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Shvoong Home>Science>Mathematics>Problem: Quadratic Equations 03 Summary

( 21 ) The equation x + 1 - x - 1 = 4x - 1 has
( a ) no solution ( b ) one solution
( c ) two solutions ( d ) more than two solutions [ IIT 1997 ]
( 22 ) If p, q, r are positive and are in A. P., then the roots of the quadratic equation
px2 + qx + r = 0 are real for
( a ) 7 4 3
p
r - ≥ ( b ) 7 4 3
r
p - ≥
( c ) all p and r ( d ) no p and r [ IIT 1995 ]
( 23 ) Let f ( x ) be a quadratic expression which is positive for all real x. If g ( x ) = f ( x ) +
f ’ ( x ) + f ” ( x ), then for any real x
( a ) g ( x ) < 0 ( b ) g ( x ) > 0 ( c ) g ( x ) = 0 ( d ) g ( x ) ≥ 0 [ IIT 1990 ]
( 24 ) If α and β are the roots of x2 + px + q = 0 and α4 and β4 are the roots of
x2 - rx + s = 0, then the equation x2 - 4qx + 2q2 - r = 0 has always
( a ) two real roots ( b ) two positive roots
( c ) two negative roots ( d ) one positive and one negative root [ IIT 1989 ]
( 25 ) Let a, b, c be real numbers, a ≠ 0. If α is a root of a2 x2 + bx + c = 0, β is a
root of a2 x2 - bx - c = 0 and 0 < α < β, then the equation a2 x2 + 2bx + 2c = 0
has a root γ that always satisfies
( a )
2
α + β
γ = ( b )
2
γ = α + β ( c ) γ = α ( d ) α < γ < β [ IIT 1989 ]
( 26 ) The equation 2
4
x ( log x ) log x 5 2
2
2
4
3
+ - = has
( a ) at least one real solution ( b ) exactly three real solutions
( c ) exactly one irrational solution ( d ) complex roots [ IIT 1989 ]
( 27 ) The equation x -
x 1
2- =
1
-
x 1
2-
has
( a ) no root ( b ) one root
( c ) two equal roots ( d ) infinitely many roots [ IIT 1984 ]
( 28 ) For real x, the function
( x c )
( x a ) ( x b )
-
- - will assume all real values provided
( a ) a > b > c ( b ) a > b > c ( c ) a > c > b ( d ) a < c < b [ IIT 1984 ]
04 - QUADRATIC EQUATIONS Page 5
( Answers at the end of all questions )
( 29 ) If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0 has
( a ) at least one root in [ 0, 1 ]
( b ) one root in [ 2, 3 ] and the other in [ - 2, - 1 ]
( c ) imaginary roots ( d ) none of these [ IIT 1983 ]
( 30 ) The number of real solutions of the equation l x l2 - 3 l x l + 2 = 0 is
( a ) 4 ( b ) 1 ( c ) 3 ( d ) 2 [ IIT 1982 ]
Published: June 12, 2010
 Please Rate this Summary : 1 2 3 4 5
1. Answer   Question  :    if a,b,c,d are positive reals such that a+b+c+d=2 and M =(a+b)(c+d) ,than m belongs to which interval.
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