ACSI 2017 Promo Paper 1 ans

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 / Solutions 1 FINAL EXAMINATION 2017 YEAR 5 IB DIPLOMA PROGRAMME MATHEMATICS HIGHER LEVEL Paper 1 Qn Solution 1. (a) Find the term independent of x in the expansion of 6 2 2x x   . (b) Prove that 11 nn nr rr r          . 62 ( 6 ) 6 1 2 3 1 64 4 4 2 2 12 3 0 4 6543Independent term 2 2 30 8 2404321 r rr r rr rTC x C x x rr C                 ! 1! ( ) ! 1 ! (1 ) ! ( (1 ) ) ! 1 n nr n nrLHS r rr n r r n rn r n RHSr         

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 / Solutions 2 Qn Solution 2. The diagram shows the graph of f . (a) On the same diagram, sketch the graph of 1f . (b) Hence find the point of intersection of ()f x and 1()f x .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 / Solutions 3 Qn Solution At the intersection of ()fx and 1()fx , 1() ()fx f x x  . Point of intersection is (4, 4). 3. The cubic polynomial ()f x and quadratic polynomial ()p x are related by the following equation: () ()( 1 ) 4fx p x x   . The roots of ()fx are at 2x and 3x . Find 32()fx x A x B x C  , where A , B and C are constants to be determined. 32 32 () ( 1 ) () 4 ( 2 ) ( 3 ) ( ) (1) 4 2(1 ) 1 () ( 2 ) ( 3 ) ( 1 ) (2 3 1) 4 (2 3 ( 1)) 6 (2 ) (3 ) (3 ) ( 1 ) ( 1 ) (2 ) 1 () 4 6 f xx p x x x x k fk k fx x A x B x C x x x A C B fx x x x                                 4. The graph of function    () c o s2 s i n2fx x x for 14 x  is shown below. (a) The function f can also be expressed as   () s i n2fx a x b where a  and 02 b . Find the value of a and b . (b) Find '( )fx , hence show that  cos 2 sin 2 2 cos 2 8xx x      .   () s i n2fx a x b Amplitude of f(x) = 2 = a Function is translated b units, b = 8 

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 / Solutions 4 Qn Solution Differentiating f(x),    '( ) 2 sin 2 2 cos 2 '( ) 2 cos 2 28 f xx x fx x          cos 2 sin 2 2 cos 2 8xx x      5. Determine the relationship between the following lines. Explain your answer clearly. 1 11 2 :8 5 , 34 lk k              r  2 23 :1 1 , 95 lh h     r  . Their direction vectors are not parallel, they are not parallel lines. 11 2 2 3 8 5 1 (1) & (3) 19 13 34 95 8332, 2 83(2) : 8 5 2 13 2 kh kh h kh hk LHS RHS                  The lines do not intersect. The lines are skew. 6. Use mathema