Quadratic Equation & Expression

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1. Solution set of

21 2 4

0 is2 3

x x x

x x

a. , 4 2,1 2 3, b. 4, 2 1,2 3,

c. , 4 2 3, d. None of these

2. Total number of real values of x such that

1/ 7 1/ 7

1/ 712 12 64

,12 3

x x

x

is equal to

a. One b. Two

c. Zero d. None of these

3. Let a,b and c are unequal positive real numbers such that 2b=a+c, then the roots of

2 2 0ax bx c are

a. Real and equal b. Real and distinct

c. Imaginary d. None of these

4. The equation x2+ax-a2-1=0 will have roots of opposite signs if

a. ,a b. 1,1a

c. , 1 1,a d. None of these

5. If a,b,c are odd integers and ax2+bx+c=0 has real roots, then

a. Both roots are rational b. Both roots are irrational

c. Both roots are positive d. Roots are of opposite signs

6. If both the roots of x2-ax+a=0 are greater than 2, then

a. ,4a b. 0,2a

c. 4,a d. None of these

7. If both roots of ax2+ax+1=0 are less then one, then complete set of values of ‘a’ is

a. 1

,0 4,2

b. ,0 4,

c. 1

,0 0,2

d. 1

, 4,2

8. Total number of integral values of a such that x2+ax+a+1=0 has integral roots is equal

to

a. One b. Two

c. Three d. Four

Quadratic Equation & Expression

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9. If x2+2ax+a<0 1,2x , then

a. 4

,5

a

b. 4 1

,5 3

a

c. 1

,3

a

d. None of these

10. If both roots of x2+ax+2=0 belong to the interval (0,3), then exhaustive range of ‘a’ is

a. 6,0 b. 11

, 2 23

c. 11

,03

d. None of these

11. If 2 3 5 0x x and

2 0ax bx c have a common root and a,b,c ,N then minimum

value of a+b+c is equal to

a. 3 b. 9

c. 6 d. 12

12. If two roots of 3 2 0x ax bx c are equal in magnitude but opposite in signs, then

a. a+bc=0 b. a2=bc

c. ab=c d. a-b+c=0

13. If a, b,cR such that a+b+c=0 and ,a c then the roots of (b+c-a)x2+(c+a-b)x+

(a+b-c)=0 are

a. Real and equal b. Real and distinct

c. Imaginary d. None of these

14. If sinθ,cosθ are the roots of 2 0,ax bx c then

a. 2 2 2a b ab b.

2 2 2b a ab

c. 2 2 2a b ac d.

2 2 2b a ac

15. Total number of real solutions of

2

1 1

x xx

x x

is equal to

a. 4 b. 2

c. 8 d. None of these

Quadratic Equation & Expression

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16. Both roots of (a2-1)x2+2ax+1=0 belong to the interval (0,1) then exhaustive set of values

of ‘a’ is

a. , 2 0, b. , 2

c. 1 5

,0 0,2

d.

1 52, 0

2

17. The complete set of values of ‘a’ such that x2+ax+a2+6a<0x 1,1 is

a. 5 21 7 45

,2 2

b. 7 45 5 21

,2 2

c. 5 21 7 45

,2 2

d. None of these

18. The complete set of values of x satisfying 2log 1 0,x

x is

a. 1, b. 1, 2

c. 2,

d. None of these

19. The complete set of values of x satisfying 2log 1 0,x

x is

a. 1, b. 1,2 0

c. 1,1 0 d. 1,0 1,

20. If t-1 and-t-1, tR, are the roots of 22 2 1 0,a x ax then complete set of values

of ‘a’ is

a. b. ,

c. ,0 d. 0,

21. If ax2+bx+c=0 and bx2+cx+a=0 have a common root and a,b,c are non zero real

numbers, then

3 3 3a b c

abc

is equal to

a. 1 b. 2

c. 3 d. None of these

22. If roots of the equation

ax3+bx2+cx+d=0 remain unchanged by increasing each coefficient by one unit. Then

a. a=b=c=d 0 b. 0a b c d

c. 0a b c d d. 0a b c d

Quadratic Equation & Expression

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23. If tanθ and secθ are the roots of equation 2 0,ax bx c then

a. 4 2 22a b ac b b. 4 2 22b a ac a

c. 4 2 24a b ac b d. 4 2 24b a ac a

24. If x3+ax+1=0 and x4+ax2+1=0 have a common root, then complete set of values a is

a. , 2 b. 2

c. 2, d. None of these

25. If a,b,c are distinct positive real number such that b (a+c)=2ac, then the roots of

2 2 0ax bx c are

a. real and equal b. real and distinct

c. imaginary d. none of these

Quadratic Equation & Expression

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1. Complete set of values of ‘a’ such that

2

1

x x

ax

attains all real values is

a. [1,4] b. 0,4

c. 0,1 d. 1,

2. The equations 2 22 0, 2 0ax bx c bx cx a and

2 2 0cx ax b will have only

positive roots, provided

a. a=b=c b. a b c

c. a b c d. a b c

3. If a R and 2,a then the equation 2 1 0x a x

a. cannot have any real root

b. must have exactly two real roots

c. must have either exactly two real roots or no real roots

d. must have either four real roots or no real roots.

4. If 2 1x ax is a factor of

3 ,ax bx c then

a. b+a+a3=0, a2+c=0 b. b-a+a3=0,a2+c=0

c. b+a-a3=0, a2+c=0 d. None of these

5. If the sum of the roots of the equation 2 0ax bx c is equal to the sum of the

squares of their reciprocals, then

2

2

b bc

ac a is equal to

a. 1 b. -1

c. 2 d. -2

6. The least value of ‘n’ such that (n-2)x2+8x+n+4>0 ,x R where ,n N is

a. 3 b. 5

c. 4 d. None of these

7. If both the roots of 2 22 3 0x ax a a are less than three, then

a. ,2a b. 4,a

c. 2,3a d. 3,4a

Quadratic Equation & Expression

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8. If the roots of (x-a) (x-b)=f are c and d, then the roots of (x-c)+(x-d)+f=0 are

a. a and b b. ,a b

f f

c. ,f f

a b d. ,

c d

f f

9. If the roots of x2-ax+b=0 differ by unity then

a. b2=1+4a b. a2=1+4b

c. b2+4a=1 d. a2+4b=1

10. Let a, b, c are positive real number forming a G.P. If ax2+2bx+c=0 and dx2+2ex+f=0

have a common root, then , ,d e f

a b c are in

a. G.P b. A.P

c. H.P d. None of these

11. The maximum value of the expression y=2 (a-x) 2 2x x b is equal to

a. 2 2a b b.

2 22a b

c. 2 22a b d.

2 2a b

12. If x2+ax+bc=0 and x2+bx+ca=0 a b have a common root, then their other roots

satisfy the equation

a. x2+ax+ab=0 b. x2+bx+ab=0

c. x2+cx+ab=0 d. x2+abx+c=0

13. If x2-(a-3)x+a=0 has atleast one positive root then

a. ,0 7,9a b. ,0 7,a

c. ,0 9,a d. None of these

14. x1,x2 are the roots of x2-3x+a=0 and x3,x4 are the roots of x2-12x+b=0. If x1,x2,x3,x4,

form an increasing G.P., then ordered pair (a,b) is

a. (1,16) b. (2,32)

c. (2,16) d. (1,32)

15. If roots of x2-(a-3)x+a=0 are such that atleast one of the roots is greater than 2, then

a. 7,9a b. 7,a

c. 9,a d. 7,9a

Quadratic Equation & Expression

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16.

,

x b x cf x

x a

where a,b,c are distinct real numbers, will assume all real values

provided.

a. c lies between a and b b. a lies between b and c

c. b lies between a and c d. None of these

17. Let x1 and x2 are the roots of ax2+bx+c=0 and x1.x2<0, x1+x2 is non zero. Roots of x1

(x-x2)2+x2 (x-x1)

2=0 are

a. negative b. Real and of opposite signs

c. Positive d. None real

18. Solution set of 2log 0,x

xx

x

is

a. ,0 1,2 b. ,1 2,

c. , 1 0,1 d. , 2 0,1

19. Consider the equation x2+2x-n=0, where nN and 5,100 .n Total number of

different values of ‘n’ so that the given equation has integral roots is

a. 4 b. 8

c. 3 d. 6

20. a,b,cR such that abc 0. If x1 is a root of a2x2+bx+c=0, x2 is a root of a2x2-bx-c=0

and x1>x2>0, then the equation a2x2+2bx+2c=0 has a root x3 such that

a. x1>x3>x2 b. x3>x1>x2

c. x1>x2>x3 d. None of these

21. If sin sin and cos are in G.P., then roots of 2 2 cos 1 0x x are always

a. Real b. Real and negative

c. Greater than one d. Non real

22. If the roots of the equation 2 2 2 2

1 1 1 1 1 1 1 1 10ax bx a b c a b a c bc are non real, then

a. 2

1 12 0a b a b b.

2

1 12 0a b a b

c. 2

1 12 0a b a b d. None of these

Quadratic Equation & Expression

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23. If 2

2log 1 ,ax x a x R then exhaustive set of value of ‘a’ is

a. 5

0,12

b.

5 51 ,1

2 2

c. 5

0,12

d.

51 ,

2

24. Total number of values of x. satisfying 2 2

33 1 3 1 2 ,x x

x is equal to

a. 2 b. 3

c. 1 d. None of these

25. If 2 3 0x x a for atleast one negative value of x, then complete set of values of

‘a’ is

a. ,4 b. ,2

c. ,3 d. ,1

Quadratic Equation & Expression

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1. Total number of values of a so that x2-x-a=0 has integral roots, where a N and 6

100,a is equal to

a. 2 b. 4 c. 6 d. 8

2. Total number of integral values of ‘a’ so that x2-(a+1)x+a=1=0 has integral roots, is equal

to

a. 2 b. 1 c. 4 d. None of these

3. Let

2

2

3 1,

1

x xy x R

x x

then

a. 5

1,3

y

b. 1 5

,3 3

y

c. 1

,33

y

d. 1,3y

4. If the quadratic equations 3x2+ax+1=0 and 2x2+bx+1=0 have a common root then

the value of 5ab-2a2-3b2, where a,bR, is equal to

a. Zero b. 1 c. -1 d. None of these

5. If tan and tan are the roots of the equation x2+ax+b=0, then the value of

2 2sin sin .cos cosa b a 2cosb a is equal to

a. a b. ab c. b d. a/b

6. If the quadratic equation ax2+bx+6=0 does not have real roots and ,b R then

a.

2

max , 624

ba i b

b.

2

max , 624

ba i b

c.

2

min , 624

ba i b

d.

2

min , 624

ba i b

7. If 2cos

8

is a root of the equation

2 0x ax b where a,b rational numbers, then

ordered pair (a,b) is

a. 1

1,8

b.

11,

8

c. 1

1,8

d. 1

1,8

Quadratic Equation & Expression

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8. If x1 and x2 are the roots of 2 21sin 1 cos 0,

2x then maximum value of

2 2

1 2x x is equal to

a. 2 b. 3

c. 9

4 d. 4

9. If 3 3 0 1,3 ,x a x a then exhaustive set of values of ‘a’ is

a. 1

0,3

b. 2,3

c. 1

,33

d. 2,0

10. If 2 3

tan , tan , tan are the real roots of 3 21 0,x a x b a x b then

1 2 3 1 2 30, , then is equal to is equal to

a. 2

b.

4

c. 3

4

d.

11. If 1 2,x x are the roots of

2 0ax bx c and 1 2

,x d x d are the roots of 2 0,px qx r

Where d 0 , then

a. 2 24 4b ac q pr b.

a b c

p d q d d r

c. 1

2d bp aq

ap d. a+d=p

12. If x2+5=2x-4cos(a+bx), where a,b (0,5), is satisfied for atleast one real x, then the

maximum value of a+b is equal to

a. 3 b. 2

c. d. None of these

13. If the equations 2 2 1 2x b bx and

2 2 1 2x a ax have exactly one common root,

then

a. a=b+2 b. a=b2+2

c. a+b=2 d. a+b+2=0

Quadratic Equation & Expression

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14. If 16-x2> x a is to be satisfied by at least one negative value of x, then complete set

of values of ‘a’ is

a. 65

16,4

b. 65

8,2

c. 16,16 d. 8,8

15. If 1-cosx=3

2x a has no solution, then complete set of values of ‘a’ is

a. 1

,2

b.

3,4

2 3

c. 3

,2 3

d. 4,

16. The equation 4 2sin sin 1 0x a x will have a solution if

a. , 2 b. 2,2

c. , 2 2, d. 2,

17. If and are the real and distinct roots of 2 0,x bx c where , 0 ;a c b then

a. 0a a b. 0a a

c. 0 a d. 0a

18. If 1 2 3, , ...................

nx x x x are the roots of 0,nx ax b then the value of

1 2 1 3 1 4 1.......

nx x x x x x x x is equal to

a. 1

nx b b. 1

1

nn x

c. 1

1

nn x a

d.

1

1

nn x b

19. If 2 2 22 ,2 5 4 0x xa a has atleast one real solution, then complete set of values of

‘a’ is

a. ,1 b. 3,1

c. 1, d. 8

, 17

Quadratic Equation & Expression

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20. If 2 2 1 5 0x a x a has real roots belonging to the interval (1,3), then complete

set of values of ‘a’ is

a. 8

,7

b. 4,

c. 48

,3

d. None of these

21. If 2 2

2 2 2 22 3 1 2 4 1 0x x a x x x x a x x has at least one

root, then complete set of value of ‘a’ is

a. 7

1,3

b. 7

,53

c. 3

,17

d.

195,

3

22. Let a,b,c are positive real numbers forming an A.P. if ax2+bx+c=0 has real roots, then

a. 2 3a c

c a b. 2 3

a c

c a

c. 2 3a c

c a d. 2 3

a c

c a

23. If the equations x2+ax+12=0, x2+bx+15=0 and x2+(a+b)x+36=0 have a common

positive root, then ordered pair (a,b) is

a. (-6,-8) b. (-7,-8)

c. (-6,-7) d. (-7,-9)

24. The minimum value of x2+2xy+3y-6x-2y, where x, yR, is equal to

a. -9 b. -11

c. -12 d. -10

25. If , , are the roots of x3-x2-1=0, then the value of 1 1 1

1 1 1

is equal to

a. -5 b.-6

c.-7 d. -2

Quadratic Equation & Expression

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Answers Exercise-1 Quadratic Equation and Expressions

1. a 2. a 3. b 4. a 5. b 6. d 7. d 8. b 9. a 10. b

11. b 12. c 13. b 14.d 15. d 16.b 17. d 18. b 19. d 20. a

21. c 22. a 23. c 24. b 25. c

Answers Exercise-2 Quadratic Equation and Expressions

1. d 2. a 3. d 4. b 5. c 6. b 7. a 8. a 9. b 10. b

11. d 12. c 13. c 14.b 15. c 16.b 17. b 18. d 19. b 20. a

21. a 22. a 23. d 24. c 25. c

Answers Exercise-3 Quadratic Equation and Expressions

1. d 2. b 3. a 4. b 5. c 6. a 7. c 8. d 9. a 10. b

11. c 12. a 13. a 14.a 15. c 16. a 17. b 18. c 19. a 20. d

21. d 22. a 23. b 24. b 25. a

Quadratic Equation & Expression

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