04 - QUADRATIC EQUATIONS
Page 1
( Answers at the end of all questions )
The value of a for which the sum of the squares of the roots of the equation 2 x - ( a - 2 ) x - a - 1 = 0 assume the least value is
(b) 3
om
2
[ AIEEE 2005 ]
- b x + c = 0 be two consecutive integers, then
(c) 2
(d) 1
If both the roots of the quadratic equation x then k lies in the interval ( b ) ( 6, ∞ )
( a ) ( 5, 6 ]
(4)
(d) 2
If the roots of the equation x 2 b - 4 c equals (a) -2
(3)
(c) 3
2
(c) (
∞, 4)
( d ) [ 4, 5 ]
2
2
( b ) x - 18x + 16 = 0 2 ( d ) x - 18x - 16 = 0
xa
.e
If ( 1 - p ) is a oot of quadratic equation x ( b ) - 1, 1
w w
( a ) 0, 1
w
(6)
(7)
[ AIEEE 2005 ]
Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic e uation ( a ) x + 18x + 16 = 0 2 ( c ) x + 18x - 16 = 0
(5)
[ AIEEE 2005 ]
- 2kx + k + k - 5 = 0 are less than 5,
ra
(2)
(b) 0
ce .c
(a) 1
m
(1)
( c ) 0, - 1
2
[ AIEEE 2004 ]
+ px + ( 1 - p ) = 0, then the roots are
( d ) - 1, 2
[ AIEEE 2004 ]
2
If one root of the equation x + px + 12 = 0 is 4, while the equation 2 x + px + 12 = 0 has equal roots, then the value of q is (a)
49 4
( b ) 12
(c) 3
(d) 4
The number of real solutions of the equation x (a) 2
(b) 4
(c) 1
(d) 3
[ AIEEE 2004 ]
2
- 3 l x l + 2 = 0 is [ AIEEE 2003 ]
04 - QUADRATIC EQUATIONS
Page 2
( Answers at the end of all questions ) The value of ‘ a ’ for which one root of quadratic equation 2 2 ( a - 5a + 3 ) x + ( 3a - 1 ) x + 2 = 0 is twice as large as the other is
(9)
(b)
-
2 3
If roots of the equation 2
x
2
2
x + px + q = 0 are α + β
1 3
[ AIEEE 2003 ]
- 5x + 16 = 0 are α, β and roots of the equation
2
αβ
and
2
, then
( b ) p = - 1 and q = - 56 ( d ) p = - 1 and q = 56
[ AIEEE 2002 ]
( b ) b and c ( d ) ( a + b ) and ( b + c )
2
3
2
3
2
( b ) p - q ( 3p + 1 ) + q = 0 3 2 ( d ) p + q ( 3p + 1 ) + q = 0
xa
( a ) p - q ( 3p - 1 ) + q 0 3 2 ( c ) p + q ( 3p - 1 ) + q = 0
[ IIT 2004 ]
+ 2ax + 10 - 3a > 0 for every real value of x, then
.e
2
[ AIEEE 2002, IIT 1992 ]
q = 0 is square of the other, then for any p
m
If one root of the equation x + px and q it will satisfy the relation
( 12 ) If x
, c ≠ 0, then the roots
ce
If α and β be the roots of the equation ( x - a ) ( x - b ) = of the equation ( x - α ) ( x - β ) = c are ( a ) a and c ( c ) a and b
( 11 )
-
(d)
.c
( a ) p = 1 and q = - 56 ( c ) p = 1 and q = 56
( 10 )
1 3
(c)
om
2 3
(a)
ra
(8)
w w
(a) a > 5
(b) a < -5
(c) -5 < a < 2
2
(d) 2 < a < 5
[ IIT 2004 ]
2
( 13 ) If minimu value of f ( x ) = x + 2bx + 2c is greater than the maximum value of 2 2 g ( x ) = - x - 2cx + b , then for real value of x
w
(a) lcl > lbl (c) 0 < c <
2
2b
(b) lcl
2 > b
( d ) no real value of a
( 14 ) The set of all real numbers x for which x ( a ) ( - ∞, - 2 ) ∪ ( 2, ∞ )
( b ) ( - ∞, -
( c ) ( - ∞, - 1 ) ∪ ( 1, ∞ )
(d) (
2
- l x + 2 l + x > 0, is
2)∪(
2, ∞)
[ IIT 2003 ]
2, ∞) [ IIT 2002 ]
04 - QUADRATIC EQUATIONS
Page 3
( Answers at the end of all questions )
log4 ( x - 1 ) = log2 ( x - 3 ) is
( 15 ) The number of solutions of (a) 3
(b) 1
(c) 2
(d) 0
[ IIT 2001 ]
2
( 16 ) If α and β are the roots of the equation x + bx + c = 0, where c < 0 < b, then
(a)
1 3
2
om
For the equation 3x then p is equal to
[ IIT 2000]
+ px + 3 = 0, p > 0, if one of the roots is square of the other,
(b) 1
(c) 3
(d)
2 3
[ IIT 2000 ]
0 has
ra
( 18 ) If b > a, the equation ( x - a ) ( x - b ) - 1
ro t in ( - ∞, a ) and the other in ( b, + ∞ )
(b) o
m
( a ) both roots in ( a, b )
.c
( 17 )
(b) α < 0 < β < lαl (d) α < 0 < lαl < β
ce
(a) 0 < α < β (c) α < β < 0
( d ) both roots in ( - ∞, a )
xa
( c ) both roots in ( b, + ∞ )
[ IIT 2000 ]
( 19 ) The harmonic mean of the oots of the equation 2
2 )x - (4 +
(5 +
(b
4
w w
.e
(a) 2
5 )x + 8 + 2
( 20 ) If the roots
w
(a) a < 2
2
The equation
(c) 6
f the equation x
2
(d) 8
( a ) no solution ( c ) two solutions
[ IIT 1999 ]
- 2ax + a2 + a - 3 = 0 are real and less than 3, then
(b) 2 ≤ a ≤ 3
x + 1 -
5 = 0 is
x - 1 =
(c) 3 < a ≤ 4
(d) a > 4
[ IIT 1999 ]
4x - 1 has
( b ) one solution ( d ) more than two solutions
[ IIT 1997 ]
04 - QUADRATIC EQUATIONS
Page 4
( Answers at the end of all questions )
If p, q, r are positive and are in A. P., then the roots of the quadratic equation 2 px + qx + r = 0 are real for r p (b) (a) - 7 ≥ 4 3 - 7 ≥ 4 3 p r
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
( 24 )
(b) g(x) > 0
(d) g(x) ≥ 0
(c) g(x) = 0
2
4
[ IIT 1990 ]
4
If α and β are the roots of x + px + q = 0 and α and β are the roots of 2 2 2 x - rx + s = 0, then the equation x - 4qx + 2q - r = 0 has always ( b ) two positive oots ( d ) one positiv and one negative root
ra
( a ) two real roots ( c ) two negative roots
2
[ IIT 1989 ]
2
Let a, b, c be real numbers, a ≠ 0. If α is a root of a x + bx + c = 0, β is a 2 2 2 2 root of a x - bx - c = 0 and 0 < α < β, then the equation a x + 2bx + 2c = 0 has a root γ that always satis ies α + β β (a) γ = (b) γ = α + (c) γ = α (d) α < γ < β [ IIT 1989 ] 2 2
xa
m
( 25 )
[ IIT 1995 ]
.c
( 23 )
( d ) no p and r
om
( c ) all p and r
ce
( 22 )
3
x 4 ( og x ) 2 + log x -
.e
( 26 ) The equation
2
2
5 4
=
2
has
w w
( a ) at le st one real solution ( b ) exactly three real solutions ( c ) exactly one irrational solution ( d ) complex roots
w
( 27 ) The equation x -
2 2 = 1 has x - 1 x - 1
( a ) no root ( c ) two equal roots
( b ) one root ( d ) infinitely many roots
( 28 ) For real x, the function (a) a > b >c
[ IIT 1989 ]
[ IIT 1984 ]
( x - a )( x - b) will assume all real values provided (x - 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 3ax
2
+ 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 ]
om
2
( 30 ) The number of real solutions of the equation l x l - 3 l x l + 2 = 0 is (b) 1
(c) 3
(d) 2
[ IIT 1982 ]
.c
(a) 4
( 31 ) If a > 0, b > 0 and c > 0, then both the roots of the equation ax ( b ) have negative rea pa ts
ce
( a ) are real and negative ( c ) none of these
( 32 ) Both the roots of the equation ( x - b ) ( x are always
ra
( b ) negative
c
(c
re l
[ IIT 1980 ]
( x - a ) ( x - c ) + ( x - a )( x - b ) = 0
( d ) none of these
[ IIT 1980 ]
m
( a ) positive
+ bx + c = 0
xa
( 33 ) If l, m, n are real, l ≠ m, th n the roots of the equation 2 ( l - m ) x - 5 ( l + m ) x - 2 ( l - m ) = 0 are ( b ) complex d ) none of these
.e
( a ) real and equal ( c ) real and unequal
2
+ kx - x + 9 is strictly above the X-axis if
w w
( 34 ) The entire graph of the equation y = x and on y if (a) k < 7
w
( 35
(b) -5 < k < 7
(c) k > -5
If α and β are roots of the equation ax 2 2 (1 + α + α )(1 + β + β ) = (a) 0
( b ) positive
[ IIT 1979 ]
( c ) negative
2
( d ) none of these
+ bx + c = 0, then
( d ) none of these
[ IIT 1979 ]
04 - QUADRATIC EQUATIONS
Page 6
( Answers at the end of all questions ) 2
( 36 ) If the two equations ax + bx + c = 0 and px then the value of ( aq - bp ) ( br - cq ) is ( a ) - ( ar - cp )
2
( c ) ( ac - pr )
( d ) ( ar - cp )
( c ) ( 1,
+ 2x + p ( p -
∞)
2
( 39 ) The value of p for which 2 x + px + 8 = 0 is 2 are
( 40 ) If a > 0, then 1 2
4a - 1
( d ) none of these
differen e
(c) ±6
a +
a +
etween
the
roots
of
the
equation
(d) ±8
a + ..... ∞
[
1 1+ 2
(b)
.e
(a)
the
are equal, then
m
(b) ±4
xa
(a) ±2
ce
( c ) A. P.
) = 0
∞
( d ) ( 0,
If the roots of the equation a ( b - c ) x + b ( c - a )x + c( a - b ) = 0 a, b, c are in ( b ) G. P.
2
om
( b ) ( 0, 1 )
2
.c
∞, 0)
( a ) H. P.
4a - 1
=
]
(c)
[
1 12
4a - 1
]
( d ) none of these
2
If for the quadratic equation ax + bx + c = 0, the difference of the roots is the same as their product, then the ratio of the roots is
w w
( 41 )
2
+ qx + r = 0 have a common root,
The set of values of p for which the roots of the equation 3x are of opposite signs is (a) (-
( 38 )
( b ) ( ap - cr )
ra
( 37 )
2
2
a - b a + b
(b)
b - c b + c
(c)
c - a c + a
( d ) none of these
w
(a)
( 2 ) The integral values of m for which the roots of the equation 2 mx + ( 2m - 1 ) x + ( m - 2 ) = 0 are rational for rational k are given by (a) k(k + 1)
(b)
k2 - 1 4
(c)
k (k + 2) 4
( d ) none of these
04 - QUADRATIC EQUATIONS
Page 7
( Answers at the end of all questions ) 2
+ 6x - 27 > 0 and - x + 3x + 4 > 0, the x lies in the interval
( a ) ( 3, 4 )
( c ) ( - 9, 3 ] ∪ [ 4, 9 )
( b ) [ 3, 4 ]
log
( 44 ) The roots of the equation 7 ( a ) 2, 3
( x2 - 4x + 5 )
( c ) - 2, - 3
(b) 7
If 2, 3 are roots of the equation 2x n are ( a ) - 5, - 30
( b ) - 5, 30
3
= x - 1 are
( d ) 2, - 3
2
+ mx - 13x + n = 0, hen the values of m and
( c ) 5, 30
(d
2
(b) a - b + 2 c = 0 2 2 2 (d) (a c) = b + c
m
2
xa
If the equations ax + 2cx + b = 0 and ax root, then a + 4b + 4c = ) -1
(b) 1
2
+ 2bx + c = 0 ( b ≠ c ) have a common
( d ) none of these
Answers
1 a
2 d
3 c
4 b
5 c
w
w w
.e
(a) 0
+ bx + c = 0, then
2
( a ) a + b - 2ac = 0 2 2 2 (c) (a + c) = b + c
( 47 )
2
of these
ra
( 46 ) If sin α and cos α are the roots of the equat n ax 2
non
ce
( 45 )
7
( d ) ( - 9, 4 )
om
2
.c
( 43 ) If x
6 a
21 a
22 b
23 b
24 a
25 d
26 a,b
41 b
42 a
43 a
44 a
45 b
46 b,c
7 b 27 a 47 a
8 a
9 b
28 c,d 48
10 c 29 a
49
30 a 50
11 a 31 c 51
12 c 32 c 52
13 a 33 c 53
14 b 34 b 54
15 b 35 b 55
16 b
17 c
18 b
19 b
20 a
36 d
37 b
38 a
39 c
40 b
56
57
58
59
60
