Title IX_E-TECHNO_VOL-2_ADDITIONAL PROBLEMS_221-260.pdf ✅

e-Techno Text Book IX-Mathematics (Vol-2) 221 ADDITIONAL PROBLEMS QUADRATIC EXPRESSIONS & EQUATIONS SYNOPSIS - 1 1. The harmonic mean of the roots of the equation 5 2  4 5 8 2 5 0 2  x   x    is 1) 2 2) 4 3) 6 4) 8 2. If ,  are the roots of 3x2 + 5x - 7 = 0, then the value of 2 2 3 5 1 3 5 1                      is 1) 63 67 2) 441 67 3) 441 109 4) 63 109 3. The value of 8 2 8 2 8 2 8 .....      is 1) 10 2) 6 3) 8 4) 4 4. The equation     2 2 log 3 log 1 3     x x has 1) One root 2) Two roots 3) Infinite roots 4) No root 5. If k > 0 and the product of the roots of the equation x2 -3kx+2e2logk-1=0 is 7 then the sum of the roots is 1) 2 2) 4 3) 6 4) 8 6. The quadratic equation whose roots are i i 2 3 2 3   and i i 2 3 2 3   is 1) 5x2 -2x+5=0 2) 5x2+2x+5=0 3) 5x2+2x-5=0 4) 5x2 -2x-5=0 7.     2 2 10 10 log log 3 1 1 x x x x     then x _________. 1) 1 10 2) 2 3) 3 10 4)1 HINTS & SOLUTIONS 1. (2) H.M. = 2 2 8 2 5   4 5        = 4 2. (2)     2 2 2 2 2 1 1 2 b ac a b a b   a c      25 42 9 49    = 441 67 3. (4) Let x to       8 2 8 2 8 2 8 .....  8 2x 2     x x2 8 0

e-Techno Text Book IX-Mathematics (Vol-2) 223 6. The graph of the function   2 y x a x a      16 8 5 7 5 is strictly above the x axis then ‘a’ must satisfy the inequality 1)     15 2 a 2)     2 1 a 3) 5 7   a 4) 2 15   a COMPREHENSION TYPE: If    and x x x            0 ,   If    and x x x            0 ,   If    and x x x               0 , ,     If a   and x x x               0 , ,     7. If   and are real and distinct roots of 2 2 2 1 1 | | x x satisfy a a       then a belongs to 1) 1 1 ,0 0, 2 2 U              2) 1 1 ,0 0, 2 5 U              3) 1 1 ,0 0, 5 5 U              4) None 8. The equation 1 1 b x a x a x     has no real roots then 1) 0 1   a 2) 0 1   b 3) 0 2   a 4) 0 2   b 9. The set of all real numbers x for which 2 x x x     2 0 , is 1)    ,2 2,    2)     , 2 2,    3)     , 1 1,    4)  2, HINTS & SOLUTIONS 1. [2] 2 8 2 0 2 x  x   ; 4 1 0 2 x  x   root = 2 3 2 4 16 4      , 2  3 2  3,  2. (2) Let 3 x  t 1 2 3 3 4 0 4 3 0 x x t t              t t 1 3 0   1 3 1 3 3 0,1   x         t x 3. (2); Let  , be the roots of x2 + abx + c = 0 and  , be the roots of x2 + acx + b = 0,  being the common root.       ab ------- (1)   c ------- (2)
IX-Mathematics (Vol-2) e-Techno Text Book 224      ac ------- (3)   b --------(4) From (1) – (3),      a c b   From (2) – (4),         c b      c b a c b           or 1 a   .  from (2) and (4), c a   , i.e.,   ac and b, a   i.e.,  = ab.  The quadratic equation whose roots are  , is   2 x x 0        or x2 – (ac + ab) x + ac.ab = 0 or x2 – a(b + c)x + a2bc = 0. 4. (4)     116 0 Roots are imaginary and conjugate to each other Both roots are common 2 3 3 8 15 a b c    a b c    3 , 4 , 5 ABC is right angle triangle 2 2 2 c a b   5. [3]  x x    11 7 0   and  x x    2 2 0   x     11, 2 2,7    6. (1)     2         16 8 5 7 5 0 x a x a x R   0     2      64 5 64 7 5 0 a a         a a a 15 2 0 15 2   7. (2) 2 a ax x a     0 0 has two roots 1 and , 1 a          disc =    2 2 1 4 0;4 1 0 2 1 2 1 0 1 1 1 2 2             a a a a a    1 a           1 1 2 4 a a       2 1 4 1 a    2 1 5 a   ; 2 1 1 1 5 5 5 a a      8. (4) 1 1 b x a x a x      2 2 2x b x a x    2 2 2 2x bx a b    2 2 2 a b x b   For the above equation have no real roots 2 x  0   2 0 0 2 0 2 2 a b b b b b b          ; 0 2   b