Quadratic equations

Quadratic Droppers test Q.1 Consider a polynomial equation x 4  63x3  ax 2  bx  7!  0 whose roots are four distinct natural numbers of which three are odd. Then the value of a is (A*) 791

(B)

971

((C)

719

(D)

None of these

Q.2 If the equation sin 4 x   k  2 sin 2 x   k  3  0 has a solution then k must lie in the interval (A)  4, 2

(B) [ 3, 2)

(C) ( 4, 3)

(D*) None of these

Q.3 The value of ‘a’ for which the equation 9 x  a.3x  a  5  0 is satisfied for at least

4

one x  R . (A)

(, 5]  [1, )

(B)

[5,1]

(C) [

5 , ) 4

(D*) [1, )

Q.4 If x3  4x2  qx  r  0 has roots  ,  ,  &

    3     3      3   256 . Then the value of q  r is equal to (A*)

2

(B)

4

(C)

16

(D) 256

Q.5 If  ,  be the roots of x 2  a  x  1  b  0 , then the value of

1 1 2 is    2  a  2  a a  b (A*)

4  a  b

(B)

(C)

0

(D)

1  a  b

2  a  b

Q.6 The value of ‘a’ for which the equation x3  ax  1  0 & x4  ax2  1  0 have a common root is (A) a  2

(B*)

a  2

(C) a  0

(D) None of these

Q.7 The number of integer values of ‘a’ for which the equation ax2   3a  1 x  5  0 has integral roots is (A)

1

(B*)

2

(C)

3

(D)

4

Q.8 A cubic polynomial p(x) is such that  x  1 is a factor of p  x   2 and  x  1 is a 2

2

factor of p  x   2 . The coefficient of x in the p ( x ) is (A)

1

(B)

2

(C)

3

(D*)

None of these

(D)

1  1  ,

Q.9 If  ,  are the roots of ax 2  bx  c  0 then the equation

ax 2  bx  x  1  c  x  1  0 has roots 2

(A)

  (B*) , 1 1 

  (C) , 1  1 

1 1  ,









MORE THAN ONE CORRECT Q.10 The integral values of m for which the root of the equation

mx2   2m  1 x   m  2   0 are rational are given by (A*)

15

(B)

12

(C*)

6

(D)

4

Q.11 The set of values of ‘a’ for which the equation

1  a   x2  x  1

2









2

  2a  1 x 4  x 2  1  3a x 2  x  1  0 has real roots. Then the

value of ‘a’ can be (A)



1 2

(B*)

1 4

(C*)

1 8

(D)

1 16

Q.12 If x n  ax  1  0 has roots 1 ,  2 ....  n then (A*)

(C*)

1



1

1  2

 .... 

1

n

(B) 1  12  1 ...n  1  a  2 if n is even

a

If 1   2 then 1n 1 

a n

(D*) 1  12  1 ...n  1  a  2 if n is odd

COMPREHENSION TYPE Let P(x) be a polynomial such that P(1) = 1 and

P(2 x) 56 . For all real x for 8 P( x  1) x7

which both sides are defined. Q.13 The degree of P(x) is (A)

2

(B*)

3

(C)

4

(D)

-4/21

(C*)

-5/21

5

Q.14 The value of P(-1) is (A)

2/7

(B)

(D)

-11/21

Q.15 MATCH THE COLUMN



(P) The range of the expression

(1) 2, 2

sec   3tan  is sec4   tan 2  4

2

mx 2  3x  4  5 for all real x then range of m is x 2  3x  4 2 2 (R) If     2 x     2  x  1  0, x  R , then  (Q) If





belongs to the interval



 



2 2 2 (S) If the roots of 3 x  2 a  1 x  a  3a  2  0

5



(2) ( ,11 )

4

(3) [ 1 ,1]

3

(4) (1, 2)

are of opposite signs then, ‘a’ lies in the interval (A*) P-(3),Q-(2),R-(1),S-(4)

(B) P-(3),Q-(4),R-(1),S-(2)

(C) P-(1),Q-(2),R-(3),S-(4)

(D) P-(1),Q-(4),R-(3),S-(2)

INTEGER TYPE Q.16 If the equation x 6  2 x3  5  ax3  a  0 has 4 real roots then a  ? [Ans: 5] Q.17 If p and q are the roots of the equation x 2  2 x  A  0 & r and s are the roots of the equation x 2  18 x  B  0 . If p < q < r < s are in A.P, then the value of

A B is 37

[Ans: 2] Q.18 Let N =  be a 6 digit number (all digits equal) and N is divisible by 924. Let

 ,  Be the roots of the equation x 2  11x    0 , then the product of all possible values of  is found to be112M . Find the value of M. [Ans: 6] Q.19 If one of the root of the equation  l  m  x 2  lx  1  0 is double the other  l  m and if l is real then the greatest value of 8m is. [Ans: 9] Q.20 Let p, q  {1, 2,3, 4} , If the number of equations of the form px2  qx  1  0 having real roots is M. The value of M is. [Ans: 7]