x iy5(x + iy) 2(x - iy) = 3 14i3x + 7iy = 3 14i Equating real.

8 3x = 3 so x = 1 Equating imaginary : 7y = -14 so y= -2z = 1 CalculusDifferentiating from first principles Gradient of curve or tangent at x is f (x) = You may need to use the binomial expansionDifferentiate from first principles to find the gradient of the curvey = x4 at the point (2,16)f(x) = 24 f(2 + h) = (2 + h) 4= 24 + 4(23h) + 6(22h2)+ 4(2h3)+ h4= 16 + 32h + 24h2+ 8h3+ h4As h approaches zero Gradient = 32f(2 + h) f(2)h= 16 + 32h + 24h2 + 8h3 + h4 16h= 32 + 24h + 8h2 + h3 You may need to give the equation of the tangent/normal to the curve easy to doonce you know the gradient and have the coordinates of the pointImproper IntegralsImproper if one or both of the limits is infinity the integrand isundefined at one of the limits or somewhere in between thelimitsVery important toinclude thesestatementsVery important toinclude Trigonometry GENERAL SOLUTION don t just give one answer there should be an n somewhere!

9 ! SKETCH the graph of the basic Trig function before you start Check the question forDegrees orRadians MARK the first solution (from your calculator/knowledge) on your graph mark afew more to see the pattern Find the general solutionbefore rearranging to get x or on it s the general solution, in radians, of the equation 2cos2 x=3sin x2(1- sin2 x) = 3sin x (Usingcos2 x + sin2 x =1)2sin2 x + 3 sin x 2 = 0(sin x + 2)(2sin x 1) = 0sinx = p65p62p +p62p +5p6no solutionsfor sin x = -2 General Solutionsx = 2p n +p6, x = 2p n +5p6 You may need to use the fact that tan q =sin q to solve equations of the formcos qsin (2x ) = cos (2x )10. Numerical solution of equations Rearrange into the form f(x) = 0 To show the root lies within a given interval evaluate f(x) for the upper and lowerinterval boundsOne should be positive and one negativechange of sign indicates a root within the interval Interval Bisection- Determine thenature of f(Lower) and f(upper) sketch the graph of the interval- Investigate f(midpoint)- positive or negative ?

10 - Continue investigating new midpoints until you have an interval to the degree ofaccuracy required Linear Interpolation- Determine the Value of f(Lower) and f(upper) sketch the graph of the interval- Join the Lower and Upper points together with a straight line- - Mark p the approximate root- - Use similar triangles to calculate p (equal ratios) Newton- Raphson Method- given in formula book asvalue of value ofxn + 1 = xn f (xn)f ' (xn)new approximation previous approximation- you may be required to draw a diagram to illustrate your methodtangentto the curveat xnGradient of thetangent = f (xn)f (xn) =f(xNB : When the initial approximation is not close to f(x) the method may fail!)