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 mathematical induction to prove that 74 n is divisible by 3for n   .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 / Solutions 5 Qn Solution Let Pn be the proposition: 74 3 ,n pp   To prove P1 is true, 743 . P1 is true. Assume Pn is true for n=k: 74 3 ,k qq   To prove Pk+1 is true: 174 3 ,k rr      174 74 ( 4 )74 73 74 74 3 34 7 3 kkLHS q q qr        Since P1 is true and Pk is true  Pk+1 is true, 74 n is divisible by 3 for n   . 7. The functions f and g are defined by   2 1fx x   , x and  1g xx  , ,1x x . (a) State the ranges for f and g . (b) Give a reason why fg exists. (c) Find the composite function fg and state its domain. [1, [ ], 0 ] f g R R    ], 0 ] ], [gfRD     Hence fg exists.      2 () 1 1 1 1 11 fg x x x xx          Since ,1x x  [1, [fg gDD  8. (ai) Given the equation of the curve 22 22 4xx y y  , find dy dx in terms of x and y . (aii) Hence find the value(s) of x when the tangent is parallel to the x-axis. (b) Differentiate arctan x with respect to x .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 / Solutions 6 Qn Solution ai) 22 22 4 22 24 0 2 xx y y dy dyxx yy dx dx dy y x dx y x      aii) When the tangent if parallel to the x-axis, 0dy dx  . 02 dy y x dx y x xy    22 222 22 4 224 2 xx y y xxx x      b)   1 2 11tan 21 1 21 d xdx xx xx      Application of arctanx formula and chain rule. Section B 9. Consider the function () , ax bfx cx d   x , dx c for non-zero constants a , b , c and d . (a) i) Find an expression for  f x . ii) Given that ad bc , show that  f x is an increasing function for all values of x , dx c . 2 2 () ()'( ) () ,() ac x d ca x bfx cx d ad bc d xcx d c      Since ,0ad bc ad bc  , 2'( ) 0() ad bcfx cx d   ,  f x is an increasing function. (b) Given 2a , 0b , 3c and 2d  , find   1f x for 2, 3xx  . (i) Hence or otherwise show that   2f xx and find   201f x . (ii) With the given values of ,,abc and d , solve the inequality ()f xx .

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 / Solutions 7 Qn Solution Let 2 32 xy x  1 23 2 2 32 2() 32 x xy y yx y xfx x         1 1 2() ()32 () () xfx f x x f fx f f x x       201 2 100 2(( ) ) ( ) 32 xfx f f x f x x      2 2 2 32 2( 3 2 ) 032 43 032 43 320 x xx xxx x xx x xx x        Roots at 240, , 33xx x  . 240, 33xx  (c) The function h is defined by   sinhx p x q x r  , x, where ,pq and r are real constants. Given that h is an odd function, find the value of r .   () s i n s i n () () sin sin( ) 0 hx h x p xq xr p x q x r px q x r p x q x r r            

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 / Solutions 8 Qn Solution 10. The graph of a polynomial function 42() 4 4fx x x  of degree four is shown below. ()fx has roots at (2 . 2 0 , 0 ) and (2.20,0) . (a) Determine the interval(s) on the domain of f where f is concave downwards. (b) ()fx is translated by h units in the positive y-direction such that the equation () 0fx  will have four real and distinct roots. Find all possible value(s) of h . 42 3 2 () 4 4 '( ) 4 8 ''( ) 12 8 0 22 33 fx x x fx x x fx x x        42 3 2 () 4 4 '( ) 4 8 0 (2 ) 0 fx x x fx x x xx      ()fx has turning points at 0x , 2x . When 0x , 4y  . When 2x , 224 ( 2 ) 4 8y    . 48 h (b) Sketch the graph of 1 ()fx for the function ()fx shown above. State clearly the coordinates of any asymptotes, intercepts and turning points on the graph.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 / Solutions 9 Qn Solution Vertical asymptotes: x a and x a Horizontal asymptote: 0y  Min point/ y-intercept: (0, 0.25) Max points: 1(2 , )8 and 1(2 , )8 11. Let 32() 8 8 6 1 0 ,px x x x x    . (a) The roots of the equation () 0px  are  ,  and  . Find the values of (i)   ; (ii)  , (iii)    . 32() 8 8 6 1 0 ,px x x x x    Sum of roots = 8 18  Product of roots = 10 5 84   63 84     (b) Find all three roots of () 0px  . Show that the roots form a triangle with an area of 1 unit2 when plotted on an Argand diagram.

ACS (Independent) / Mathematics Department / Mathematics HL / Year 5 / 2017 Final Exam / Paper 1 / Solutions 1 0 Qn Solution 32 32 2 2 2 () 8 8 6 1 0 (1 ) 0 886 1 0 ( 1 ) ( ) 8 10 61 6 81 6 1 0 0 48 5 0 86 4 4 2 0 1182 px x