Diploma Programme 
                                                                                                                                                         
                                                                                                                                                         
                                                                                                                                                         
                                                                                        Mathematics HL and 
                                                                              further mathematics HL 
                                                                                                      formula booklet 
                                                                                                                                                         
                                                                                              For use during the course and in the examinations 
                                                                                                                            First examinations 2014 
                                                                                                                                                         
                                                                                                                          Edited in 2015 (version 2) 
                                                                                                                                                         
                                                                                                                                                                    
                  © International Baccalaureate Organization 2012                                                                        5048 
                 Contents 
                 Prior learning                                                                                       2 
                 Core                                                                                                 3 
                 Topic 1: Algebra                                                                                     3 
                 Topic 2: Functions and equations                                                                     4 
                 Topic 3: Circular functions and trigonometry                                                         4 
                 Topic 4: Vectors                                                                                     5 
                 Topic 5: Statistics and probability                                                                  6 
                 Topic 6: Calculus                                                                                    8 
                 Options                                                                                             10 
                 Topic 7: Statistics and probability                                                                 10 
                      Further mathematics HL topic 3 
                 Topic 8: Sets, relations and groups                                                                 11 
                      Further mathematics HL topic 4 
                 Topic 9: Calculus                                                                                   11 
                      Further mathematics HL topic 5 
                 Topic 10: Discrete mathematics                                                                      12 
                      Further mathematics HL topic 6 
                 Formulae for distributions                                                                          13 
                      Topics 5.6, 5.7, 7.1, further mathematics HL topic 3.1 
                 Discrete distributions                                                                              13 
                 Continuous distributions                                                                            13 
                 Further mathematics                                                                                 14 
                 Topic 1: Linear algebra                                                                             14 
                 Mathematics HL and further mathematics formula booklet                                                1 
                  
                  Formulae 
                 Prior learning 
                  Area of a parallelogram                    , where b is the base, h is the height 
                                                    A=bh×
                  Area of a triangle                    1       , where b is the base, h is the height 
                                                    A=()bh×
                                                        2
                  Area of a trapezium                   1         , where a and b are the parallel sides, h is the height 
                                                    A=()a bh+
                                                        2
                  Area of a circle                        2 , where r is the radius 
                                                    Ar=π
                  Circumference of a circle                 , where r is the radius 
                                                   Cr=2 π
                  Volume of a pyramid                   1                                  
                                                   V =(  area of base × vertical height)
                                                        3
                  Volume of a cuboid                            , where l is the length, w is the width, h is the height 
                                                   V=l×wh×
                  Volume of a cylinder                    2  , where r is the radius, h is the height 
                                                   V=πrh
                  Area of the curved surface of     A=2 rhπ , where r is the radius, h is the height 
                  a cylinder 
                  Volume of a sphere                    4   3 , where r is the radius 
                                                   V=rπ
                                                        3
                  Volume of a cone                      1   2 , where r is the radius, h is the height 
                                                   V=rhπ
                                                        3
                  Distance between two                           22
                                                                                
                                                   d=(xx−)(+−yy)
                  points        and                        12 12
                         (,xy) (,xy)
                           11 22
                  Coordinates of the midpoint of     xx++yy
                                                   
                                                      1212
                  a line segment with endpoints             ,          
                                                   
                                                        22
                          and                      
                   (,xy) (,xy)
                    11 22
                  Solutions of a quadratic                                                   −b±b2−4ac
                  equation                         The solutions of ax2 +bx+=c    0 are  x =                  
                                                                                                   2a
                  
                 Mathematics HL and further mathematics formula booklet                                                 2 
                        Core 
                       Topic 1: Algebra 
                         1.1        The nth term of an                                                  
                                                                              uu=+−(n1)d
                                    arithmetic sequence                         n      1
                                    The sum of n terms of an                          nn
                                                                                                                                  
                                                                                   =2 (+−1)                        =(          ) +
                                                                              S und uu
                                                                                        (                    )
                                    arithmetic sequence                         nn11
                                                                                      22
                                    The nth term of a                                    n−1  
                                                                              u =ur
                                    geometric sequence                          n      1
                                    The sum of n terms of a                                nn
                                                                                      ur(     −1)      u(1−r)
                                                                                       11
                                                                                                                    ,          
                                    finite geometric sequence                 Sn =                                 =r≠1
                                                                                         rr−11−
                                    The sum of an infinite                              u
                                    geometric sequence                        S = 1 ,  r <1 
                                                                                ∞     1−r
                         1.2        Exponents and logarithms                    x                            , where                               
                                                                              ab=⇔=xlogab                                ab>>0,        0,a≠1
                                                                                x xaln
                                                                              a =e           
                                                                                       x           loga x  
                                                                              loga a=xa=
                                                                              log a = logc a  
                                                                                   b       log b
                                                                                               c
                         1.3                                                    n
                                                                               n!  
                                    Combinations                                     =
                                                                              
                                                                                r       rn!(  −r)!
                                                                              
                                    Permutations                              nP =          n!       
                                                                                  r      (nr− )!
                                                                                                      nn
                                                                                                                        
                                                                                        n n                 n−−1                  nrr              n
                                                                                                                                                      
                                    Binomial theorem                          ()ab+ =a+ ab++ab++b
                                                                                                                        
                                                                                                      1                     r
                                                                                                                        
                         1.5        Complex numbers                                                                         iθ              
                                                                               z=a+=ibr(cosθθ+isin )=re=rcisθ
                         1.7                                                                           n      n                              nni θ       n          
                                    De Moivre’s theorem                         r(cosθθ+isin )            =r(cosn+θisinnθ=) r=e                        rcisnθ
                                                                              [                       ]
                        
                        
                       Mathematics HL and further mathematics formula booklet                                                                                            3