Mathematics Quadratic Equation MCQ (2)

04 - QUADRATIC EQUATIONS

Page 1

( Answers at the end e nd of all questions que stions )

(1)

The value of a for which the the sum of the squares squares of the roots roots of the equation equation 2 x - ( a - 2 ) x - a - 1 = 0 assume the the least least value value is (a) 1

(2)

(b) 0

(b) 3

( b ) ( 6,

[ AIEEE 2005 ]

)

(c) (

2

- 2kx 2kx + k + k - 5 = 0 are less than than 5,

, 4)

( d ) [ 4, 5 ]

[ AIEEE 2005 ]

2

( b ) x - 18x + 16 = 0 2 ( d ) x - 18x - 16 = 0

[ AIEEE 2004 ]

2

If ( 1 - p ) is a oot of quadratic equation x + px + ( 1 - p ) = 0, then the roots are ( a ) 0, 1

( b ) - 1, 1

( c ) 0, - 1

( d ) - 1, 2

[ AIEEE 2004 ]

2

If one root of the equation x + px + 12 = 0 is 4, while the equation equation 2 x + px px + 12 = 0 has equal roots, then the the value of q is (a)

(7)

(d) 1

Let two numbers have have arithmetic mean 9 and geometric geometric mean 4. Then these numbers are the the roots roots of the the quadratic quadratic e uation uation 2

(6)

- b x + c = 0 be two consecutive integers, then

(c) 2

( a ) x + 18x + 16 = 0 2 ( c ) x + 18x - 16 = 0

(5)

2

[ AIEEE 2005 ]

If both the roots of the quadratic equation x then k lies in the interval ( a ) ( 5, 6 ]

(4)

(d) 2

If the roots of the equation x 2 b - 4 c equals (a) - 2

(3)

(c) 3

49 4

( b ) 12

(c) 3

(d) 4

The number of real solutions solutions of the equation equation x (a) 2

(b) 4

(c) 1

(d) 3

[ AIEEE 2004 ]

2

- 3 l x l + 2 = 0 is [ AIEEE 2003 ]

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Page 2

( Answers at the end e nd of all questions que stions ) (8)

The value of ‘ a ’ for which one root of quadratic equation equation 2 2 ( a - 5a + 3 ) x + ( 3a - 1 ) x + 2 = 0 is twice as large as the the other is 2 3

(a)

(9)

(b)

2 3

If roots of the equation 2

x + px + q = 0 are

2

x +

( a ) p = 1 and q = - 56 ( c ) p = 1 and q = 56

( 10 )

2

1 3

(d)

- 5x + 16 = 0

2

and

2

[ AIEEE AIEEE 2003 ]

are

and roots of the equation

, then

( b ) p = - 1 and q = - 56 ( d ) p = - 1 and q = 56

If and be the roots of the equation ( x - a ) ( x - b ) = of the equation ( x - ) ( x ) = c are ( a ) a and c ( c ) a and b

( 11 )

1 3

(c)

, c ≠ 0, then the roots

( b ) b and c ( d ) ( a + b ) and ( b + c )

2

If one root of the equation equation x + px and q it will satisfy the relation 3

[ AIEEE 2002 ]

2

( a ) p - q ( 3p - 1 ) + q 0 3 2 ( c ) p + q ( 3p - 1 ) + q = 0

[ AIEEE 2002, IIT IIT 1992 ]

q = 0 is square of the other, then for any p 3

2

( b ) p - q ( 3p + 1 ) + q = 0 3 2 ( d ) p + q ( 3p + 1 ) + q = 0

[ IIT 2004 ]

2

( 12 ) If x + 2ax + 10 - 3a > 0 for every real value of x, then (a) a > 5

( 13 ) If minimu minimu g (x )

(b) a < -5

(c) -5 < a < 2

2

2

(d) 2 < a < 5

[ IIT 2004 ]

value value of f ( x ) = x + 2bx + 2c is greater than the maximum value of 2 2 2cx + b , then for real value of






Page 3


log 4 ( x - 1 ) = log2 ( x - 3 ) is

( 15 ) The number of solutions of (a) 3

( 16 ) If

(b) 1

( 17 )

(d) 0

[ IIT 2001 ]

2

and

(a) 0 < (c) <

(c) 2

are the roots of the equation equation x + bx bx + c = 0, where c < 0 < b, then < < 0

(b) (d)

< 0 < < 0 <

l

<

l

l

<

l

[ IIT 2000]

2

For the equation 3x + px + 3 = 0, p > 0, if one of the roots is square of of the other, then p is equal to (a)

1 3

(b) 1

(c) 3

(d)

2 3

( 18 ) If b > a, the equation ( x - a ) ( x - b ) - 1 ( a ) both both root roots s in ( a, b ) ( c ) both roots in ( b, +

0 has

ro t in ( -

(b) o )

[ IIT 2000 ]

, a ) and a nd the other in ( b, +

( d ) both roots in ( -

, a)

)

[ IIT 2000 ]

( 19 ) The harmonic mean of the oots of the equation (5 +

2 )x

(a) 2

( 20 ) If the the roots roots (a) a < 2

2

- ( 4 +

(b

4

5 ) x + 8 + 2

(c) 6

f the equation x

5 = 0 is

(d) 8

2

(b) 2 ≤ a ≤ 3

[ IIT 1999 ]

- 2ax + a2 + a - 3 = 0 are real real and less than 3, then (c) 3 < a ≤ 4

(d) a > 4

[ IIT 1999 ]






Page 4


( 22 )

If p, q, r are positive po sitive and are in A. P., then the roots r oots of the quadratic qua dratic equation equat ion 2 px + qx + r = 0 are real for r p (a) 7 4 3 (b) 7 4 3 p r ( c ) all p and r

( 23 )

If 2 x -

(b) g(x) > 0

(d) g(x) ≥ 0

(c) g(x) = 0

2

4

[ IIT 1990 ]

4

and are the roots of x + px + q = 0 and and are the roots of 2 2 rx + s = 0, then the equation x - 4qx + 2q - r = 0 has always

( a ) two real roots ( c ) two negative roots

( 25 )

[ IIT 1995 ]

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 )

( d ) no p and r

( b ) two positive oots ( d ) one positiv and one negative root

2

[ IIT 1989 ]

2

Let a, b, c be real numbers, numbers, a ≠ 0. If is a root of a x + bx + c = 0, is a 2 2 2 2 root of a x - bx - c = 0 and 0 < < , then the equation a x + 2bx + 2c = 0 has a root that always always satis ies α β β α (a) (b) (c) = (d) < < [ IIT 1989 ] 2 2 3

( 26 ) The equation

x 4 ( og x ) 2 2

log x 2

5 4

2

has

( a ) at le st one real solution ( b ) exactly three real solutions ( c ) exactly one irrational solution ( d ) complex roots

( 27 ) The equation x -

2

= 1 -

2

has

[ IIT 1989 ]






Page 5

( Answers at the end e nd of all questions que stions ) 2

( 29 ) If a + b + c = 0, then the quadratic equation equation 3ax + 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

( 30 ) The number of real solutions solut ions of the equation (a) 4

(b) 1

(c) 3

l

2

xl

[ IIT 1983 ]

- 3 l x l + 2 = 0 is

(d) 2

[ IIT 1982 ]

( 31 ) If a > 0, b > 0 and c > 0, then both both the roots roots of the equation equation ax ( a ) are real and negative negative ( c ) none of these

( b ) have negative negative rea pa ts [ IIT 1980 ]

( 32 ) Both the roots of the equation ( x - b ) ( x are always ( a ) positive positive

+ bx + c = 0

( b ) negative negative

(c

c

re l

( x - a ) ( x - c ) + ( x - a )( x - b ) = 0

( d ) none of these

[ IIT 1980 ]

( 33 ) If l , m, n are real, l ≠ m, th n the roots roots of the the equation equation 2 ( l - m ) x - 5 ( l + m ) x - 2 ( l - m ) = 0 are ( a ) real and equal ( c ) real and unequal

( 34 )

( b ) complex d ) none of these

The entire graph of the equation equat ion and on y if (a) k < 7

(b) -5 < k < 7

[ IIT 1979 ]

2

y = x + kx - x + 9 is strictly above the X-axis if

(c) k > -5

( d ) none of these

[ IIT 1979 ]






Page 6


2

( 36 ) If the two equations ax + bx + c = 0 and px + qx + r = 0 have a common root, root, then the value of ( aq - bp ) ( br - cq ) is 2

2

( a ) - ( ar - cp )

( 37 )

( d ) ( ar - cp )

2

, 0)

( b ) ( 0, 1 )

( c ) ( 1,

2

If the roots of the equation a ( b - c ) x a, b, c are in ( a ) H. P.

( b ) G. P.

(a) ±2

(b) ±4

( 40 ) If a > 0, then

(a)

1 2

4a

1

a

1 1 2

a

2

( d ) ( 0,

are equal, then

etween

the

roots

of

the

equation

(d) ±8

.....

4a

) = 0

( d ) none of these

differen e

(c) ±6

a

(b)

the

)

+ b ( c - a )x + c( a - b ) = 0

( c ) A. P.

( 39 ) The value of p for which 2 x + px + 8 = 0 is 2 are

( 41 )

2

( c ) ( ac - pr )

The set of values of p for which which the roots of the equation equation 3x + 2x + p ( p are of opposite signs is (a) (-

( 38 )

2

( b ) ( ap - cr )

=

1

(c)

1 1 2

4a

1

( d ) none of these

If for the quadratic quadratic equation equation ax + bx + c = 0, the difference of the roots is the same as their product, then the ratio of the roots is b

b








Page 7


2

( 43 ) If x + 6x - 27 > 0 and - x + 3x + 4 > 0, the x lies in the interval interval ( a ) ( 3, 4 )

( c ) ( - 9, 3 ]

( b ) [ 3, 4 ]

log

( 44 ) The roots of the equation ( a ) 2, 3

( 45 )

(b) 7

( x2

( 46 ) If sin

and cos 2

4x

( c ) - 2, - 3

( b ) - 5, 30

5)

x

If 2, 3 are roots roots of of the the equation equation 2x n are ( a ) - 5, - 30

3

1

are

( d ) 2, - 3

2

- 13x + n = 0, hen the values of m and

+ mx

( c ) 5, 30

(d

non non

of these these

2

are the roots roots of the the equat equat n ax + bx + c = 0, then

2

( a ) a + b - 2ac = 0 2 2 2 (c) (a + c) = b + c

( 47 )

7

7

( d ) ( - 9, 4 )

[ 4, 9 )

2

2

(b) a - b + 2 c = 0 2 2 2 (d) (a c) = b + c

2

If the equations ax + 2cx + b = 0 and ax + 2bx + c = 0 ( b ≠ c ) have a common root, then a + 4b + 4c =

Mathematics Quadratic Equation MCQ (2)

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