Quadratic Equation Assignment

Solution of Quadratic equations Basi c L evel

1.

A real root of the equation log 4 {log 2 ( x + 8 (a) 1

−

x )} = 0 is

(b) 2 log 7 ( x

2

3.

4.

 1 , 2   2  x The T he solut solution ion of the the equat uation 3 log a + 3 xlog a 3 = 2 (a) (a)

2

(a) (a) 3 5.

−

2

,2

2

(b) (b)

log 2 a

If 3 x+1

(b) (b) 3

7.

+|

x≥0

log 3 a

(d) (d) 2

− log 3 a

(d) (d) log 2 3

x ∈ (1, α )

(d) None of these

+ 5|

x|

+4 = 0

(d) 3

are are

[MNR 1993]

(c) (c) {−4 , 4}

(b) (b) {1, 4 }

(d) None of these these [Roorkee 1982; Rajasthan PET 1992]

(c) (c)

3, 2, − 2

(d) (d) 4, 4 , 3

{ x ∈ R :| x − 2 | = x 2 } =

If ax2

[EAMCET 2000]

+ bx + c = 0,

(c) (c) {−1, − 2}

(b) (b) {1, 2}

− 7 x1 / 3 + 10 = 0,

{1, − 2} [MP PET 1995]

(b) (b)

−b±

b2 2a

− ac

(c) (c)

2c

− b±

b2

− 4 ac

(d) None of these these

then x =

[BIT Ranchi 199 1992] 2]

(b) (b) {8}

(c) (c)

The T he roots roots of the give given equa quation tion ( p − q)x2 p− q ,1 r− p

(d) (d)

then x =

b ± b2 − 4 ac 2a

If x2 / 3

(a) (a) 14.

2


(c) 2

− x − 6 | = x + 2, then the values of of x x are −2, 2, − 4 (b) (b) −2, 2, 4

(a) (a) {125 } 13.

(c) (c)

 

22 

If | x 2

(a) (a) 12.

1 ,  4

(c) (c)

(b) 0

(a) (a) {−1, 2} 11.

(c) (c)

x| = x2 / | x − 1 | is

The real real root roots s of the the equa quation tion x2

(a) (a) 10.

(d) 3, 5

If 2 log( x + 1) − log( x2 − 1) = log 2, then x equals

(a) (a) {−1, − 4} 9.

(c) 2, 3

(c) log 3 2

(b) (b) x > 0

(a) 1 8.

4

is given by

− log 2 a

(b) 2

The T he solut solution ion of | x /(x − 1)| (a) (a)

(d)

= 6 log 2 3 , then x is

(a) 3 6.

(c) 3

− 4 x+ 5 )

The root roots s of the equa quation tion 7 are = x − 1 are (a) 4, 5 (b) 2, – 3 The solut solution ion set set of the equa quation tion log x 2 . log 2 x 2 = log 4 x 2 is

2.

[AMU [AM U 1999 1999]]

(b) (b)

The solut solution ion of the the equat uation x +

+ (q − r)x + (r − p) = 0

q− r ,1 p− q 1

x

=2

will be

φ

are (c) (c)

(d) (d) {125 , 8} [Rajasthan PET 1986; MP PET 1999]

r− p ,1 p− q

(d) (d) 1,

q−r p− q [MNR 1983]

Trusted by over 1 million members

Try Scribd FREE for 30 days to access over 125 million titles without ads or interruptions! Start Free Trial Cancel Anytime.













(b) (b) 0, − 1, −

(a) 2, –1 15. 16. 17.

If 9

x

−4×3

+ 3 = 0, 5

21.

= 2 2 x+ 3 + 48 ,

−3

23.

2x − 2

+

x− 3

=0

(d)

7

(c) 3, 1, 3, 1

(d) 1, 2, 1, 2

[MP PET 1986] [T.S. [T.S . Rajendra R ajendra 1991] 1991]

(c) 1

(d) 0

(c) (2, 4)

(d) (1, 3)

(c) – 3

(d) 1

are are

[AMU [AM U 1985 1985]]

(c) 3 and 4

=2

x+1

The solut solution ion of the the equat uation

(d) 4 and 5

is

[Roorkee [Roork ee 1979] 1979]

(b) 19

(c) 3, 19

+

x−1

=0

(d) 3, –19

is

[II T 1978 1978]]

(b) – 1

(c) 5/4

(d) None None of these

If x = 6 + 6 + 6 + ..... to ∞ , then (a) (a) x is an irrational irrational number number

24.

(c) 5

the value of x of x will be

The root roots s of the equa quation tion 4 x − 3 .2 x+3 + 128 (a) 1 and 2 (b) 2 and 3

(a) 1

[MP PET 1985 1985]]

2

(b) – 2

2

The root root of the the equa quation tion

is

then the solution pair is

(a) 3 22.

3


5

− 14 x 4 + 31 x3 − 64 x2 + 19 x + 130 = 0

(b) (2, 3)

In the equation 4 x+ 2 (a) (a)

20.

5

(b) 2 x+ 2

(a) (1, 2) 19.

5

The root roots s of the equa quation tion x − 4 x + 6 x − 4 x + 1 = 0 are are (a) 1, 1, 1, 1 (b) 2, 2, 2, 2 One root of the equation equation (x + 1)( x + 3 )(x + 2)( x + 4 ) = 120 is 4

− 1, − 1

(c) (c)

One root of the following given equation 2 (a) 1 (b) 3

(a) –1 18.

1

[Pb.CE T 1999 1999]]

(b)

2 2

The rea real va values lues of x of x which satisfy the equation (5 + 2 6 )x −3

< x< 3

+ (5 − 2

2

6 )x

(c) (c) −3

= 10

x=3

(d) (d)

are are [Kurukshetra [Kuruks hetra CEE CE E 1995; Karnataka CE T 1993 1993]]

±2

(a) (a) 25.

26. 27.

x∈ R

± 2, ±

(c) – 3

+ [x − 1] 2 = ( x − 1)2 + [x + 1]2

(b) (b) x ∈ N

 1  +  1 + 1  +  1 + 1  + .....  1 + 199   4   4 200   4 100   4 200 

2

(d) (d) 2, 2

(b) 50

(c) (c)

[Rajasthan PET P ET 1986 1986]]

(d) None of these these [Roorkee [Roork ee 1986] 1986]

(d) None of these [Rajasthan PET P ET 1997 1997]]

(d)

±3

is x∈ I

(d) (d) x ∈ Q

is (c) 51


The value value of x = 2 + 2 + 2 + ..... is (a) –1

31.

(b) 3

The solut solution ion set set of the equa quation tion (x + 1)2

(a) 49 30.

(c) (c)

2

+ b(c − a)x + c(a − b) = 0 is 1 then, its other roots is a(b − c) c(a − b) b(c − a) (a) (a) (b) (b) (c) (c) b(c − a) a(b − c) a(b − c) The ima imagina ginary root roots s of of the the equat uation (x 2 + 2)2 + 8 x2 = 6 x( x2 + 2) are are (a) (a) 1 ± i (b) (b) 2 ± i (c) (c) −1 ± i 2 GM of the roots of the equation x − 18 x + 9 = 0 is

(a) (a) 29.

±

If one root of the equation a(b − c)x 2

(a) 6 28.

(b) (b)

If x

2

− x + 1 = 0,

(a) –1,1

(b) 1 then value of x

3n

(b) 1

[Karnataka CET CE T 2001 2001]]

(c) 2

(d) 3

is

[DCE 1995]

(c) –1

(d) 0













(a) 4, – 2 34.

If expre expression ssion (a) (a)

35.

(b) – 4, – 2 2

4

6

1 1+ 3

The root roots s of equat quation ion

If x2

2 x + 31 9

38. 39.

42. 43.

47.

3, − 3

(c)

[II T 1991 1991]]

(d) (d)

5, − 5 [BIT Ranchi 1992]

(c) {3, 4, – 3, – 4}

+ | x − 2 | −2 = 0

2

(d) {– 3, – 3}

is

+ 4 x+ 3| + 2x+ 5 = 0

[II T 1997 1997;; Himachal Hi machal CET C ET 2002 2002]]

are are

[II T 1988 1988]]

(c) 3

The num number ber of the the real real value values s of x of x for which the equality | 3 x

(d) 4

+ 12 x + 6 | = 5 x + 16 holds good is

2

(b) 3

(c) 2

The num number ber of real real solut solution ions s of the equa quation tion sin (a) 0 (b) 1 The num number ber of the the real real solut solution ions s of the equa quation tion (a) 1 (b) 2 | x| The num number ber of solut solution ions s of cos x = is

x

x

= 5 +5

−x

(d) 1

is (c) 2

− x + x − 1 = sin 2

[AMU [AM U 1999 1999]]

[II T 1990 1990,, 2002 2002]]

(d) Infinitely nfi nitely many many

4

x is (c) 0

(d) 4

(c) 53


80

The equa quation tion

(b) 52 (x + 1)

−

( x − 1)

=

has (4 x − 1) has

[II T 1997 1997]]

(b) One solution

The num number ber of real real root roots s of 5 x 2

− 6x+ 8 −

5 x2

(c) T wo solutions

− 6x − 7 = 1

(d) More than than two solution

is

(b) 2


(c) 3

The num number ber of roots roots of the quad quadra ratic tic equat quation ion 8 sec (a) I nfinite nfi nite (b) 1

2

θ − 6 sec θ + 1 = 0

(d) 4 is

[Pb. CET 1989,94]

(c) 2

(d) 0

The num number ber of value values s of x of x in the interval [0, 5π ] satisfying the equation 3 sin x − 7 sin x + 2 = 0 is 2

(b) 5

(c) 6 2n

The maxim aximum um num number ber of of real real root roots s of of the the equat quation ion x (a) 2

49.

2


(b) 2

(a) 0 48.

π


1− 2

are are

9

The num number ber of real real solut solution ions s of the the equa quation tion | x2

(a) 1 46.

2 x + 47

< x<

(a) 2 (b) 4 (c) 1 (d) None None of these A two digit digi t number number is four times the sum and three times the product of its i ts digits. digi ts. T he numbe numberr is is [MP PET 1994] (a) 42 (b) 24 (c) 12 (d) 21

(a) No solution 45.

=

The som some of all rea real root roots s of of the the equat quatio ion n | x− 2|

(a) 50 44.

x2 + 7 x2 − 7

(b) {3, – 3}

(a) 4 41.

1− 3

+ y2 = 25 , xy = 12, then x =

(a) 1 40.

+

2

(c) (c)

(b) 5, – 5

(a) {3, 4} 37.

1

(b) (b)

(a) 3, – 3 36.

(d) – 4, 2 cos x , 0 satisfies the equation x2 − 9 x + 8 = 0, find the value of cos x + sin x

{(sin x+ sin x+ sin x+..... ∞) ln 2}

(c) 4, 2

The equa quation tion x + (a) No real root

− 1 = 0,

(b) 3 2 1− x

=1+

2 1−x

(d) 10

is

[MP PET 2001]

(c) n

, has has

(b) One real root

(c) T wo equal equal roots

The num number ber of real real root roots s of equat quation ion (x − 1)

51.

(a) 2 (b) 1 The num number ber of roots roots of the equa quation tion log( −2 x) = 2 log( x + 1) are (b) 2

2

+ ( x − 2) + (x − 3) = 0 2

2

is

(c) 0

∑

(d) Infinitely nfi nitely many many roots [II T 1990 1990;; Karnataka Kar nataka CE T 1998 1998]]

(d) 3 [AMU [AM U 2001 2001]]

(c) 1 10

(d) 2n 2n [II T 1983 1983;; MNR MN R 1998; 1998; Kurukshetra Kur ukshetra CEE C EE 1993 1993]]

50.

(a) 3

[IIT 1998, MP PET 2000]














54.

1

Rationalised denominator of 2

(a) (a) 55.

2 3

+3

2

−

30

(b) (b)

12

If x = 7 + 4 3 , then x +

1

x

(a) 4 56.

+

is

+ 5 2 −2 3 −

3

3

30

15

10 3

(c) 3

(c) (c)

58.

+ x− = 0 (b) (b) x2 + − 1 = 0 If f (x) = 2 x 3 + mx 2 − 13 x + n and 2, 3 are roots of the equation

61. 62. 63.

The num number ber of value values s of a of a for which (a2

(d) (d) x2

+x − =0

5, 30


(d) None of these

is

=0

(c) 0


(c) – 1


is

− 3a + 2)x2 + (a2 − 5 a + 6)x + a2 − 4 = 0

is an identity in x is

(c) 1

The num number ber of value values s of the pair pair (a (a, b) for which a(x + 1)

2

(d) 3

+ b( x − 3 x − 2) + x + 1 = 0 2

(b) 1

is an identity in x is

(c) 2

(d) I nfinite nfi nite

(c) 1


of x is 13 )x / 2 then the number of values of x (b) 4 6

The num number ber of real real solut solution ions s of the the equa quation tion (a) T wo

65.

=

(b) 2

=(

20

15

f (x) = 0, then the value of of m m and n are are

(b) 0

(a) 2 64.

x

The sum sum of the the real real root roots s of of the the equa quation tion x2 + | x| −6

If ( 2 )x + ( 3 )x

−


+ x+1 = 0

x2

(c)

The num number ber of real real solut solution ions s of the the equa quation tion (a) 1 (b) 2

(a) 0

2

[Kurukshetra CEE 1993; MP PET 1989]

(b) – 5, 30

(a) 0

+3

[EAMCET 1994]

x2

(a) 4

2 3

x ≠ y, then x + y =

57.

60.

(d) (d)

(d) 2

(c) 37/6

59.

40

[EAMCET 1994]

= log 2 y + log y 2 and and

(a) – 5, – 30

+

10

(a) 2 (b) 65/8 The equa quation tion log e x + log e (1 + x) = 0 can be writte wri tten n as (a) (a)

2

= (b) 6

If log 2 x + log x 2 =

−3

2 3

(c) (c)

x

2

−x =2+ x x+2 −4

(b) One

The num number ber of real real solut solution ions s of x2 − 4 x + 3 (a) One (b) T wo

is (c) Zero

+

x2 − 9

=

4 x2


− 14 x + 6

is (c) T hree


A dvance Level 66.

If −1 ≤ x < 0 , then solution of the equation | x + 1 | (a) 1, 5/3

67.

The rea real root roots s of | x| 3 −3 x2

70.

+ 3| (b)

+ 3 | x − 1 ||

x − 2|

= x+ 2

is

(c) 1/3

[II T 1976 1976]]


are x| −2 = 0 are

[DCE 1997]

±1

The num number ber of real real solut solution ions s of the the equa quation tion 2 (a) One

69.

x|

(b) 5/3

(a) 0, 2 68.

−|

±2

(c) x/2

+(

x

+ 1) = (5 + 2

2

(b) T wo

(d) 1, 2

x/ 2

2)

is

(c) Four 2

The num number ber of nega negativ tive e inte integral gral solut solution ions s of of x .2 (a) 0 (b) 1

x +1

| x− 3| +2

+2

=x

2

| x− 3| +4

.2

(d) Infini Inf inite te

+2

x− 1

is

(c) 2

(d) 4

(c) Exactly Exactl y two real roots

(d) Infi I nfinitely nitely many many real roots

x

− x − 1 = 0 has The equa quation tion has (a) Only Onl y one real root x =0 (b) At A t least two real real roots

[DCE 1993] [Kurukshetra [Kuruks hetra CEE CE E 1998] 1998]













73.

(a) 4 74.

75. 76.

+

The num number ber of real real solut solution ions s of equa quation tion log 10 [98

[ x3

− x2 − 12 x + 36 ]] = 2

(b) 1

are are

(c) 2

2

The equa quation tion x(3 / 4 )(log 2 x) + (log 2 x)−5 / 4 = (a) A t least one real solution solution (c) Exactly one irrational solution The num number ber of solut solution ions s of | [ x] − 2 x|

(d) 3

2 has has

[II T 1989 1989]]

(b) (d) A ll the above above = 4 , where [x [x] is the greate greatest st integ i nteger er is ≤ x , is

(a) 2 (b) 4 Let Let f (x) be a function defined by f ( x) = x − [ x] , 0

Exactly Exactl y three real solutions solutions

(c) 1

(d) Infinite nfi nite

[x] is the greatest integer less than or equal to ≠ x ∈ R , where [x

x. then the

 1  = 1   x 

number of solutions of f (x) + f  (a) 0 77.

(b) I nfinite nfi nite

If m If m be the number of integral solutions of equation 2 x x3

(d) 2

− 3 xy − 9 y − 11 = 0 2

and and n be the number of real solutions of equation

− [x] − 3 = 0 , then m =

(a) (a) n 78.

(c) 1 2

(b) 2n 2n 3

The set set of value alues of of c c for which x (a) {0}

(c) (c) n/2

− 6 x + 9x − c 2

(d) 3n

is of the form form (x − α ) ( x − β ) (α, β real) real) is i s given by 2

(b) {4}

(c) {0, 4}

(d) Null set k

79.

If 0

< ar < 1

for for r =1, 2, 3, ….., k and and m be the number of real solutions of equation

∑

(ar )

x

=1

and n be the number of real

r =1

k

∑

solution of equation

( x − ar )

101

= 0 , then

r =1

(a) (a) 80.

Let Let

=n (b) (b) ≤ 2 Pn(x) = 1 + 2 x + 3 x + ..... + (n + 1)xn

≥n

(c) (c)

(d) (d)

>n

be a polynomial such that n is even. even. T hen the number of rea reall roots of Pn(x) = 0 is [DCE 1994]

(a) 0

(b) n

(c) 1


81.

The num number ber of all possib ossible le triple triplets ts (a1, a2 , a3 ) such that a1 + a2 cos 2 x + a3 sin x = 0 for all x is

82.

(a) Zero (b) One (c) T hree (d) Infinite nfi nite − = The solut solution ions s of the the equa quation tion 2 x 2[ x] 1 , where [x [x] =the greatest integer less than or equal to x, are

2

(a) (a) 83.

x = n+

1 2

, n∈ N

1 2

, n∈ N

The num number ber of real real solut solution ions s of 1 + | ex − 1 | (a) 0

84.

(b) (b) x = n −

=

(c) (c)

x 2

1 2

, n∈ Z

(d) (d) n < x < n + 1, n ∈ Z

ex(ex − 2) is

(b) 1

The equa quation tion 2 sin 2

x = n+

(c) 2

⋅ cos 2 x = x + 1 , 0 < x ≤ π x

2

(d) 4

has has

(a) One real solution soluti on (c) I nfinitely nfi nitely many real solutions sol utions 85.

86.

If y ≠

87.

(b) No real solution soluti on (d) x 1 x then the number number of values of the pair (x, y) such that x + y + = and and (x + y) 0 then y 2 y

(a) 1 (b) 2 The num number ber of real real solut solution ions s of the the equa quation tion log 0.5 x =| x| is (a) 1

(b) 2

The prod produ uct of all all the the solut solution ions s of the equa quation tion (x − 2)

2

(a) 2

(b) – 4

[IIT [II T 1987 1987 ]

None of these these

=−

1 2

, is

(c) 0


(c) 0


− 3 | x − 2| + 2 = 0 (c) 0

is (d) None None of these













(b) (– 5, – 4] ∪ (2, 3]

(a) (2, 4) 90.

= [x + 2] , where [x [x] =the greatest integer less than or equal to x, then x must be such that x = 2, − 1 (b) (b) x ∈ [2, 3) (c) (c) x ∈ [−1, 0) (d)

The solut solution ion set set of

x+1 x

+ | x + 1| =

(a) (a) { x| x ≥ 0} 92.


If [ x] (a) (a)

91.

(c) [– 4, – 3) ∪ (3, 4]

2

( x + 1)2 | x|

is

(b) (b) {x| x > 0} ∪ {−1}

If a.3 tan x + a. 3 − tan x − 2 = 0 has real real solutions, x ≠ (a) [–1, 1]

None of these these

π 2

(d) { x| x ≥ 1 or x ≤ −1}

(c) {–1, 1} ,0

≤ x ≤ π , then the set of possible values of the parameter a is

(b) [–1, 0)

(d) (0, +∞)

(c) (0, 1]

Nature of roots Basi c L evel 93. 94. 95.

The root roots s of the quad quadra ratic tic equa quation tion 2 x2 + 3 x + 1 = 0 , are (a) I rrational rrati onal (b) Rational Rati onal

(b) Real and distinct dis tinct

99.

(b) I rrational

[Pb. CE T 1988 1988]]

(d) Equal

are are

[DCE 2002]

(c) Real and equal

If b1b2

(b) Real and equal

(d) One real and one

= 2(c1 + c2 ) , then at least one of the equations

(c) I magin maginary ary 2

x

+ b1 x + c1 = 0

(b) Purely imaginary imagin ary roots

In the equation x + 3 Hx + G = 0 , if G if G and and H are real and G (a) All Al l real and equal equal (b) All Al l real and distinct dis tinct 3

For the equation | x2 |

2

and and x

+ b2 x + c2 = 0

(d) None of these these has has

(c) I magin maginary ary roots 2


+ 4 H > 0 , then the roots are 3


(c) One On e real and two imaginary

(d) All A ll real and two equal equal

+ (x − b)3 + (x − c)3 = 0 , has

+|

x|

(b) One real and two imaginary roots (d) None of these these

− 6 = 0 , the roots are

(a) One and only one real number number (c) Real with wi th sum zero

[EAMC ET 1988 1988,, 93]

(b) Real with wi th sum one (d) Real with wi th product zero

If a > 0 , b > 0, c > 0 , then both the roots of the equation ax2 + bx + c = 0 (a) A re real and negative negative

105.


+ (2a + b)x + b = 0 , a ≠ 0 , will be

+ 2k + 4 = 0

(a) A ll the roots real (c) T hree real roots namely namely x = a, x = b, x = c

104.

[IIT 1979; Rajasthan PET 1983]

Let Let a, b and c be real real numbers numbers such s uch that 4 a + 2b + c = 0 and and ab > 0 . Then T hen the the quadratic quadratic equation ax2 + bx + c = 0 has has [II T 1990 1990]] (a) Real roots (b) Complex roots (c) Purely imaginary imagin ary roots (d) Only Onl y one root If a < b < c < d , then the roots of the equation (x − a)( x − c) + 2( x − b)(x − d) = 0 are are [II T 1984 1984]]

102. The equa quation tion (x − a)3

103.

are are

(c) Non-real 2

(b) Real and unequal unequal

(a) Real roots 101.

(d) Irrational rrati onal and unequal unequal

(c) Real and equal

If a If a and and b are the odd integers, then the roots of the equation 2ax

(a) Real and distinct dis tinct 100.

− 5(l + m)x − 2(l − m) = 0

2

(a) Complex imaginary 98.

(c) I rrational rrati onal and equal equal

If l If l, m, n are real and l ≠ m, then the roots of the equation (l − m)x

If k ∈ (−∞, − 2) ∪ (2, ∞) , then the roots of the equation

(d) None of these these [Rajasthan PET P ET 1986 1986]]

2

(a) Rational 97.

(c) I magin maginary ary

The root roots s of the equa quation tion x2 + 2 3 x + 3 = 0 are are (a) Real and equal (b) Rational Rati onal and equal equal (a) Complex

96.

[II T 1983 1983]]

Let one root of ax (a) (a) 3 − 5

2

(b) Have H ave negative negative real parts

+ bx + c = 0 , where a, b, c are integers be 3 + (b) 3

[II T 1980 1980]]

(c) A re rational numbers numbers

(d) None N one of these these

5 , then the other root is

(c)

5

[MNR 1982]
















λ

(a) Real and unequal unequal roots for all λ (c) Real roots for λ > 0 only = 0 only

109.

(b) Real roots for (d)

If the roots of the equation ax2 + x + (a) Rational

111. 112.

113.

=0

be real, then the roots of the equation x2

−4

Both negative negative

ab x + 1 = 0 will be

(c) Real

(d) Imaginary

− 8 x + (a2 − 6 a) = 0 are real, then (a) (a) −2 < a < 8 (b) (b) 2 < a < 8 (c) (c) −2 ≤ a ≤ 8 2 If the roots of the given equation (cos p − 1)x + (cos p)x + sin p = 0 are real, then  π π  (a) (a) p ∈ (− π , 0) (b) (b) p ∈  − ,  (c) (c) p ∈ (0, π )  2 2  The gre greatest test value value of a nonnon-ne nega gattive rea real num number ber λ for which both the equations If the roots of the equation x2

r −7 p

≥4

[Rajasthan P ET 1987 1987,, 97]

(b) 12

(b) (b)

3

[IIT 1990; Rajasthan PET 1995]

(d) (d) p ∈ (0, 2π ) 2 x2

+ (λ − 1)x + 8 = 0

(c) 15

p −7 r

≥4

2

+ qx + r = 0

(b) 9

[IIT 1995]

+ qx + 1 = 0

(d) For no value of p of p, r

having real real roots is is

(c) 7

117.

If 0 < a < b < c , and the roots α, β of the equation ax (a) (a) | α | =| β | (b) (b) | α | > 1

118.

If roots of the equation a(b − c)x2 (a) A .P.

120. 121.

2

If the equation (m− n)x

+ b(c − a)x + c(a − b) = 0

+ (n − l)x + l − m = 0 (b) (b) 2 = + l

(d) 8 [Kerala (Engg.) 2002]

(d) 7

are non-real complex numbers, numbers, then (c) (c) | β | < 1 (d) None of these these

are equal,, then a, b, c are in

[Roorkee 1993; Rajasthan PET 2001]

(c) H.P. H.P .

(d) None of these

has equal roots, r oots, then l, m and and n satisfy

= n+ l The cond conditio ition n for for the the root roots s of the the equa quation tion (c − ab)x − 2(a − bc)x + (b − ac) = 0 (a) (a) a = 0 (b) (b) b = 0 (c) (c) c = 0 2 2 2 2 2 If the roots of the equation (a + b )t − 2(ac + bd)t + (c + d ) = 0 are equal, then 2

2

2

[DCE 2002; EAMCET 1990]

(d) (d) l =

(c) (c)

2

[IIT 1994]

+ bx + c = 0

(b) G.P. G.P . 2

+n

−8x+ λ + 4 = 0

are real if

116. The leas leastt inte integer ger k which makes makes the roots of the equation equation x2 + 5 + k = 0 imaginary is (a) 4 (b) 5 (c) 6

(a) (a) 2l =

and and x2

(d) 16

(c) For all all values values of p of p, r

3

Let Let p, q ∈ {1, 2, 3, 4 } . T he number number of equation equations s of the form px2 (a) 15

119.

≤ a≤ 8

(d) (d) 2

[AMU [AM U 1990 1990]]

If p If p, q, r are positive and are in A.P., then roots of the equation px (a) (a)

115.

Real and unequal unequal roots for

(b) (d) Both nonreal complex complex

(b) I rrational

have real roots is (a) 9 114.

only

If a > 1 , roots of the equation (1 − a)x2 + 3 ax − 1 = 0 are are (a) One positive posi tive and one negative negative (c) Both positive positi ve

110.

λ<0

to be equal is

+n [TS Rajendra 1982 1982]]

(d) None of these these [MP PET 1996]













(a) 144

(b) 12

128. The value value of k of k for which 2 (a) – 9 and – 7 2

129. The root roots s of 4 x (a) (a) 130.

2

−k + x+8 = 0 (b) 9 and 7

+ 6 px + 1 = 0

4

If the equation x2

has equal and real roots are (c) – 9 and 7

[BIT Ranchi 1990]

(d) 9 and – 7 [MP PET 2003]

1

(c) (c)

3

2

(d) (d)

3

− (2 + m)x + (m2 − 4 m+ 4 ) = 0

has coincident roots, then

(b) (b) m = 0, m = 2

(c) (c) m =

(a) (a) m = 0, m = 1

± 12

(d)

are equal, then the value of p of p is

(b) (b)

5

(c) – 12

4 3 [Roorkee [Roork ee 1991] 1991]

2 3

,m= 6

(d) (d) m =

If two roots of the equation x3 − 3 x + 2 = 0 are same, then the roots will be (a) 2, 2, 3 (b) 1, 1, – 2 (c) – 2, 3, 3 132. The equa quation tion || x − 1 | + a| = 4 can have real solutions for x if a if a belongs to the interval

2 3

, m= 1

131.

(a) (–∞, 4]

(b) (–∞, – 4]

[MP PET 1985]

(d) – 2, – 2, 1

(c) (4, ∞) 2

133. The se set of of value values s of m of m for which both roots of the equation x

(d) [– 4, 4]

− (m+ 1)x + m+ 4 = 0

are real and negative consists of all m

such that [AMU [AM U 1992 1992]]

134.

(a) (a) −3 < ≤ −1 (b) (b) −4 < ≤ −3 (c) (c) −3 ≤ ≤ 5 Both the roots of the given equation ( x − a)(x − b) + (x − b)(x − c) + ( x − c)(x − a) = 0 are always

(d) (d)

−3 ≥

or

≥5

[MNR 1986 1986;; I IT 1980 1980;; Kuruks K urukshetra hetra CEE CE E 1998] 1998]

(a) Positive osi tive 135.

(b) Negative

If P(x) = ax2 + bx + c and and Q(x) = −ax2 (a) Four real roots

+ dx + c

(c) Real

(d) Imaginary

where ac ≠ 0 , then P( x).Q( x) = 0 , has at least

(b) T wo real roots

(c) Four imaginary roots

[II T 1985 1985]]


136. The cond conditio itions ns that hat the the equat quation ion ax + bx + c = 0 has both the roots positive is that [SCRA 1990] (a) (a) a, b and and c are of the same sign sign (b) a and and b are of the same sign (c) (c) b and and c have the same sign opposite to that of a of a (d) (d) a and and c have the same sign opposite to that of b of b 2

137.

If [x] denotes the integral part of x and and k = sin −1 (x − [k])(x + α ) − 1

(a) 1 138.

=0

1 + t2 2t

>0,

α

for which the equation

has integral integral roots is is (b) 2

(c) 4

If the roots of the equation ax2 + bx + c = 0 are real and of the form (a) (a) b2 − 4 ac

then the integral value of

(b) (b) b2 − 2ac

(c) (c)


α α −1 2

2b

and and

− ac

α +1 , then the value of (a + b + c)2 α (d) None of these these

is [AMU [AM U 2000 2000]]













143. The equa quation tion (a + 2)x2

+ (a − 3)x = 2a − 1, a ≠ −2

has roots rational rational for

(a) A ll rational rational values values of a of a except a = −2 (c) Rational Rati onal values of a >

(b) A ll real real value values s of a of a except a = −2

1

(d)

2

− 2 x − λ = 0, λ ≠ 0 Cannot have a real root if λ < 1 Can have a rational root if λ is a perfect square Cannot have have an integral root if n2 − 1 < λ < n2 + 2n

None of these these

144. The quad quadra ratic tic equat quation ion x2 (a) (b) (c)

where n = 0 , 1, 2, 3,....

(d) None of these these 145.

If the roots of the equation x2 the equation x2

+ px + q = 0

− 4 qx + 2q2 − r = 0

are are α and β and roots of the equation x2

are are

α 4 , β 4 , then the roots of

will be

[II T 1989 1989]]

(a) Both negative negative (c) Both real 146.

− xr + = 0

(b) Both positive positi ve (d) One negative negative and one positive positi ve

If equat equation ion a(b − c)x2 + b(c − a)x + c(a − b) = 0 has equal roots, a, b, c >0 >0,, n ∈ N, then (a) (a) an +

n

≥ 2bn

(b) (b) an + cn

>2

n

(c) (c) an +

n

≤ 2bn

(d) (d) an + cn

< 2bn

k− 1

147.

∑ ∑

x2r

If rk=−01

is a polynomial in x for two values of p of p and and q of k of k, then roots of equation equation x2 + px + q = 0 cannot be xr

r =0

(a) Real 148.

(b) I magin maginary ary n 1/n

If for x >0, f (x) = (a − x )

, g(x) = x

2

(c) Rational Rati onal

+ px + q, p, q ∈ R

(d) I rrational rrati onal

and equation g(x) − x = 0 has imaginary roots, then number of real

roots of equation g (g(x)) − f ( f (x)) = 0 is (a) 0 149.

(b) 2

Let Let p, q ∈ {1, 2, 3, 4}. T he number number of equations of the the form px (a) 15

150.

If α1 , α 2 and and

α1 y + α 2 z = 0

(b) 9

having real and unequal roots is

(c) 7 2

(b) (b) p2br


= q2ac

and and px

(d) 8 2

+ qx + r = 0

respectively and system of equations [II T 1987 1987]]

(c) (c) c2 ar = r 2 pb


If a If a, b, c, d are four consecutive terms of an increasing AP then the roots of the equation (x − a)(x − c) + 2( x − b)( x − d) = 0 are are (a) Real and distinct dis tinct

152.

+ qx + 1 = 0

β1, β 2 are the roots of the equations ax + bx + = 0 and and β1 y + β 2 z = 0 has a non-zero non-zero solution. soluti on. The T hen n

(a) (a) a2qc = p2br 151.

(c) 4 2

(b) Nonreal Nonreal complex complex

(c) Real and equal 2

If a If a, b, c are three three distinct disti nct positive posi tive real numbers numbers then the number number of real real roots of ax

(d) Integers ntegers

+ 2b|

x|

−c = 0

is













(c) I s equal equal to 1 157.

(d) I s equal equal to –1

If x If x, y, z are real and distinct then f (x, y) = x2 (a) Non-nega Non-negative tive

158.

(b) Nonpositive Nonposi tive

If a If a ∈ R, b ∈ R then then the factors of the expressi expression on a( x

− y ) − bxy 2

(b) Real and identical

If a If a, b, c are in H.P. then the expression a(b − c)x

2


are are (c) Complex


+ b(c − a)x + c(a − b)

(a) Has real and distinct disti nct factors (c) Has no real factor 160.

is always

(c) Zero 2

(a) Real and differe dif ferent nt 159.

+ 4 y2 + 9 z2 − 6 yz − 3 zx − 2 xy

(b) I s a perfect square (d) None of these these

If a If a, b, c are in G.P., where where a, c are positive, positive, then the equa equation tion ax2 + bx + c = 0 has has (a) Real roots (b) I magin maginary ary roots (c) Ratio of roots =1 : w where w is a nonreal cube root of unity (d) 2

161. The polyn polynom omial ial (ax (a) Four real zeros zeros

+ bx + c)(ax − dx − c)

Ratio Rati o of roots =b : ac

ac ≠ 0 , has

2

(b) A t least two real real zeros

(c) At most two real zeros

(d) No real zeros zeros

Relation between Roots and Coefficient Basi c L evel 162.

If α, β are roots of the equation equation ax2

+ bx + c = 0 , then the value of α 3 + β 3

is

[Kurukshetra [Kuruks hetra CEE 1991 1991;; BIT BI T Ranchi 1998 1998;; MP PET PE T 1994 1994;; Rajasthan Rajas than PET PE T 1989 1989,, 96] 96]

3 abc + b

3

(a) (a) 163.

a3

(b) (b)

a

+b

3

(c) (c) n3 2

+ bx + c = 0

(d) (d) 2n2

(c) Negative 1

165.

If α, β are the roots of the equation 8 x (a) (a)

166.

1 3

(c) (c)

4

If

β are the roots of the equation

(

1)

is

7

0 , then (

1) (β

[AMU [AM U 1990 1990]]

(d) 4

2

(c) q 2


+ q = 0 , then the value of α 2 + αβ + β 2 + q

(b) 1

[DCE 2000]

1

3 3   α 2  β 2     = 0 , then the value of   +     β   α 

1

If α, β are the roots of the equation x2 + px + p2 (a) 0

167.

(b) (b)

− 3 x + 27


(a ≠ 0; a, b, c being different) di fferent),, then then (1 + α + α 2 )(1 + β + β 2 ) =

(b) Positive osi tive 2

b3 − 3 abc a3

is equal to

2

(b) (b) n2

(d) (d)

a3

− (1 + n2 )x + 1 (1 + n2 + n4 ) = 0 , then α 2 + β 2

If α and and β are the roots of the equation a (a) Zero

3 abc − b

3

(c) (c)

3ab

If α, β are roots of the equation x2 (a) 2n 2n

164.

3

is equal qual to

[AMU [AM U 1993 1993]]

(d) 2q 2q 1)

[BIT S Ranchi

; Him. C

]













173.

If α and and β are roots of ax2 + 2bx + (a) (a)

174.

(b) (b)

ac

−1

(b) (b)

2

is equal to

2b

(c) (c)

ac

If α, β are the roots of the equation x2 + 2 x + 4 (a) (a)

175.

2b

α β + β α

= 0 , then

1

= 0 , then

α

3

1

+

1

β3

[BITS Ranchi 1990]

2b

−

(d) (d)

ac

2

is equal to

[Kerala (Engg.) 2002]

(c) 32

2

b

−

(d)

+ a 2 + bx + c = 0 , then α −1 + β −1 + γ −1 = (b) (b) −b / c (c) (c) b / a 3 3 + 1 = 0 , then the value of α + β is

1 4

If α, β , γ are roots of equation x3 (a) (a) a / c

[EAMCET 2002]

(d) (d) c / a

176.

If α, β are roots of x

177.

(a) 9 (b) 18 (c) – 9 (d) –18 If A.M. of the roots of a quadratic equation is 8/5 and A.M. of their reciprocals is 8/7, then the equation is

2

−3

(a) (a) 5 x − 16 x + 7 = 0 (b) (b) 7 x − 16 x + 5 = 0 (c) (c) 7 x 178. The quad quadra ratic tic in t, such that A.M. of its roots is A and and G.M. is i s G, is 2

2

179.

2

180.

2

2

− 8 x + 16 = 0 (b) (b) 15 x2 + 8 x − 16 = 0 If x2 + px + q = 0 is the quadratic whose roots are a − 2

2

(d) (d) 3 x

[AMU [AM U 2001 2001]]

− 12 x + 7 = 0 [IIT 1968, 74]

(c) (c) t + 2 At + G = 0 (d) None of these these = 0 , then the equation whose roots are sin A and tan A will be[Roorkee [Roork ee 1972 1972

3

(a) (a) 15

− 16 x + 8 = 0

2

(a) (a) t − 2 At + G = 0 (b) (b) t − 2 At − G = 0 In a triangle ABC, ABC, the value of ∠A is given by 5 cos A + 3 2

[MP 1994; 1994; BIT Ranchi 1990 1990]]

2

2

− 8 x − 16 = 0 b − 2 where a and and b are the roots of x2 − 3 x + 1 = 0 , then (c) (c) 15 x2

and and

−8

2 x + 16

=0

2

(d) (d) 15

[Kerala (Engg.) 2002]

(a) (a)

p = 1, q = 5

(b) (b) p = 1, q = −5

181. The root roots s of the equa quation tion x2 (a) (a)

x2

+ abx +

3

=0

183.

184.

p = −1, q = 1

+ ax + b = 0 arep re p and and q, then the equation whose roots are (b) (b) x2 − ab + b3 = 0 (c) (c) bx2 + + a = 0

182. The equa quation tion whos whose e root roots s are are (a) (a) 7 x2 − 6 x + 1 = 0

(c) (c)

1 3+ 2

(b) (b) 6 x2

If α, β are the roots of the equation l

and and

1 3− 2

−7x +1 = 0 2 + mx + n = 0

− nl(m2 − 2nl )x + n4 = 0 l 4 x2 + nl(m2 − 2nl)x − n4 = 0

(c) (c)

x2

− 6x + 7 = 0

+ 6 x + 1 = 0 , then the equation equation wi w ith the roots + 6x − 9 = 0

(d) (d) x2

− 7x + 6 = 0

(c) (c)

[MP PET 1997]

+ nl(m2 − 2nl)x + n4 = 0 l4 x2 − nl(m2 + 2nl)x + n4 = 0

(d) (d)

(b) (b) x2

+ ax + ab = 0

then then the equation equation whose roots are α 3 β and and αβ 3 is

(c) (c)

+ 3 x + 18 = 0

(d) (d) x2

[MP PET 1980]

[MP PET 1994]

(b) (b) l4 x2

(a) (a) 2 x2

p2 q and and pq2 will be

is

(a) (a) l 4 x2

If α, β are the roots of 9 x2


x2

1

,

1

α β

is

+ 6x + 9 = 0

[EAMCET 2000]

(d) (d) x2

− 6x + 9 = 0















189.

Let Let

be the roots of

x2

− x+1 = 0

2

+ x + 1 = 0 , then the equation equation whose whose roots are α 31 , α 62

is

[AMU [AM U 1999 1999]]

+ x−1 = 0 (c) (c) x2 + x + 1 = 0 If α, β are roots of the equation x2 − 2 x cos 2θ + 1 = 0 then the equation equation with w ith roots α n / 2 , β n / 2 (a) (a) x2 − 2nx cos θ + 1 = 0 (b) (b) x2 + 2nx cos nθ + 1 = 0 (c) (c) x2 + 2 x cos nθ + 1 = 0 (a) (a)

190.

α, α 2

(b) (b) x2

191. The equa quation tion whos whose e root roots s are are recip recipro roca call of of the the root roots of of the the equat quation ion 3 x2 − 20 x + 17 (a) (a) 3 x2

+ 20 x − 17 = 0

(b) (b) 17 x2

− 20 x + 3 = 0

(c) (c)

=0 2 17 x + 20 x + 3 = 0

(d) (d) x60

+ x30 + 1 = 0

will be


(d) (d) x2

− 2 x cos nθ + 1 = 0

is

[DCE 2002]


192. The sum sum of the the root roots s of of a equat quation ion is 2 and and sum sum of the their cub cubes is 98, the then the the equat quation ion is (a) (a) 193.

195.

x

+ 2 x + 15 = 0

x2

+ x− 6 = 0

x

+α x − β = 0

(c) (c)

x2

+ [(α + β ) + αβ ]x + αβ (α + β ) = 0

If α, β are roots of x2 − 5 x − 3

p2

p2 p2

In the equation

2x

+ q2

− 2 x + 15 = 0

[MP PET 1986] 2

(d) (d) x

− 2 x − 15 = 0

, then equation equation is is (c) (c)


+ x+1 = 0

x2

(d) (d) x2

− 6x + 1 = 0

+ bx − c = 0 , then the equation equation whose w hose roots are b and and c is 2 (b) (b) x − [(α + β ) + αβ ]x − αβ (α + β ) = 0

(b) (b) 33 x2 − 4 x + 1 = 0

If 2 + i 3 is a root of the equation x2

1 2α − 3

and and

1 2β

−3

is

(c) (c) 33 x2 − 4 x − 1 = 0


(d) (d) 33 x2 + 4

− px + q = 0 , then the value of sin 2 (α + β ) = (c) (c)

[Pb. CET C ET 1989 1989]]

+ [αβ + (α + β )]x − αβ (α + β ) = 0

q2 p2

+ (1 − q)2

(c) (4, 7)

+1 = 0


(d) (d)

p2 ( p + q)2

+ px + q = 0 , then (p (p, q) is equal to

(b) (– 4, 7) 2

6

= 0 , then the equation equation with w ith roots

(b) (b)

+ (1 − q)2

1

2

(d) (d) x2

Given Gi ven that tan tanα and tanβ are the roots of x2 p2

(c) (c)

− x+6 = 0

(a) (a)

2

(a) (7, – 4) 198.

+ 15 x + 2 = 0

If α, β are the roots of the quadratic equation equation x2

(a) (a) 197.

(b) (b) x

(b) (b) x2

(a) (a) 33 x2 + 4 x − 1 = 0 196.

2

Sum of roots is –1 and sum of their their reciprocals reciprocals is (a) (a)

194.

2

[II T 1982 1982;; MP 1997]

(d) (7, 4)

, the coefficient of was taken as 17 in place of 13 and its roots were found to be –2 and –15.













206.

If the sum of the two roots of the equation 4 (a) 1, 2, – 2

207.

+ 16 x2 − 9 x − 36 = 0

− 2, 2 , − 2 3

= 9 ac

3

= 9 ac 2 x − px + 36 = 0

(b) (b) 2 b2

If α, β are the roots of the equation

2

and and

(d) (d)

2

p r

=

q s

+ px + q = 0

= −4 ac

− 4, 3 , − 3 2

2

(c) 2 and and

(b) (b) 2 h =

α + h, β + h

2

(c) (c)

p2

− 14 x − 74 = 0

x2

212. The quad quadra ratic tic equat quation ion with with one one root root as the the squa square re root root of − 47 + 8

−3

x2

+ 2 x + 49 = 0

(b) (b) x2

+ 14 x + 74 = 0

(c) (c)

(a) (a)

x2

(b) (b) x2

− 2 x + 49 = 0 1

213.

The quad quadra ratic tic equat quation ion whos whose e one one root root is

214.

− 4x−1 = 0 The quad quadra ratic tic equat quation ion with with one one root root 2 − 3 is 2 (a) (a) x − 4 x + 1 = 0 (b) (b) x2 − 4 x − 1 = 0 (a) (a)

x2

+ 4x−1 = 0

(b) (b) x2

2+ 5

[AMU [AM U 1991 1991]]

(d)

±9

(d) 1

are the roots of

p r  q + s  

= c2

[EAMCET 2002]

+ rx + = 0 , then

− 4 q = r2 − 4 s

211. The quad quadra ratic tic equat quation ion with with real real coeffic oefficie ient nts s wh whose one one root root is 7 + 5 i will be (a) (a)

(d) (d) a2

of p p are are α 2 + β 2 = 9 , then the value of (c) ± 8

(b) 3

If α, β be the roots of x2 (a) (a)

215.

[MP PET 1986]

[MP PET 1986]

(c) (c) b2

±3 (b) ± 6 3 If α, β , γ are the roots of 2 x − 2 x − 1 = 0 , then (∑ αβ )2 =

(a) – 1 210.

− 3, 3 , − 3

(c) (c)

(a) (a) 209.

is zero, then the roots are

If the roots of the equation ax2 + bx + c = 0 are are l and 2l 2l, then (a) (a) b2

208.

(b)

3

(c) (c)

x2

[AMU [AM U 2001 2001]]

(d) (d) pr 2

= qs2

[Kerala (Engg.) (E ngg.) 2001 2001,, 02; Rajasthan Rajas than PET P ET 1992 1992]]

+ 14 x − 74 = 0

(d) (d) x2

− 14 x + 74 = 0

is

± 2 x + 49 = 0

[II T 1995 1995]]

(d) (d) x2

± 2 − 49 = 0

will be


(c) (c)

x2

+ 4x+1 = 0

(d) None of these these [Rajasthan PET P ET 1985 1985]]

+ 4x+1 = 0 (d) (d) 2 The quad quadra ratic tic equat quation ion whose whose root roots s are are thre hree time times the the root roots s of of the the equa quation tion 3ax + 3bx + = 0 is (a) (a) ax2 + bx + c = 0 (b) (b) ax2 + 3bx + c = 0 (c) (c) ax2 + bx + 3c = 0 (d) (d) (c) (c)

2

x

2

x

+ 4 x−1 = 0 [AMU [AM U 1990 1990]]

ax2

+ 3 bx + 3 c = 0













= 4q + 1 The num numerical rical diffe differe renc nce e of the the root roots s of of x2 − 7 x − 9 = 0 (a) (a)

225.

p2

= 4q

(b) (b) p2

(a) 5 226.

227.

±2

−1 = 0

− px + 8 = 0 ±4

If the roots of the equations x

− bx + c = 0

(d) (d)

85

be 2, then the value of p of p is (c)

2

− bx + c = 0

±6

(c) (c) b2 − 4 2

and and x


(d)

±8

be 1, then

(b) (b) b2 − 4 c = 0 2


(c) (c) 9 7

If the difference of the roots of the equation (a) (a) b2 − 4

228.

(b)

= 4q − 1

is

(b) 2 85

If the difference of the roots of x2 (a) (a)

p2

(c) (c)

− cx + b = 0


+1 = 0

(d) (d) b2 + 4

−1 = 0

differ by the same quantity, then b + c is equal to [BIT Ranchi 1969; 1969; MP PET P ET 1993 1993]]

(a) 4

(b) 1 2

− bx + c = 0

229.

I f the roots roots of x (a) 1

230.

If α, β are the roots of x

231.

are two consecutive integers, then b − is (b) 2 (c) 3

[Kurukshetra [Kuruks hetra CEE CE E 1998] 1998]

(d) 4

2

(b) 6

(c) 2 2

If X If X denotes the set of real numbers p for which the equation x (a) (a)

233.

(d) – 4

2

and α < 1 < β then − 3 x + a = 0, a∈ R and 9    9  (a) (a) a ∈ (−∞, 2) (b) (b) a ∈  − ∞,  (c) (c) a ∈  2,  4    4  If α, β be the roots of 4 x2 − 16 x + λ = 0, λ ∈ R such that 1 < α < 2 and and 2 β < 3 (a) 5

232.

(c) 0

 − 2, − 1    2  

(b) (b)

 − 1 , 1     2 4 

If one root of the quadratic equation a 1

1

= p(x + p)

+ bx + c = 0

then the number of integral solutions of λ is (d) 3

has its roots greater than p then X is equal to (d) (– ∞, 0)

(c) Null Nul l set 2


th

is equal to the n

power of the other root, then the value of













242.

(a) (a) k = 1 (b) (b) k = 2 (c) (c) k = 3 (d) None of these these If a If a and and b are rational and b is not a perfect square then the quadratic equation with rational coefficients whose one root is 1

a+ i s (a) (a) 243.

If

− 2ax + (a2 − b) = 0

1 4

(a) (a) 244.

x2

If α ,

is a root of a

2

− 3i a = 25 , b = −8 β , γ

246.

− b)x 2 − 2ax + 1 = 0

− b)x 2 − 2bx + 1 = 0

(c) (c) (a2

be the roots of the equation x(1 + x 2 ) + x 2 (6 + x) + 2 (b)

(c) (c) a = 5, b = 4

=0

1 2

then the value of

(c) (c)


α −1 + β −1 + γ −1

−1

− 12 x 2 + 39 x − 28 = 0 are in A.P. then their common difference is (b) (b) ±2 (c) (c) ±3 The root roots s of the equa quation tion x3 + 14 x 2 − 84 − 216 = 0 are in (b) G.P. G.P .

(c) H.P. H.P .

If 3 and 1 + 2 are two roots of a cubic equation with rational coefficients, then the equation is (c) (c)

248.

− 3 x 2 − 4 x + 12 = 0 What is the sum of the squares of roots of x 2 − 3 x + 1 = 0 (a) 5

(c) 9

249.

If α (a) (a)

x3

− 5 x2 + 9 x − 9 = 0

(b) (b) x3 (b) 7

+ β = 3 and and α + β = 27 , then α and and β are the roots of 2 (b) (b) 9 x 2 − 27 x + 20 = 0 3x + 9x + 7 = 0 3

x3

(d) (d)

±4

(d) None of these

247.

(a) (a)

is


2

I f the roots roots of x3 (a) (a) ±1 (a) A .P.


+ bx + 1 = 0 , where a, b are real, then (b) (b) a = 25 , b = 8

(a) – 3 245.

(b) (b) (a2

− 5x2 + 7 x + 3 = 0

(d) None of these these [Karnataka CET CE T 1993 1993]]

(d) 10

3

(c) (c)

2x

2

− 6 x + 15 = 0














and β = −2 α = 1 and If p If p and and q are the roots of x 2 + px + q = 0 , then (a) (a) p = 1 (b) (b) p = −2 If roots of the equation 2 x 2 − (a2 + 8 a + 1)x + a2 − 4 a = 0 (a) (a) 0 < a < 4 (b) (b) a > 0 (a) (a)

259. 260. 261.

x2

− −2=0

m n

+

n m

+1 = 0

(c) (c)

and

β =1

(d) (d)

α=2

and and

β = −2

[IIT 1995, AIEEE 2002]

(c) (c)

p = 1 or 0

(d) p = −2 or o

are in opposite sign, then

[AMU [AM U 1998 1998]]

(c) (c) a < 8

(d) (d)

−4 < a < 0 [SCRA 1999]

+ x− 2 = 0 (c) (c) + x + 1 = 0 are in the ratio m : n then

2

(b) (b)

m+ n +1

+ nx + n = 0

If the roots of the equation 12 x

2

=0

(c) (c)

(c) (c)

− mx + 5 = 0

m n + +1 n m p q

±

(c) (c)

=0

+ x+ 2 = 0

+ n+ 1 = 0

(d) (d)

q is equal to [Rajasthan PET 1997; BIT Ranchi 1999] p

+

(d) (d)

n/ l

are in the ratio 2 : 3 , then

(b) (b) 3 10

(d) (d) x 2


are in the ratio p : q then

l/n

− +2=0

x2

−

l/n

=



2 10

+ bx + c = 0 be p : q , then − (P + q) 2 ac = 0 (c) (c) pqa2 − (P + q)2 bc = 0 + 14 + 24 = 0 are in the ratio 3 : 2 . The T he roots roots will wil l be

If the ratio of the roots of the equation ax 2 (a) (a)

pqb 2

+ (P + q)2 ac = 0

(b) (b) pqb 2

266. The two two root roots s of of an an equa quation tion x3 − 9 x 2 (a) 6, 4, –1 (b) 6, 4, 1

267. The cond conditio ition n tha that one one root root of the equa quation tion ax (a) (a) b

2

= 8 ac

(b) (b) 3 b x − bx ax − c 2

268.

α=2

(b) (b) x 2

(b) (b)

n/ l

(a) (a) 5 10 265.

(b) (b)

If the roots of the equation lx2 (a) (a)

264.

β = −1

If the roots of the equation x (a) (a)

263.

and and

Which of the following equation has 1 and –2 as the roots (a) (a)

262.

α =1

If the roots of the equation

(c) –6, 4, 1 2

+b +c= 0

(d) None of these these [UPSEAT 1999]

(d) –6, –4, 1

is three times the other other is is

+ 16 a = 0 (c) (c) 3 b = 16 ac λ −1 = are such that α + β = 0 , then the value of λ λ +1

2

[Pb. CET C ET 1994 1994]]

[DCE 2002]

2

(d) (d) b is

2

+ 3a = 0















275.

276. 277.

∠R = π

 p  and tan  Q  are the roots of the equation ax 2 + bx + c = 0 (a ≠ 0) , then [II T 1999 1999]]    2  2   2  (a) (a) a + b = c (b) (b) b + c = 0 (c) (c) a + c = b (d) (d) b = c 2 The prod produ uct of all all real root roots s of the equa quation tion x − | x | −6 = 0 is [Roorkee [Roork ee 2000] 2000] (a) (a) −9 (b) 6 (c) 9 (d) 36 2 If the sum of the roots of the equation ax + bx + = 0 is equal to the sum of the squares of their reciprocals then PQR, In a triangle PQR

. If tan 

bc 2 , c a r e in (a) A .P. 278. The root roots s of the equa quation tion x 2

[II T 1976 1976]]

(b) G.P. G.P .

− 2x + A = 0

(c) H.P. H.P . are are p, q and the roots of the equation equation x 2

(d) None of these

− 18 x + B = 0

are are r, s. If p < q < r < s are are i













(a) (a) 287.

1 4

(a + b + c + d)

1

+b+ 2

If α 1 , α 2 are the roots of equation equation x 2 parallel to α 2 ˆi (a) (a)

288.

a

(b) (b)

c 3

+d

(c) (c)

4

− px + 1 = 0

and and

β1 , β 2

a 1

−b+ 2

c 3

−d


4

be those of equation x 2

− qx + 1 = 0

and vector

α 1ˆi + β 1 jˆ

+ β 2 jˆ , then

p = ±q

(b) (b) p = ±2 q

I f the roots roots of a1 x 2

+ b1 x + c1 = 0

(c) (c)

are are α 1 and and

p = 2q

β 1 and those of a2 x 2 + b2 x + c2 = 0

(d) None of these these are are α 2 and and

β 2 such that α 1α 2 = β 1 β 2 = 1 ,

then (a) (a) 289.

=

b1 b2

=

c1 c2

(b) (b)

a1 c2

=

b1 b2

If the sum of the roots of the equation qx 2 (a) (a)

290.

a1 a2

−2

(a) 0

(b) (b)

3

If x = (β

is

− γ )(α − δ ),

y = (γ − α )(β

(b)

=

c1 a2

+ 2 x + 3q = 0

3 2

(c) (c) a1 a2

= b1 b2 = c1 c2

is equal to their product, then the value of q of q is equal to (c) 3

− δ ), z = (α − β )(γ − δ ) , then the value of x3 + y3 + z 3 − 3 xyz

α 6 + β 6 + γ 6 + δ 6


(c) (c)

α 6 β 6 γ 6 δ 6

(d) –6 is (d) None of these these













r i s

300.

given by (a) (a) x3 − 3 Ax2

(b) (b) x3

(c) (c)

(d) (d)

Let Let

302.

7

x2

− +2=0

7

and B = a3

+ a5 + a6

− 3 Ax2 + 3(G 3 / H)x − G 3 = 0 x3 − 3 Ax2 − 3(G 3 / H )x + G 3 = 0

then A and and B are roots of the equation


− x− 2 = 0 (c) (c) x 2 + x + 2 = 0 (d) None of these these 2 2 2 If α , β are the roots of the equation x − px + q = 0 , then the quadratic quadratic equation equation whose whose roots are (α − β )(α 3 − β 3 ) and and α 3 β 2 + α 2 β 3 is [Roorkee [Roork ee 1994] 1994] (a) (a) x 2 − Sx + P = 0 (b) (b) x 2 + Sx + P = 0 (c) (c) x 2 + Sx − P = 0 (d) None of these these 4 2 2 2 2 4 2 2 [Where S = p( p − 5 p q + 5 q ) and and P = p q (p − 5 p q + 4 q ) ] (a) (a)

301.

+ G 3 (3 x − 1) = 0 x3 + 3 Ax2 + 3(G 3 / H )x − G 3 = 0 π 2π a = cos + i sin 2 , A = a + a2 + a4 (b) (b) x 2

Let Let A, G and and H are the the A .M., .M ., G.M. G.M . and H.M. H.M . respectively respectively of two unequal positive posi tive integers. integers. T hen the the equation equation Ax2 − |













311.

312.

If α ,

β

(a) (a)

α −1 , β −1

I f the roots roots of ax 2 (a) (a)

313.

are the roots of ax 2

If α ,

b2

− bx − = 0

− 4 ac are the roots of x2

p(p 2

then the equation (a + cy)2

α 2, β 2

= b2 y (c) (c)

in y has the roots

αβ −1 , α −1 β

(d) (d)

α −2 , β −2

change by the same quantity then the expression in a, b, c that does does not change is

(b) (b)

a

α3 − β3 (a) (a)

(b) (b)

2

β

+ c = bx

b − 4c a

− px + q = 0

(c) (c)

b2

+ 4 ac a2


then then the product of the roots of the quadratic equation whose roots are α 2

−β2

and and

is

− q)2

(b) (b) p( p2

− q)( p2 − 4 q)

(c) (c)

p( p2

− 4 q)( p2 + q)

314. The quad quadra ratic tic equa quation tion whos whose e root roots s are are the A.M. and and H.M. of the root roots s of the equat quation ion x 2

315.

(b) (b) 45 x 2 − 14 + 14 = 0 + 14 x − 45 = 0 If z0 = α + iβ , i = − 1 , then the roots of the cubic equation (a) (a) 2, z0 , z0 (b) (b) 1, z0 , − z0

x3

316.

Let Let a, b, c be real numbers and a ≠ 0 . If α is a root of a2 x 2

+ bx + c = 0 , β

(a) (a) 14 x 2

2

2


+ 7x−1 = 0

is

(d) None of these these + 45 x − 14 = 0 − 2(1 + α )x2 + (4α + α 2 + β 2 )x + 2(α 2 + β 2 ) = 0 are are (c) (c) 2, z0 , − z0 (d) (d) 2, − z0 , z0 (c) (c) 14 x 2

is a root of of a2 x2 − bx − c = 0 , and 0

<α < β













326.

Let Let A, G and H be the A.M., G.M. and H.M. of two positive number a and b. T he quadratic equation whose roots are A and and H i s

− ( A 2 + G 2 )x + AG 2 = 0 (b) (b) Ax2 − ( A 2 + H 2 )x + AH 2 = 0 (c) (c) Hx 2 − (H 2 + G 2 )x + HG 2 = 0 (d) None of these these 2 2 2 If x + y + z = 1 , then the value of xy + yz + zx lies in the interval 1 1   1   (a) (a)  , 2  (b) (b) [−1, 2] (c) (c)  − , 1 (d) (d)  − 1,  2 2   2   If px 2 + qx + r = 0 has no real roots and p, q, r are real real such that p + r 0 , then (a) (a) p − q + r < 0 (b) (b) p − q + r > 0 (c) (c) p + r = q (d) A ll of of these these 2 The quad quadra ratic tic equa quation tion x − 2 x − λ = 0, λ ≠ 0 (a) Cannot have a real root if λ < −1 (b) Can have a rational root if λ is a perfect square (a) (a)

327.

328. 329.

Ax2















Condition Condi tion for commo common n roots Basic L evel 337.

I f equations equations

2

+ bx + a = 0

and and x 2

+ ax + = 0

have one root common and a ≠ b , then [Rajasthan PET PE T 1992 1992;; II T 1986 1986]]













350.

If every pair from among the equation x 2

+ px + qr = 0 ,

x2

+ qx + rp = 0

and and x 2

+ rx + pq = 0

has a common root, then the

product of three common common roots is is (a) (a) 351.

pqr

If the equation x 2 respectively (a) (a) r, pq

(b) (b) 2 pqr

+ px + qr = 0

and and x 2

(b) –r, –r, pq

(c) (c)

+ qx + pr = 0

p 2 q2 r 2


have a common root, then the sum and product of their other roots are (c) (c) pq, r

(d) –pq, –pq, r













(a) (a) r ∈ (−∞, 1) 364. 365.

(b) (b) r

If x If x is real, then the value value of x 2 − 6 (a) 4 (b) 6 If x If x be real, the least value val ue of

2

=

24

(c) (c) r ∈ (3,

5

+ 13

− 6 x + 10

+ ∞)

(d) (d) r

will wil l not be less less then

1 2 [Rajasthan PET P ET 1986 1986]]

(c) 7 is

=

(d) 8 [Kurukshetra [Kuruks hetra CEE CE E 1998] 1998]













(a) 24 378.

If

2

(a) (a) 379.

If x + 1 is a factor factor of x4 If x2

(c) 3, 24

− 3 x + 2 is a factor factor of x4 − px 2 + q , then p = 4, q = 5 (b) (b) p = 5 , q = 4

(a) – 4 380.

(b) 0, 24

+ px + 1

(d) 0, 3 [IIT 1974; MP PET 1995]

(c) (c)

p = −5 , q = −4


− (p − 3)x 3 − (3 p − 5)x 2 + (2 p − 7)x + 6 , then p is equal to (b) 4

is a factor of the expression a

(c) –1 3

+ bx + c , then

[II T 1975 1975]]

(d) 1 [II T 1980 1980]]













100

393.

If f (x) =

∑

ar xr and and f (0) and and f (1) are odd numbers, then for any integer x

r =0

394.

(a) (a)

f (x) is odd or even according as x is odd or even even

(b)

f (x) is even or odd according as x is odd or even

(c) (c)

f (x) is even for all integral values of x of x

(d) (d)

f (x) is odd for all i ntegra ntegrall values values of x of x

If x ∈ [2, 4 ] then for the expressi expression on x2 − 6 x + 5 (a) T he least value

= −4

=0

(b) T he greatest value

=4

(c) T he least value

=3

(d) T he greatest value = −3































































Quadratic Equation Assignment

Recommend Documents