ZETA MATHS Higher Mathematics Revision Checklist Contents: Epressions & Functions Page Logarithmic & Eponential Functions Addition Formulae. 3 Wave Function.... 4 Graphs of Functions. 5 Sets of Functions 6 Vectors. 8 Relationships & Calculus Polnomials.. Quadratic Functions.. Trigonometr. 3 Further Calculus.. 5 Applications The Straight Line. 6 The Circle. 7 Recurrence Relations.. 7 Differentiation. 8 Integration.. 9 V..
Epressions & Functions Topic Skills Notes Logs and Eponentials Equation of a Line Eponential Functions Know and use = m + c to determine the equation of a line An eponential function is written in the form = a where a is the base, and is the inde or eponent = a (, a) The Logarithmic Function The logarithmic function is the inverse of the eponential function. It is written as = log a where a is the base NB: On our calculator the log button is log 0 (a, ) = log a Convert Between Logarithmic and Eponential Form The Eponential Function Natural Logarithms Laws of Logs If = a then = log a e.g. 3 = log a 8 a 3 = 8 a = The eponential function is written as = e where e is the base, which is approimatel.78 The natural log function is the inverse of the eponential function = e, it is written as = ln, which means = log e log a = log a + log a e.g. log 4 8 + log 4 = log 4 (8 ) = log 4 6 = (since 4 = 6) log a = log a log a e.g. log 4 8 log 4 = log 4 8 log a n = n log a = log 4 4 = (since 4 = 4) e.g. 3 log 9 7 = log 9 7 3 = log 9 3 = (since 9 = 3) log a a = Use the Laws of Logs to Solve Log Equations e.g. log 5 5 = (since 5 = 5) E. Solve: log 5 ( + ) + log 5 ( 3) = Soln. log 5 ( + ) ( 3) = (using first law) ( + )( 3) = 5 (since 5 = 5) 3 = 5 (solve for ) 8 = 0 ( + )( 4) = 0 =, = 4 www.zetamaths.com Zeta Maths Limited
Use Laws of Logs to Solve Eponential Growth or Deca Problems E. Find if 4 log 6 log 4 = Soln. log 6 4 log 4 = 6 log 4 = 4 6 4 = 4 = 4 3 4 4 (since 6 4 = 4 3 4 and 4 = 6 = 4 ) = 3 4 = 8 For finding an initial value; substitue given values in to equation to determine the initial value For finding a half-life, make the equation equal to one half e.g. In the equation, where A represents micrograms of a radioactive substance remaining over time t. Find: (a) the initial value if there are 500 microgram after 00 ears (b) the half-life of the substance (a) (b) A t = A 0 e 0.004t 500 = A 0 e 0.004 00 500 = 0.67 A 0 A 0 = 746 micrgrams 373 = 746e 0.004t ln = e 0.004t = ln e 0.004t Formulae for Eperimental Data 0.004t = ln t = 73 ears In eperimental data questions, two tpes of eponential functions are considered, = k n and = ab = k n Taking logs of boths sides, this equation ma be epressed as log = n log + log k. To find the unknown values n and k: If the data given is and data, then take logs of two sets of the data for and and form a new table with log and log Substitute new values into log = n log + log k and solve simultaneousl to find values for n and log k Find k b solving log k Write = k n with values of k and n = ab Taking logs of boths sides, this equation ma be epressed as log = log b + log a. To find the unknown values a and b: If the data given is and data, then take logs of the data for Substitute values into log = log b + log a and solve simultaneousl to find values for log a and log b Find a and b b solving log a and log b Write = ab with values of a and b Sketch the Graph of the Inverse Function of a Log or Eponential Function See Graphs of Functions www.zetamaths.com Zeta Maths Limited
Addition Formulae Prior skills Pthogoras Theorem a = b + c a c b SOHCAHTOA sin θ = Opp Hp Adj Opp, cos θ =, tan θ = Hp Adj Hpotenuse θ Adjacent Opposite Solve Trig Equations Eact Values Convert from Degrees to Radians and Vice-Versa Use Eact Values to Calculate Related Obtuse Angles Use the CAST diagram or graphical method to solve equations (see Relationships in National 5 checklist) See Trigonometr See Trigonometr E. Find the eact value of cos 5⁰ Soln. The related acute angle is 45⁰ since 80⁰ + 45⁰ = 5⁰ From the graph or CAST diagram cos 5⁰ is negative. cos 5 = cos 45 = E. Find the eact value of sin π 3 Soln. The related acute angle is π 3 since π π 3 = π 3 From the graph or CAST diagram sin π 3 is positive. Use Addition Formulae to Epand Epressions Use Addition Formulae to Evaluate Eact Values of Epressions sin π 3 = sin π 3 = 3 sin(α + β) = sin α cos β + cos α sin β sin(α β) = sin α cos β cos α sin β cos(α + β) = cos α cos β sin α sin β cos(α β) = cos α cos β + sin α sin β NB: For sin functions the signs are the same, for cos functions the signs are different E. Find the eact value of cos 75⁰ Soln. cos 75 = cos(45 + 30) (use addition formulae epansion) = cos 45 cos 30 sin 45 sin 30 (use eact values) = 3 = 3 E. Given sin A = 3 56 and cos B =, show that sin(a + B) = 5 3 65 Soln. Use SOHCAHTOA to sketch triangles from the info given and use Pthagoras to find unknowns 5 3 3 5 A 4 Epand using Addition Formulae sin(a + B) = sin A cos B + cos A sin B = 3 + 4 5 5 3 5 3 = 56 65 B www.zetamaths.com 3 Zeta Maths Limited
Solve Trig Equations using Trig Identities Determine which part of the equation has a related identit Replace trigonometric term with related identit and solve e.g. Solve sin + sin = 0, for 0 80 Soln. As sin = sin cos, then sin cos + sin = 0, solve b factorising Factorise sin (cos + ) = 0 sin = 0 cos + = 0 (using graph) cos = = 0, 80, 360 A = 60 = 0, 0, 80 (using CAST) = 0, 40 Wave Function Solve Trig Equations Eact Values Use Addition Formulae to Epand Epressions Write an Epression Form k sin( ± α) or k cos( ± α) Use the CAST diagram or graphical method to solve equations (see Relationships in National 5 checklist) See Trigonometr See Addition Formulae a cos + b sin can be written in one of the following forms: k sin( + α) k sin( α) k cos( + α) k cos ( α) Where k = a + b and tan α is derived from a and b E. Epress 3 sin + cos in the form k sin( + α) where k > 0 and 0 360 Soln. k sin( + α) = k(sin cos α + cos sin α) = k cos α sin + k sin α cos = 3 sin + cos To find k: k = 3 + = To find α: k cos α = 3 and k sin α = tan α = k sin α k cos α (Epand) tan α = (use eact values) 3 α = 30 (NB: k sin α and k cos α are both positive, therefore the angle is in quadrant, i.e. less than 90⁰) 3 sin + cos = sin( + 30) www.zetamaths.com 4 Zeta Maths Limited
E. Epress 8 cos 6 sin in the form k cos( + α) where k > 0 and 0 360 Soln. k cos( + α) = k(cos cos α sin sin α) = k cos α cos k sin α sin = 8 cos 6 sin k cos α = 8 and k sin α = 6 To find k: k = 8 + 6 = 0 To find α: tan α = k sin α k cos α tan α = 6 (use calculator) 8 α = 36.9 8 cos 6 sin = 0 cos( + 36.9) (Epand) Graphs of Functions Sketch Quadratics Sketch Trig Graphs Sketch Related Graphs Sketch quadratics of the form = k and = a ( + b) + c Sketch graphs of the form = a sin b + c and = a cos b + c Ensure all given coordinates are translated and marked on the new graph and aes and graphs are labelled = f() + a Graph moves up or down b a Up for f() + a Down for f() a = f( + a) Graph moves left or right Left when f( + a) Right for f( a) = f() Graph reflects in -ais = f() = f( ) Graph reflects in -ais = kf() Graph is stretched verticall for k > Graph is squashed verticall for 0 < k < www.zetamaths.com 5 Zeta Maths Limited
= f(k) Graph is squashed horizontall for k > Graph is stretched horizontall for 0 < k < Sketch Log and Eponential Graphs Log graphs of the form = log a alwas cut the -ais at the point (, 0) and will pass through (a, ) (a, ) = log a Eponential graphs of the form = a alwas cut the -ais at the point (0, ) and will pass through (, a) (, a) = a Sketch the Graph of the Inverse Function of a Log or Eponential Function All of the related graph transformations above appl to log and eponential graphs The graph of an inverse function is reflected along the line =. The logarithmic graph is the inverse of the eponential graph and viceversa e.g. For the graph of the function = the inverse function is = log = (, ) = log (, ) Sketch a Trig Graph of the Form = k sin( ± α) or = k cos( ± α) Sets and Functions Identif the Turning Point of a Quadratic Find Composite Functions See Trigonometr Sketch a Trig Graph from its Equation From completed square form = a ( + b) + c, turning point is (-b, c) Composite functions consist of one function within another. e.g. If f() = 3 and g() = 4, find (a) f(g()) (b) g(f()) Soln. (a) f(g()) = 3( 4) = 3 = 3 4 (b) g(f()) = (3 ) 4 = 9 + 4 4 = 9 www.zetamaths.com 6 Zeta Maths Limited
Evaluate Using Composite Functions e.g. Find H( ) where H() = g(f()) and f() = 3, g() = 4 Soln. Method : H() = g(f()) = = 9 (from eample above) H( ) = 9( ) ( ) = Determine a Suitable Domain of a Function State the Range of a Function Find an Inverse Function Method : f( ) = 3( ) = 5 g( 5) = ( 5) 4 = Restrictions on the domain of a function occur in two instances at Higher Mathematics. A restriction will occur when a denominator is zero, which is undefined or when a square root is negative, which is non-real e.g. For f() = (4 ) and R, write a restriction on the domain of f() Soln. 4 as this would make the denominator zero e.g. State the minimum turning point of the function f() = + 5 and hence state the range of the function (see prior skills) Soln. Minimum turning point is (0, 5) as the -coordinate of the turning point is 5, the range of the function is f() > 5 For a function f() there is an inverse function f (), such that f(f ()) = To find an inverse function: Replace with in the function and f() with Change the subject to e.g. For the function f() = Soln. f() = = 4 = 3 3 4 3 4 4 3 = 3 4 find the inverse function f () = 4 3 f () = 4 3 Common Terms Domain Range Number Sets The domain of a function is the set of numbers that can be input into the function (see Determine a Suitable Domain above) The range of a function is what comes out of the function after the -values have been put in (see State the Range of a Function) There are five standard number sets to consider at Higher Mathematics The set of natural numbers N (counting numbers) N = {,, 3, 4, } The set of whole numbers W (the same as natural numbers, but inclusive of zero) W = {0,,, 3, 4, } The set of integers Z (the same as whole numbers, but inclusive of negative numbers) Z = {,,, 0,,, 3, 4, } www.zetamaths.com 7 Zeta Maths Limited
Vectors Vector Notation The set of rational numbers Q (all numbers that can be written as fractions) The set of real numbers R (inclusive of both rational and irrational numbers i.e. π, 3, etc.) Vectors can be named in one of two was. Either b using the letters of the points at the end of the line segment AB or b using a single letter in lower case u. When writing the lower case name, underline the letter B A u D Line Segments Add or subtract D line Segments Vectors end-to-end Arrows in same direction a a+b b Finding a Vector from Two Coordinates 3D Vectors Know that to find a vector between two points A and B then AB = b a A b - a NB: Vector notation for a -a B vector between two points b A and B is AB Determine coordinates of a point from a diagram representing a 3D object Look at difference in, and z aes individuall e.g. The cuboid is parallel to the, and z aes. Find the coordinates of C z C (5, 9, 0) A (4, 5, 0) B (5, 9, 6) C Position Vectors The position vector of a coordinate is the vector from the origin to the coordinate. E.g. A (4, -3) has the position vector a = ( 4 3 ) The Zero Vector Add and Subtrac Vector Components Multipl Vector Components ( 0 ) is called the zero vector, written 0 0 u + ( u) = 0 Add and Subtract D and 3D vector components. 3 + 3 a = ( ) and b = ( ) a + b = ( + ) 4 5 4 + 5 Multipl vector components b a scalar a = ( ) = ( ) 4 8 www.zetamaths.com 8 Zeta Maths Limited
Find the Magnitude of a vector Find the magnitude of a D or 3D vector: For vector u = ( ), u = + For vector v = ( ), v = + + z z Writing Vectors Parallel Vectors The Unit Vector Vectors can be written in component form i.e. a = ( ) or in terms z of i, j and k,where each of these represents the unit vector in the, and z direction. 4 e.g. AB = 4i 3j + 6kcan be written as AB = ( 3) 6 Vectors are parallel if one vector is a scalar multiple of the other 4 e.g. a = ( ) and b = ( 4 ) = 4 ( ) 4 6 4 b = 4a vectors are parallel For an vector, there is a parallel vector u of magnitude. This is called the unit vector e.g. Find the unit vector u parallel to vector a = ( 5 ) a = 5 + = 3 u = 3 ( 5 5 ) = ( 3 ) 3 Collinearit Points are said to be collinear if the line on the same line. To show points are collinear using vectors; show (a) the are parallel b demonstrating one vector is a scalar multiple of the other and (b) that the share a common point Divide Vectors into a Given Ratio to Find an Unknown Point e.g. Show that A(-3, 4, 7), B(-, 8, 3) and C(0, 0, ) are collinear 3 Soln. AB = b a = ( 8 ) ( 4 ) = ( 4 ) = ( ) 3 7 4 0 BC = c b = ( 0) ( 8 ) = ( ) 3 AB = BC and point B is common A, B and C are collinear e.g. P is the point (6, 3, 9) and R is (, 6, 0). Find the coordinates of Q, such that Q divides PR in the ratio : Soln. PQ QR = PQ = QR q p = (r q) q p = r q 3q = r + p 6 30 3q = ( 6 ) + ( 3) = ( 5) 0 9 9 0 q = ( 5 ) Q (0, 5, 3) 3 NB: this could also be calculated using section formula www.zetamaths.com 9 Zeta Maths Limited
Find the Ratio in which a Point Divides a Line Segment e.g. A(-, -, 4), B(, 5, 7) and C(7, 7, 3) are collinear. What is the ratio in which B divides AC? 3 Soln. AB = b a = ( 5) ( ) = ( 6) 7 4 3 7 6 3 BC = c b = ( 7) ( 5) = ( ) = ( 6)=AB 3 7 6 3 Scalar Product AB = BC AB BC = AB : BC : When given an angle between two vectors, the scalar product is calculated using a a. b = a b cos θ θ b NB: To find the angle between the two vectors θ, the vectors must be pointing awa from each other and 0 θ 80 a b When given component form, i.e. if a = ( a ) and b = ( b ), the a 3 b 3 scalar product is calculated using a. b = a b + a b + a 3 b 3 Angle Between Two Vectors The angle between two vectors is calculated b rearranging the scalar product formula cos θ = a.b a b which can be epanded to cos θ = a b +a b +a 3 b 3 a b NB: to find the angle between two vectors, the vectors must be pointing awa from or towards each other. The must not be going in the same direction e.g. a θ b Not b θ a Perpendicular Vectors Common Terms Vector Vectors are perpendicular when a. b = 0 A vector is a quantit that contains both a size and a direction www.zetamaths.com 0 Zeta Maths Limited
Relationships & Calculus Topic Skills Notes Polnomials Factorise Solving Quadratics Common Factor, Difference of Two Square and Trinomial factorising Algebraicall or using quadratic formula Factors and Roots A root is a value for which a polnomial f() = 0. These are the - coordinates at the point of intersection with the -ais A factor is the form from which a root is derived e.g. for ( 4) = 0, and ( 4) are factors, = 0 and = 4 are roots Full Factorise a Polnomial Use snthetic division e.g. Factorise f() = 3 3 + Soln. Set up snthetic division using coefficients from polnomial. - If there is no term, use 0 - The value outside the division is derived from factors of the last term (in this case factors of ) - If the remainder of the division is 0 then the value outside the division is a root - 0-3 - 4 - - 0 Remainder Theorem ( + ) is a factor and = - is a root ( + )( + ) = 0 ( + )( )( ) = 0, = - (twice) and = Find the Quotient and Remainder of a Function Use snthetic division with the given value The value at the end is the remainder e.g. Find the quotient and remainder when f() = ( 3 3 + 9)is divided b ( 3) Soln. 3 0-3 9 Identif the Equation of a Polnomial from a Graph 3 9 8 3 6 7 f() = ( 3 3 + 9) = ( 3)( + 3 + 6) + 7 Quotient is ( + 3 + 6) and remainder 7 Determine the factors from the roots on the graph Set up a polnomial with a coefficient of k outside the brackets Substitute the -intercept or given point to determine value of k e.g. 5-5 Soln. = k( + )( )( 5), When = 0, = 5 5 = k(0 + )(0 )(0 5) 5 = 0k, k = = ( + )( )( 5) www.zetamaths.com Zeta Maths Limited
Show that a Term is a Factor of a Polnomial From the factor, determine the root (see Factors and Roots above) Use snthetic division with the given value If the remainder is 0 the term is a factor Find Unknown Coefficients of a Polnomial Substitute roots into equation and solve simultaneousl e.g. Find the values of p and q if ( + ) and ( ) are factors of f() = 3 + 4 + p + q Soln. ( + ) is a factor = is a root ( ) 3 + 4( ) p + q = 0 8 p + q = 0 q = p 8 ( ) is a factor = is a root () 3 + 4() + p + q = 0 5 + p + q = 0 q = p 5 Solve two equations simultaneousl p 8 = p 5 p =, q = 6 and f() = 3 + 4 + 6 Sketch the Graph of a Polnomial Function Find the -intercepts (roots, when = 0) using snthetic division Find the -intercept (when = 0) Find stationar points and their nature (see differentiation) Find large negative and positive Common Terms Polnomial A function containing multiple terms of different powers e.g. 4 3 + 3 Quadratic Functions (see Relationships in National 5 Checklist) Solve a Quadratic Graphicall, Algebraicall or using Quadratic Formula Sketch a Quadratic Identif the Equation of a Quadratic from its Graph Complete the Square In completed square form In root form See Identif the equation of a Polnomial from a Graph (above) + a + b = ( + a ) + b a E. + 8-3 = ( + 4) 3 6 = ( + 4) 9 E. + 3 + 0 = ( + 3 ) + 0 9 = ( + 3 4 ) + 40 9 = ( + 3 4 4 ) + 3 4 Discriminant b 4ac where = a + b + c The discriminant describes the nature of the roots b 4ac > 0 two real roots b 4ac = 0 equal roots (i.e. tangent) b 4ac < 0 no real roots www.zetamaths.com Zeta Maths Limited
Show a Line is a Tangent to a Quadratic Function Determine Nature of Intersection Between a Line and a Quadratic Function Determine Points of Intersection between a Line and a Quadratic Function Use Discriminant to Find Unknown Coefficients of a Quadratic Function Equate the line and quadratic Bring to one side Use the discriminant or factorise to show repeated root e.g. Show that the line = + 5 is a tangent to the curve = + 5 + 9 Soln. + 5 + 9 = + 5 + 4 + 4 = 0 ( + )( + ) = 0 = twice line is a tangent to quadratic. Equate line and quadratic Bring to one side Use the discriminant to determine nature of intersection Equate line and quadratic Bring to one side Solve for Substitute values into line to find values Identif coefficients a, b and c Use discriminant E. Find p given that + + p = 0 has real roots Soln. a =, b =, c = p b 4ac 0 4()(p) 0 4p 0 4p p 4 E. Find p given that 4 + p + = 0 has no real roots Soln. a = 4, b = p, c = b 4ac < 0 (p) 4(4)() < 0 4p 6 < 0 4(p 4) < 0 4(p + )(p ) < 0 (Sketch a graph) For no real roots < p < Common Terms Parabola Trigonometr Eact Values The graph of a quadratic function Know eact values from table or using triangles and SOHCAHTOA 45 30 3 Solve Trig Equations 45 60 Know eact values of 0⁰, 90⁰, 80⁰, 70⁰ and 360⁰ from trig graphs Use the CAST diagram or graphical method to solve equations (see Relationships in National 5 checklist) www.zetamaths.com 3 Zeta Maths Limited
Determine Period, Shape and Ma and Min Values from a Trig Equation Adding Fractions = a cos b Amplitude = a Period = 360 b NB: in the graph = a tan b the amplitude cannot be measured For solutions in radians, the abilit to add fractions is required e.g. π π 3 = 3π 3 π 3 = π 3, π + π 3 = 6π 3 + π 3 = 7π 3 Convert from Degrees to Radians Convert from Radians to Degrees Solve Trig Equations with multiple solutions Multipl degrees b π and divide b 80 then simplif e.g. Change 35⁰ to radians Soln. 35π = 7π 80 36 Multipl radians b 80 and divide b π then simplif e.g. Change 5π 6 Soln. 5π 6 = 5 80π 6π to degrees = 5 30 = 50 Identif how man solutions from the question Solve the equation e.g. Solve cos 3 =, for 0 π Soln. cos 3 = cos 3 = (As cos 3 has 6 solutions for 0 π, 3 solutions for 0 π) 3 = π 3, π π 3, π + π 3 3 = π 3, π 3, 7π 3 = π 9, π 9, 7π 9 Solve Trig Equations b Factorising Identif the Equation of a Trig function from its Graph Factorise the equation in the same wa as an algebraic equation, b looking for common factors, difference of two square or a trinomial e.g. Solve sin cos + sin = 0, for 0 80 Soln. Factorise sin (cos + ) = 0 sin = 0 cos + = 0 = 0, 80, 360 = 0, 0, 80 e.g. Determine the equation of the graph. 5. -3 0⁰ cos = - A = 60 (using CAST) = 0, 40 π 6 7π 6 Sketch a Trig Graph from its Equation Ans: = 4 sin 3 + Ans: = cos( π 6 ) e.g. Sketch the graph of = sin( 30) for 0 360 showing clearl where the graph cuts the -ais and the -ais Ans. Amplitude of. Graph moves 30⁰ to the right. Find -intercepts when = 0. Find -intercept when = 0 - - 30⁰ 0⁰ 360⁰ www.zetamaths.com 4 Zeta Maths Limited
Common Terms Radians Further Calculus Differentiate Integrate Trig Equations Radians Differentiate Trig Functions Radians are an alternative unit for measuring angles. π radians = 80⁰ Common Conversions: π = 90, π 3 = 60, π 4 = 45, π 6 = 30, π 3 = 0, 3π = 70 See Differentiation (in Applications) See Integration (in Applications) See Trigonometr (above) See Trigonometr (above) = sin d d = cos d = cos d = sin Integrate Trig Functions Chain Rule sin d cos d = cos + C = sin + C Used for differentiating composite functions Differentiate the outer function Multipl b the derivative of the inner function e.g. If h() = f(g()) then h () = f (g()) g () E.. Find d when = 5 d Soln. Prepare function for differentiation d = ( 5) = d d = d 5 ( 5) Integration of Composite Functions E. Find d when = 3 d cos Soln. Prepare function for differentiation = 3(cos ) d = 6(cos d ) sin d = 6 cos sin d When integrating composite functions Integrate the outer function Divide b the derivative of the inner function e.g. (a + b) n d = (a+b)n+ n+ a + C E. ( 3 + 5) 4 d Soln. ( 3 + 5) 4 d = (3 +5) 5 + C = (3 +5) 5 + C 5 6 30 E. sin(4 3) d Soln. sin(4 3) d = cos(4 3) + C 4 www.zetamaths.com 5 Zeta Maths Limited
Applications Topic Skills Notes The Straight Line Distance Between Two Points Distance formula or alternative: Distance = ( ) + ( ) Gradient of a line m = m = tan θ Perpendicular gradients: m m = θ e.g. if m =, the perpendicular gradient m = Equation of a Line From Two Points Midpoint Point of Intersection Equation of a Perpendicular Bisector Equation of a Median Equation of an Altitude Collinearit Common Terms Collinear Congruent Concurrent Centroid Circumcentre Orthocentre For ever equation of line a point and gradient is required Calculate gradient and substitute point (a, b) into equation: b = m( a) Epand bracket and simplif Midpoint = ( +, + ) Solve using simultaneous equations Find the midpoint of the line joining the two points Find gradient using perpendicular gradients Substitute midpoint and inverted gradient into b = m( a) Find the midpoint of the line joining the two points Find gradient of the median Substitute into b = m( a) Find gradient of the altitude using perpendicular gradients Substitute into b = m( a) with point from verte Show that three points are collinear (i.e. on the same line) Find gradients (the same if parallel) and point in common Statement: Points A, B and C are collinear as m AB = m BC and point B is common to both Points on the same line The same size Lines that intersect at the same point NB: In a triangle, altitudes are concurrent (intersect at orthocentre), medians are concurrent (intersect at centroid) and perpendicular bisectors are concurrent (intersect at circumcentre) The point of intersection of the three medians of a triangle The point of intersection of three perpendicular bisectors in a triangle The point of intersection of three altitudes of a triangle www.zetamaths.com 6 Zeta Maths Limited
The Circle Distance between two points Distance formula or alternative: Distance = ( ) + ( ) Gradient m = Perpendicular gradients: m m = Midpoint Discriminant Midpoint = ( +, + ) b 4ac where a + b + c = 0 Equation of Circle with Centre the Origin and Radius r Equation of a Circle with Centre (a, b) and Radius r Centre and Radius of a Circle from its Equation Equation of a Tangent to a Circle Points of Intersection of a Line and a Circle Use Discriminant to Determine whether a Line and Circle Intersect Common Terms Concentric Recurrence Relations Finding Percentage Multipliers b 4ac > 0 b 4ac = 0 b 4ac < 0 + = r Determine the centre and radius NB: Finding the centre often involves finding a midpoint of a diameter, or using a coordinate diagram and smmetr Substitute into equation ( a) + ( b) = r Use the equation + + g + f + c = 0 Centre from ( g. f) Radius: r = g + f c NB: if g + f c < 0 the equation is not a circle Determine the gradient of the radius from centre and the point of contact of the tangent Find gradient using perpendicular gradients Substitute perpendicular gradient and point into equation of a line b = m( a) Rearrange the line to = m + c then substitute into the equation of a circle Solve the quadratic to find Substitute value into = m + c to find Rearrange the line to = m + c then substitute into the equation of a circle Simplif to quadratic form Use discriminant to determine intersection Circles that have the same centre Determine whether question is percentage increase or decrease Add or subtract from 00% Divide b 00 e.g. 4.3% increase 00% + 4.3% = 04.3% =.043 www.zetamaths.com 7 Zeta Maths Limited
Form Linear Recurrence Relations Use Linear Recurrence Relations to Find Values Find the Limit of a Linear Recurrence Relation Find values of a and b for relation u n+ = au n + b where a is the percentage multiplier and b is the increase Start with u 0 (initial value) and substitute into relation NB: Set up calculator b inputting u 0 and pressing = then ANS a + b and continue pressing = until answer is reached. Write down all answers Determine values of a and b Ensure - < a < Use limit formula L = a Interpret what the limit means in a specific contet b Differentiation Laws of indices Differentiate a Function Find the Gradient or Rate of Change of a Function at a Given Point Find the Equation of a Tangent to a Curve Find Stationar Points and Determine their Nature Know and use each of the laws of indices to manipulate algebraic fractions; e.g. 3 + = 3 + = + To differentiate f() = a n, f () = an n For f() = g() + h() f () = g () + h () NB: For differentiation questions, algebraic fractions need to be broken down into individual fractions (see Laws of Indices above) Know that f () = m = rate of change Differentiate the function Substitute -coordinate into derivative e.g. Find the gradient of f() = 3 when = Soln. f () = 6 f (4) = 6() = 6 Differentiate function Find gradient from derivative (see above) Substitute point and gradient into equation of a line b = m( a) Find Stationar Points Differentiate function Know that stationar points occur whenf () = 0 Solve f () = 0 to find -coordinates of stationar points Substitute -coordinates into f() to find -coordinates Determine Nature Draw nature table with values slightl above and below the stationar points (make sure the are not lower or higher than an other stationar points) Sketch the nature from positive or negative values Answer question, e.g. Ma turning point at (, ) 3-3 3 + f () - 0 + slope 3 www.zetamaths.com 8 Zeta Maths Limited
Sketch a curve Find stationar points and nature (see above) Find roots b solving function (when = 0) Find -intercept (when =0) Find large positive and large negative. e.g. as f(), and as f() +, + Sketch information on graph Sketch the derived function Sketch function Etend stationar points to other coordinate ais Determine where the gradient m is +ve and ve. (the gradient is +ve where the graph of d is above the -ais d and negative where it is below d d Closed Intervals Increasing and Decreasing Functions Find the maimum and minimum value in a closed interval Find the stationar points and determine their nature (see above) Find the -coordinates at the etents of the interval Eamine to see where the maimum and minimum values are Differentiate the function Determine where gradient is positive or negative from the derivate or a sketch of the derivative NB: A function is increasing where the gradient m is positive and decreaseing where the gradient is negative Common Terms Leibniz Notation Function Notation Rate of Change Integration Laws of indices Integrate a Function and d d f() and f () The rate at which one variable changes in relation to another. To find rate of change, differentiate function See Differentiation To integrate: a n d = an+ + C, n+ NB: When the integral is indefinite (i.e. there are no limits), remember C the constant of integration Evaluate a Definite Integral Integrate function Evaluate between two limits 3 e.g. Evaluate 4 3 Soln. 4 d d = [ ] 3 = ((3) ) (() ) = 6 www.zetamaths.com 9 Zeta Maths Limited
Area Between Curve and -ais Find the area above the -ais Find the area below the -ais (ignore the negative) Add them together 0 f() d and f() d NB: The area below the -ais will give a negative answer. Ignore the negative Area Between Two Curves Set the curves equal to each other and solve to find the limits Set up integral with: = g() [upper curve lower curve] d b [f() g()] d a Evaluate answer = f() a b Differential Equations Equations of the form d d equations. The are solved b integration = a + b are called differential e.g. The curve = f() is such that d d = 9, the curve passes through (, 5). Epress in terms of Soln. = 9 d = 3 3 + C at (, 5), 5 = 3() 3 + C 5 = 3 + C C = = 3 3 + www.zetamaths.com 0 Zeta Maths Limited