2024 RI H2 Math Prelim P2 (Soln) - new

RAFFLES INSTITUTION 2024 Year 6 H2 Mathematics Prelim Exam Paper 2 Questions and Solutions with comments ______________________ 2024 Yr 6 H2 Math Prelim Exam Paper 2 Solution with Comments 1 The function f is defined by 2f: 2 xx x , for x  , 2x . (a) Sketch the graph of f and find its range. [3] Another function g is defined by g: 3 2xx  , for x  . (b) Show that the composite function fg exists. Find fg( )x and state the domain and range of fg. [5] (a) [3] Range of f = (, ) \ { 2 }  Note that the curve passes through origin. Other possible notations for range of f:  (, ) \ { 2 }   \{ 2 } (b) [5] gR[ 3 , ) fD(, 2 ) ( 2 , )    Since fRDg  , function fg exists. 2(3 2 ) 6 2 2fg( ) f (3 2 ) 32 2 12 xxxx xx        fgD   Note that 2(3)f( 3 ) 632  . fg gDD  gR [3, ) fgR (2,6] Hence, fgR( 2 , 6 ] . Give domain and range in set notations. Some students are confused about the domain and range of a composite function. x y 2x 2y O g f

2 ______________________ 2024 Yr 6 H2 Math Prelim Exam Paper 2 Solution with Comments 2 The function f is defined by 432f( ) 4 5 ,zzA zB zC z   where A, B and C are real numbers. Given that 2i is a root of f( ) 0z  and 2()zk is a factor of f( ) ,z where k is a positive real number, find the values of A, B, C and k . [5] [5] Since all coefficients of f( )z are real and 2i is a root of f( ) 0z  , then 2i is also a root of the equation. The quadratic factor of the equation is       22i 2i 4 5zz z z  Then,    2432 2 22 45 ( 4 5) 2( 4 5 ) zA zB zC z z kz z zk z kz z          Comparing constants, 245 5 3kk   Since 0k  , therefore, 3k  . So,   432 2 43 2 45 6 9 ( 4 5) 10 38 66 45 zA zB zC z zz z z zzzz            10, 38, 66AB C   This question is generally well done. Some students tried to substitute the root in, then compare the real and imaginary parts to obtain 2 equations. However, 2 equations are not enough to solve for 3 unknowns, without using the condition that 2()zk is also a factor of f( )z .

3 ______________________ 2024 Yr 6 H2 Math Prelim Exam Paper 2 Solution with Comments 3 (a) The points A , B and C on the plane  have position vectors a , b and c respectively. Show that a vector perpendicular to  is parallel to bccaab . [3] (b) p and q are non-zero vectors and  pp q q . (i) Find the relationship between p and q . [1] (ii) Find q . [1] (c) u is the position vector of a fixed point U relative to a fixed origin O. A variable point V has position vector v relative to O. Given that  0vvu , describe geometrically the set of all possible positions of the point V. [2] (a) [3]    , since AB BC         