SAJC Check your Understanding_Inequalities_tutor

Check your Understanding (Inequalities) Section 1: Type of roots of Quadratic Equation 1. RVHS JC2 Prelim 8865/2019/Q1 Find the exact range of values of the constant k for which the equation 2 20kx x k    has 2 distinct real roots. [4] RVHS JC2 Prelim 8865/2019/Q1 (Solutions) Since the equation has 2 real distinct roots,      2 2 2 Discriminant 0 1 4 2 0 1 4 8 0 4 8 1 0 kk kk kk           551 1 022kk                            5511 22 k     2. CJC JC2 Prelim 8865/2019/Q2 Find the set of values of k for which the equation   2 3 3 0kx k x k     has real roots. Without carrying out further calculations, state the set of values of k for which   2 3 3 0kx k x k     for all real values of x. [4] CJC JC2 Prelim 8865/2019/Q2 (Solutions) 2 Since the equation has real roots, Discriminant 0        2 2 3 4 3 0 3 6 9 0 3 1 3 0 13 k k k kk kk k              The set of values is  : 1 3kk    . Side working: Let 24 8 1 0kk   .   8 64 4 4 1 8 8 80 8 51 2 k        x

If   2 3 3 0kx k x k     for all real values of x, 0k  and Discriminant 0    0 1 and 1 or 3 2k k k    Combining (1) and (2), 3k  The set of values is  :3kk . Section 2: Conditions for Quadratic Equation to be always positive or negative 3. ASRJC JC2 Prelim 8865/2019/Q1 Find algebraically the exact set of values of k for which 2 (3 1) (4 4 ) 0kx k x k     for all real values of x. [5] ASRJC JC2 Prelim 8865/2019/Q1 (Solutions) For 2 (3 1) (4 4 ) 0kx k x k     for all real values of x, 2 conditions need to be satisfied: (i) the coefficient of x2 must be positive 0k --- (1) and (ii) Discriminant < 0 2 22 2 2 (3 1) 4 (4 4 ) 0 9 6 1 16 16 0 7 10 1 0 7 10 1 0 k k k k k k k kk kk                 5 4 2 5 4 2or77kk     ---(2) Combining (1) and (2), 5 4 2 7k  The set of values of k is 5 4 2: 7kk  . 4. EJC JC2 Prelim 8865/2019/Q1 Find algebraically the set of values of k for which   2 20kx k x k    for all real values of .x [4] EJC JC2 Prelim 8865/2019/Q1 (Solutions) For   2 20kx k x k    , Discriminant 0 and 2coefficients of x > 0 Side working: Consider 27 10 1 0kk   210 10 4(7)( 1) 2(7) 10 128 5 4 2 14 7 k        

  2 2 4( )( ) 0k k k   and  02k  22 4 4 4 0k k k    23 4 4 0kk   (3 2)( 2) 0kk    22 or 1 3kk  Combining (1) and (2), the set of values is 2{ : } 3kk 5. NYJC JC2 Prelim 8865/2019/Q1 Find the exact range of values of k for which   212 k x x k 