Tiêu đề XI - maths - chapter 5 - QUADRATIC EQTS _ EXPRESSIONS (75-94).pdf ✅

NARAYANAGROUP 75 QUADRATIC EQUATIONS & EXPRESSIONS JEE-MAIN-SR-MATHS VOL-I 61. Put 1 3 x t  62. 1 1 2 4 2 2 y y y           y y 5 0   63. 2 ax bx c    0       2 a k x b k x c k       0 a k b k c k a b c       k k k a b c a b c      2     x x 1 0 64. Let   2 f x ax bx c    equation whose roots 1 1,       is 1 0 1 f x         2     px qx r 0     2 & 1 1 0 c x b x a      are same. 65. 2 2 1 2 4 3 3 2 4 x x x R x x             2 1 1 2 2 2 1 1 93. 6.3 4 3 2.3 4 93. 6.3 4 3 2.3 4 x x x x x x x x              2 2 2 4 2 4 y y y y      when 1 3x y   66. min value 2 2 4 1 4 1 ac b a b       2 1 1 m b b    Range = 0,1 67.          1 4 3 3 4 5 0 4 5 4 3 x x x x x x            18 0 4 5 4 3 x x         4 5 4 3 0; x x   5 3, 4 4 x         68. We have,   2 5 5 log log 2 x x   Put 5 log x a  ,then    2 a a a a       2 2 1 0     2 1 a or 5    2 log 1 x 2 5 5 x     or 1 5 25  x 69.      x  , 5 1,2    From the diagram of wave nature of the curve 70. solve   3 x x ax    1 0 i.e., 4 2 x ax x    0 and 4 2 x ax   1 0 71. range = , c c f f a a                     72. 4 R   3 4 P Q     tan tan 3 3 4     P Q               a b c 73. 5 2 b      and 8 2 5 5 2     given that harmonic mean between the roots of the given equation is 4. 2 4      ; 8 2 5 2 b    4 5 1. If  , are the roots of the equation 2 8 3 27 0 x x    , then the value of 1 1 2 2   3 3                is 1) 1 3 2) 1 4 3) 7 2 4) 4 2. If  , are the roots of 2 x px q    0 then the equation whose roots are      ,      is 1) 2 2 2 x qx p q     2 0 2) 2 2 2 x qx p q     2 0 3) 2 2 2 x qx q p     2 0 4) 2 2 2 x qx q p     2 0 3. If sin ,cos   are the roots of the equation 2 ax bx c    0 then 1) 2 2 a b ac    2 0 2) 2 2 a b ac    2 0 3) a b ac    2 0 4) a b c    2 0 LEVEL-II (H.W)
76 NARAYANAGROUP QUADRATIC EQUATIONS & EXPRESSIONS JEE-MAIN-SR-MATHS VOL-I 4. If  , are the roots of 2 x x   1 0 then 5 5     1) 2 2) 1 3) 12 4) 4 5. If  , are the roots of 2 3 5 7 0 x x    then the value of 2 2 1 1 3 5 3 5                  is 1) 67 63 2) 67 441 3) 109 441 4) 109 63 6. If the sum of the roots of the equation 2 ax bx c    0 is equal to sum of their squares, then 1) 2 ab b ac    2 0 2) 2 ab a ac    2 0 3) 2 ab b ac    2 0 4) 2 ab a ac    2 0 7. If  , are the roots of 2 6 6 1 0 x x    then     1 1 2 3 2 3 2 2 a ba c d a b c d              1) a b c d    2) a b c d    2 3 4 3) a b c d     / 2 / 3 / 4      4) 0 8. If  , are the roots of 2 x ax b    0 and  , are the roots of 2 x ax b    0 then                  1) 2 4b 2) 2 b 3) 2 2b 4) 2 3b 9. If the arithmetic mean of the roots of a quadratic equation is 8/5 and the arithmetic mean of their reciprocal is 8/7 then the equation is 1) 2 5 16 7 0 x x    2) 2 5 16 7 0 x x    3) 2 7 16 5 0 x x    4) 2 7 16 5 0 x x    10. If one root of the equation 2 8 6 0 x x k    is the square of the other, then k= 1) 0,3 2) -1,27 3) 0,-2 4) 1,-27 11. If the ratio of the roots of 2 ax bx c    2 0 is same as the ratio of the roots of 2 px qx r    2 0 then 1) 2 2 b p ac qr  2) b q ac pr  3) 2 2 b q ac pr  4) 2 b q ac pr  12. If the difference of the roots of the equation 2 x bx c    0 is equal to the difference of the roots of the equation 2 x cx b    0 and b c  then b c   1) 0 2) 2 3) 4 4)-4 13. Which one of the following Means of the roots of the equation 2 2 x bx a    2 0 is the A.M. of the roots of 2 2 x ax b    2 0 ? 1) A.M 2) G.M 3) H.M 4)A.G.P 14. If tanA, tanB are the roots of 2 x x    2 2 0 then   2 sin A B  1) 4/5 2) 1/2 3) 3/5 4) 1/4 15. If the roots of 2 1 1 x bx k ax c k      are numerically equal but opposite in sign then k= 1) c 2) 1 c 3) a b a b   4) a b a b   NATURE OF THE ROOTS AND PROPERTIES 16. If pr q s   2  then among the equations 2 x px q    0 and 2 x rx s    0 1) both have real roots 2) both have imaginary roots 3) at least one has real roots 4) at least one has imaginary roots. 17. The roots of the equation       2 b c x c a x a b       2 0 are always 1) real and distinct 2) real and equal 3) real 4) imaginary 18. The roots of the equation       2 a c b x cx b c a a b         2 0 are 1) real and distinct 2) real and equal 3) real 4) imaginary 19. If the roots of the equation 2 x cx ab    2 0 be real and unequal, the roots of the equation     2 2 2 2 x a b x a b c       2 2 0 are 1) real and distinct 2) real and equal 3) real 4) imaginary
NARAYANAGROUP 77 QUADRATIC EQUATIONS & EXPRESSIONS JEE-MAIN-SR-MATHS VOL-I 20. If a,b,c,d are real and no two of them are simultaneously zero, then the equation      2 2 2 3 2 0 x ax b x cx b x dx b        has 1) at least two real roots 2) at least four real roots 3) all roots real 4) at least two imaginary roots 21. a,b,c are rational. Then the roots of       2 a b c x a c x a b c         2 0 are 1) real 2) equal 3) rational 4) imaginary SOLVING EQUATIONS 22. If the equation       2 2 2 2              5 6 3 2 4 0 x is satisfied by more than two values of x then   1) 1 2) -2 3) 3 4)2 23. The number of real roots of the equation     2 2 3 3 1 3 1 2 x x x     is equal to 1) 0 2) 1 3) 2 4) more than 2 24. If a Z  and the equation  x a x      10 1 0  has integral roots, then values of ‘a’ are 1) 10,8 2) 12,10 3) 12,8 4) 10,12 25. The number of real values of x for which 2 2 4 x a x x b          , where a b, 0  1) 1 2) 2 3) 0 4) infinite 26. Number of solutions of the equation x x  cos 1) 1 2) 2 3) 0 4) 3 COMMON ROOTS 27. The value of  in order that the equations 2 2 5 2 0 x x     and 2 4 8 3 0 x x     have a common root is given by 1) 1 2) -1 3) 1 4) 3 28. If every pair from among the equations 2 x ax bc    0 , 2 x bx ca    0 and 2 x cx ab    0 has common root, thent h e sum of three common roots is 1) abc 2) 2abc 3) 3a b c    4)a b c    29. If 2 x hx    21 0 ,   2 x hx h     3 35 0 0 have a common root, then the value of h is 1) 8 2) 4 3)4 4) 2 30. The conditions that a root of the equation 2 ax bx c    0 may be reciprocal to a root of 2 1 1 1 a x b x c    0 is 1)      2 1 1 1 1 1 1 bb aa ab bc ba b c     2)      2 1 1 1 1 1 1 cb ba ac cd ab bc     3)     2 1 1 1 1 1 1 cc aa ab bc ba b c     4) a b c    0 LOCATION OF ROOTS 31. If both roots of the equation 2 x x a    0 exceed ‘a’ then 1) 2 3   a 2) a  3 3)    3 3 a 4) a  2 32. If both roots of the equation 2 2 x ax a     2 1 0 lie in the interval 3,4 then sum of the integral parts of ‘a’ is 1)0 2) 2 3) 4 4)-1 33. The expression       2 a x a x a      2 2 2 3 5 6 is positive for all real values of x, then 1) ‘a’ can be any real number 2) a>1 3) a>3 4) a=3 SIGN OF THE EXPRESSIONS AND INEQUATIONS 34. If 2 x ax b c x R      2 then 1) 2 b c a   2) 2 c a b   3) 2 a b c   4) a b c    0 35. The roots of the equation          0 x a x b x b x c x c x a          are real and equal if 1) a b c   2) a b c   3) a b c   4) a b c    0 36. The largest negative integer for which       4 2 0 1 5 x x x x      is 1) -2 2) -1 3)-3 4)-4
78 NARAYANAGROUP QUADRATIC EQUATIONS & EXPRESSIONS JEE-MAIN-SR-MATHS VOL-I MAXIMUM AND MINIMUM OF EXPRESSION 37. Let   2 2 f x x bx c    2 2 and   2 2 g x x cx b     2 , then min max  f x g x       holds for 1) 0 2  c b 2) c b  2 3) c b  2 4) no real value of b and c MODULUS FUNCTIONS 38. Number of rational roots of the equation 2 x x x     2 3 4 0 is 1) 1 2) 2 3) 3 4) 4 39. The roots of the equation 2 x x x     6 2 are 1) 1,-2,2 2) 1,2,4 3) -2,2,4 4) -2,2,3 40. For a>0, all the real roots of the equation 2 2 x a x a a     3 7 0 are 1) 4a,5a 2) -4a,5a 3) -4a, -5a 4) 4a,-5a 41. The set of real values of x satisfying x   1 3 and x   1 1 1) 2,4 2)   , 2 4,    3)   2,0 2,4    4)0,2 OTHER MODELS 42. If sin ,sin  k are the roots of 2 4 2 1 0 x x    , then k = 1) -2 2)-3 3) 3 4) 2 43. If  , are the roots of 2 x x a    3 0 and  , that of 2 x x b    12 0 and     , , , form an increasing G.P. then 1) a b   3, 12 2) a b   4, 16 3) a b   2, 32 4) a b   12, 3 44. Let f x  be a polynomial for which the remainders when divided by x x x    1, 2, 3 respectively 3,7,13. Then the remainder of f x  when divided by x x x    1 2 3    is 1) f x  2) 2 x x  1 3) 2 x 1 4) x  2 45. Let a,b,c be the real numbers, a  0 , If  is a root of 2 2 a x bx c    0,  is a root of 2 2 a x bx c    0 and 0     , then the equation 2 2 a x bx c    2 2 0 , has a root  that always satisfies 1)        / 2 2)      / 2 3)   4)     46. The equation       3 3 3 x a x b x c       0 has 1) all the roots are equal 2) one real and two imaginary 3) 3 real roots namely x a x b x c    , , 4) no real roots TRANSFORMED EQUATIONS 47. The value of ‘a’ for which the quadratic equation     2 2 2 3 2 1 3 2 0 x a x a a       possesses roots of opposite signs lies on 1) ,1 2),0 3) 1,2 4)1,2 48. If the roots of the equation 2 ax bx c    0 are the reciprocals of the roots of the equation 2 px qx r    0 then 1) 2 2 acq b pr  2) ac pr  3) 2 2 b ac q pr  4) ab pq  RANGE 49. If y x x  tan cot 3 , x R  then 1) 1 1 3   y 2) 1 1 3   y 3) 1 3 3   y 4) 1 3 3 y or y   50. If   2 5 2 3 13 x x x   then the solution set for x is 1) 2, 2) 2 3) ,2 4) 0,2 INEQUALITIES 51. If 2 2 x x x x        6 27 0; 3 4 0 then x lies in the interval 1) 3,4 2)3,4 3)    ,3 4,    4)9,4