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Quadratic Equations and Inequalities

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  1. 51.cosec18[^] is a root of the equation :
  2. 52.The number of real solutions of the equation, x2 - |x| - 12 = 0 is :
  3. 53.The number of real roots of the equation [e^6x - e^4x - 2e^3x - 12e^2x + e^x + 1 = 0] is :
  4. 54.If [ ] and [ ] are the roots of the equation 2x(2x + 1) = 1, then [ ] is equal to :
  5. 55.The product of the roots of the equation 9x2 - 18|x| + 5 = 0 is :
  6. 56.The number of real roots of the equation, e4x + e3x -- 4e2x + ex + 1 = 0 is :
  7. 57.The number of real roots of the equation 5 + |2x -- 1| = 2x (2x -- 2) is
  8. 58.If the set of all a R-1, for which the roots of the equation (1-a)x^2+2(a-3)x+9=0 are posi…
  9. 59.If the equation a(b-c)x^2+b(c-a)x+c(a-b)=0 has equal roots, where a+c=15 and b=(36)/(5), t…
  10. 60.Let , be roots of x^2+√2x-8=0.
  11. 61.Let x_1,x_2,x_3,x_4 be the solution of the equation 4x^4+8x^3-17x^2-12x+9=0 and (4+x_2^1)(…
  12. 62.The number of real solutions of the equation x|x+5|+2|x+7|-2=0 is __________.
  13. 63.Let a, b, c be the lengths of three sides of a triangle satistying the condition [(a^2+b^2…
  14. 64.The number of real solutions of the equation x(x^2+3|x|+5|x-1|+6|x-2|)=0 is _________.
  15. 65.Let , N be roots of the equation x^2-70x+ =0, where ( )/(2),( )/(3) N.
  16. 66.Let the set C=(x,y)|x^2-2^y=2023,x,y N.
  17. 67.Let [ ] denote the greatest integer .
  18. 68.The number of points, where the curve f(x)=e^8x-e^6x-3e^4x-e^2x+1,x R cuts x-axis, is equa…
  19. 69.If a and b are the roots of the equation x^2-7 x-1=0, then the value of [a^21+b^21+a^17+b^…
  20. 70.Let m and n be the numbers of real roots of the quadratic equations x^2-12x+[x]+31=0 and x…
  21. 71.If the value of real number a>0 for which x^2-5 a x+1=0 and x^2-a x-5=0 have a common real…
  22. 72.Let q be the maximum integral value of p in [0,10] for which the roots of the equation x^2…
  23. 73.Let , be the roots of the quadratic equation 12x^2 - 20x + 3 = 0, Z.
  24. 74.If , ( < ) are the roots of the equation x^2 - (+3)x + 3 = 0 such that 1 - 1 = 13, then th…
  25. 75.The sum, of the squares of all the roots of the equation x^2+|2x-3|-4=0, is
  26. 76.The region represented by z=x+iy C: |z|-Re(z)1 is also given by the inequality:
  27. 77.The number of integral values of m for which the quadratic expression (1+2m)x^2 - 2(1+3m)x…
  28. 78.The number of integral values of m for which the equation (1+m^2)x^2 - 2(1+3m)x + (1+8m) =…
  29. 79.If a, b, c are distinct positive real numbers and a^2+b^2+c^2=1, then ab+bc+ca is
  30. 80.If p and q are the roots of the equation x^2 + px + q = 0, then
  31. 81.If the sum of the roots of the quadratic equation ax^2 + bx + c = 0 is equal to the sum of…
  32. 82.The value of 'a' for which one root of the quadratic equation [ ( a^2 - 5a + 3 )x^2 + ( 3a…
  33. 83.The number of real solutions of the equation [x^2 - 3| x | + 2 = 0] is
  34. 84.If [( 1 - p )] is a root of quadratic equation [x^2 + px + ( 1 - p ) = 0] then its root ar…
  35. 85.If one root of the equation x^2 + px + 12 = 0 is 4, while the equation x^2 + px + q = 0 ha…
  36. 86.If the roots of the equation x^2 - bx + c = 0 be two consecutive integers, then b^2 - 4c e…
  37. 87.The value of a for which the sum of the squares of the roots of the equation [x^2 - ( a - …
  38. 88.In a triangle [PQR,;; R = 2.,,If,, ,( P 2 )] and [ ( Q 2 )] are the roots of [ax^2 + bx + …
  39. 89.If both the roots of the quadratic equation x^2 - 2kx + k^2 + k - 5 = 0 are less than 5, t…
  40. 90.If x is real, the maximum value of [3x^2 + 9x + 17 3x^2 + 9x + 7] is
  41. 91.If the roots of the quadratic equation x^2 + px + q = 0 are [ 30^ ] and [ 15^ ], respectiv…
  42. 92.STATEMENT - 1 : For every natural number [n 2,] [ 1 1 + 1 2 + ........
  43. 93.If the roots of the equation bx^2 + cx + a = 0 imaginary, then for all real values of x, t…
  44. 94.If [ ] and [ ] are the roots of the equation x^2 - x + 1 = 0, then [ ^2009 + ^2009 = ]
  45. 95.If the equations x^2 + 2x + 3 = 0 and ax^2 + bx + c = 0, [a,,b,,c, ,R,] have a common root…
  46. 96.Let [ ] and [ ] be the roots of equation px^2 + qx + r = 0, [p 0.] If [p,,q,,r] in A.P.
  47. 97.Let [ ] and [ ] be the roots of equation x^2 - 6x - 2 = 0.
  48. 98.The sum of all real values of x satisfying the equation [( x^2 - 5x + 5 )^x^2 + 4x - 60, =…
  49. 99.If x is a solution of the equation, [ 2x + 1 ] [ - 2x - 1 = 1,] [,,( x 1 2 ),] then [ 4x^2…
  50. 100.If the equations x2 + bx-1 = 0 and x2 + x + b = 0 have a common root different from -1, th…