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QUADRATIC EQUATIONS 1.
If one root of 5x 2 + 13x + k = 0 is reciprocal of the other, then k = (A) 0 (B) 5 (C) 61
2.
(C)
6.
If the difference of the roots of the equation, x2 - bx + c = 0 be 1, then (A) b2 - 4c - 1 = 0 (B) b2 - 4c = 0 (C) b2 - 4c + 1 = 0 (D) b2 + 4c - 1 = 0
7.
Let y =
(D) 6
If the roots of the equation, ax 2 + bx + c = 0 be α & β, then the roots of the equation cx2 + bx + a = 0 are : (A) − α , − β (B)
1
(x − 2)
, then all real
values of x for which y takes real values are : (A) - 1 ≤ x < 2 or x ≥ 3 (B) - 1 ≤ x < 3 or x > 2 (C) 1 ≤ x < 2 or x ≥ 3 (D) None
α , β1 1 α,
(x + 1) (x − 3)
1 β
(D) None of these 3.
4.
5.
If the roots of the given equation, (m 2 + 1) x 2 + 2 amx + a2 - b 2 = 0 be equal, then : (A) a2 + b2 (m 2 + 1) = 0 (B) b 2 + a2 (m 2 + 1) = 0 (C) a 2 - b2 (m 2 + 1) = 0 (D) b2 - a2 (m 2 + 1) = 0 If P(x) = ax2 + bx + c and Q(x) = - ax 2 + dx + c where ac ≠ 0, then P(x) . Q(x) = 0, has atleast : (A) Four real roots (B) Two real roots (C) Four imaginary roots (D) None of these If a root of the equation, ax 2+bx+c=0 be reciprocal of a root of the equation, a ′ x2 + b′ x + c′ = 0 then : (A) (cc ′ − aa ′ )2 = (ba′ − cb ′) (ab ′ − bc ′ ) (B) (bb ′ − aa ′ )2 = (ca′ − bc ′) (ab ′ − bc ′ ) (C) (cc ′ − aa ′)2 = (ba′ + cb ′) (ab′ + bc ′ ) (D) None of these
8.
If 2 + i 3 is a root of the equation, x2 + px + q = 0, where p & q are real, then (p, q) = (A) (- 4, 7) (B) (4 (4, - 7) (C) (4 ( 4, 7 ) ( D ) N o ne o f t h e se
9.
If (x + 1) is a factor of x 4 − (p − 3) x2 − (3p − 5) x2 + (2p − 7) x + 6, then p = (A) 4 (B ) 2 ( C) 1 ( D ) N o ne o f t h e se
1 0 . If x = 2+22/3 +2 1/3 , then, x 3-6x 2 +6x= (A) 3 (B ) 2 ( C) 1 ( D ) N o ne o f t h e se 1 1 . The co-efficient of x in the equation x 2 + px + q = 0 was taken as 17 in place of 13, its roots were found to b e - 2 and - 15 . The roots of the original equation are : (A) 3, 10 (B) - 3, - 10 (C) - 5, - 8 (D) None of these
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2 1 2 . Let a, b, c be real numbers numbers a ≠ 0 . If 2 2 α is a root of a x + bx + c = 0, β is a root of a 2 x2 − bx − c = 0 & 0 < α < β then the equation, a 2 x2 + 2bx + 2c = 0 has a root γ that always satisfies :
(A) (C)
γ = α2+β γ = α
(B) (D)
γ = α + β2 α < γ < β
1 3 . If x 2 - 3x + 2 be a factor of x4 - px 2 + q, q, then (p, q) = (A) ((3 3, 4) (B) ((4 4 , 5) (C) (4 (4, 3) (D) (5 (5, 4) 1 4 . If one root of the quadratic equation ax 2 + bx + c = 0 is equal to the n th power of the other root, then the value
(
of a c
)
1
n n +1
(
)
+ a c
(A) b (C)
n
1 n
+1
(D) -
bn + 1
n
+1
1 5 . If α, β are the roots of , (x - a) (x - b) = c, c ≠ 0, then the roots of (x −α ) (x −β )+c = 0 shall be : (A) a, c (B ) b , c (C) a, a, b (D ) a + c , b + c 1 6 . The equation, ( 3 4/) (
x (A) (B) (C) (D)
2
l o2gx ) +
(
1 8 . If the roots of the equation, x2 - bx + c = 0 and x2 - cx + b = 0 differ by the same quantity, then b + c is equal to : (A) 4 (B ) 1 ( C) 0 (D) - 4 1 9 . If x 2/3 - 7x 1/3 + 10 = 0, then x = ( A ) { 1 25 } ( B) { 8} (C) φ (D) {125, 8} 2 0 . Let one root of ax 2 + bx + c = 0,
(A) 3 1
b
Atleast one root in [0, 1] Atleast one root in [1. 2] Atleast one root in [- 1, 0] None of these
where a, b, c are integers be 3 + then the other root is :
=
(B ) - b 1
(A) (B) (C) (D)
l o2gx ) − 5 4/
= 2 has : Atleast one real solution Exactly three real solutions Exactly one irrational solution All the above .
1 7 . If a, b, c are real numbers such that a + b + c = 0, then the quadratic equation, 3ax2 + 2bx + c = 0 has :
(C)
2 2 . If
(B) 3
5
(D) None of these
5
2 1 . The x2 + (A) (C)
5,
real roots of the equation, 5 x + 4 = 0 are : {− 1, − 4} (B) {1 {1, 4} {- 4, 4} ( D ) N o ne o f t h e se 2x
1
> x+1 , then: 2x + 5x + 2
(A) (C)
2
−2 > x > −1 −2 < x < −1
( B) − 2 ≥ x ≥ − 1 (D) − 2 < x ≤ − 1
2 3 . If the roots of the equation, 1 x+p
+ x+1 q
= 1r are equal in magnitude
but opposite in sign, then the product of the roots will be :
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3
(A)
p2 + q 2 2
(B)
p2 +q2 2
(C)
p2 −q2 2
(D)
p 2 −q 2 2
2 4 . If x is real, then the maximum and minimum values of expression, x2 +14x + 9 x2 +2x + 9
(A) 4, - 5 (C) - 4, 5
will be : (B ) 5 , - 4 (D) - 4, - 5
2 5 . If both the roots of, k (6x k (6x 2 + 3) + rx + 2x 2 - 1 = 0 and 6k (2 6k (2x x2 + 1) + px + 4x 2 - 2 = 0 are common, then 2r - p is equal equal to : (A) - 1 (B) 0 ( C) 1 (D ) 2 2 6 . I f x 2 + px + 1 is a factor of the expression, ax 3 + bx + c, then : (A) a2 + c2 = - ab (B) a 2 - c 2 = - ab (C) a 2 - c2 = ab (D) None of these 2 7 . If x, y, y, z are are real and and distinct, then 2 2 2 u = x + 4y + 9z − 6yz − 3zx − 2xy is always : (A) Non-negative (B) Non-positive (C) Zero (D) None of these 2 8 . If a < b < c < d, then the the roots of the equation, (x − a) (x − c) + 2 (x − b) (x − d) = 0 are : (A) Real and distinct (B) Real and equal (C) Imaginary (D) None of these
2 9 . The number of real roots of the equation, esin x - e - sin x - 4 = 0 are : (A) 1 (B ) 2 ( C) I nf i n i t e (D) None 3 0 . If the product of roots of the equation x2 - 3kx + 2 e2 log k - 1 = 0 is 7, then its roots will be real when : (A) k = 1 ( B) k = 2 (C) k = 3 ( D ) N o ne o f t h e se 3 1 . If the roots of the equation, Ax 2 + Bx + C = 0 are α, β & the roots of the equation x 2 + px + q = 0 are α2, β2, then value value of p will be :
(A)
(C)
2 B − 2 AC
A2 B2
− 4 AC A
2
(B)
2 AC − B2 A2
(D) None of these
3 2 . If α 1 , α 2 and an d β 1, β2 are the roots of the equations, ax2 + bc + c = 0 and px 2 + qx + r = 0 respectively and the system of equations, α 1 y + α 2 z = 0 a n d β 1 y + β 2z = 0 has a non-zero solution, then : (A) a2 qc = p 2 br (B) b 2 pr = q 2 ac (C) c2 ar = r2 pb (D) None of these 3 3 . If the roots of the equation,
− bx ax − c
x2
=
−1 m+1 m
are equal but
opposite in sign, then the value of m will be : (A)
(C)
−b a +b
(B)
+b −b
(D)
a
a a
−a a +b b
+a b−a b
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4
(B) 15 x2 + 8 x - 16 = 0
3 4 . The roots of the given equation, (q - r) x2 + (r - p) x + (p - q) = 0 are
(A)
r −q p −q
(C)
p −q q−r
,1 ,1
(B )
q−r p −q
,1
(D )
r −p q− r
,1
3 5 . How many roots the equation, 2
2
x - x−1 = 1 - x−1 have ? (A) One ( C) In f i ni t e
(B) Two (D ) N o ne
3 6 . If the roots of the given equationn, (cos p - 1) x2 + (cos p) x + sin p = 0 are real, then :
∈ ( − π, 0) p ∈ (0, π)
∈ (− π2 , π2 ) p ∈ (0, 2 π)
(A) p
(B ) p
(C)
(D )
3 7 . If the roots of the equation, x2 + px + q = 0 are α & β and roots of the equation x2 − xr + s = 0 are α 4, β 4, then the roots of the equation, x2 - 4 qx + 2 q2 - r = 0 will be : (A) Both negative (B) Both positive (C) Both real (D) One negative & one positive 3 8 . If the roots of the equation, 8 x3 - 14 x2 + 7 x - 1 = 0 are in G.P., then the roots are :
(A) 1, 21 , 14 ( C) 3, 6 , 1 2
(B) 2, 4, 8 (D ) N on e of t he s e
3 9 . In a triangle ABC, the value of ∠ A is given by, 5 cos A + 3 = 0, then the equation whose whose roots are sin A and tan A will be : (A) 15 x3 - 8 x + 16 = 0
(C) 15 x2 - 8 2 x + 16 = 0 (D) 15 x2 - 8 x - 16 = 0 4 0 . Let α & β be the roots of the equation x2 + x + 1 = 0 . The equations whose roots are α19 , β7 is : (A) x2 - x - 1 = 0 (B) x2 - x + 1 = 0 (C) x2 + x - 1 = 0 (D) x2 + x + 1 = 0 4 1 . Two students while solving a quadratic equation in x, one copied the constant term incorrectly and got the roots 3 and 2 . The other copied the constant term and coefficient of x2 correctly as - 6 and 1 respectively. The correct roots are : (A) 3, - 2 (B) - 3, 2 (C) - 6, - 1 ( D ) 6, - 1 4 2 . The set of values of x which satisfy x+ 2
5 x + 2 < 3 x + 8 and x−1 < 4 is : (A) (B) (C) (D)
(2, 3) (− ∞, 1) ∪ (2, 3) ( − ∞ , 1) (1, 3)
4 3 . If α , β are the roots of x2 0 and α n + βn = V n , then : (A) Vn + 1 = a Vn + b Vn - 1 (B) V n + 1 = a V n + a Vn - 1 (C) V n + 1 = a Vn - b Vn - 1 (D) Vn + 1 = a Vn - 1 - b Vn
44 .
− ax + b =
If the inequal ity (m − 2)x 2 + 8x + m + 4 > 0 is satisfied satisfied for for all x ∈ R then the least least integral integral m is : (A) 4 (B) 5 (C) 6 ( D) n one
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5
45.
The values of k, for which the equation, x 2 + 2 (k − 1) x + k + 5 = 0 possess atleast one positive root, are : (A) [4, ∞) (B) (− ∞, − 1] ∪ [4, ∞ ) (C) [− 1, 4} (D) (− ∞, − 1]
46.
The set of values of 'a' for which the inequality, (x − 3a) (x − a − 3) < 0 is satisfied for all x ∈ [1, 3] is : (A) (A) (1/3 (1/3,, 3) (B) (B) (0, (0, 1/3 1/3) (C) (− 2, 0) (D) (− 2, 3)
47. 47.
The The val value ue of p for for whic which h bot both h the the roots of the quadratic equation, 4x2 − 20 px + (25p 2 + 15p − 66) are less than 2 lies in : ( A) ( 4 / 5, 2 ) (B) (B) (2, ∞) (C) (− 1, 4/5) 4/5) (D) (D) (− ∞ , − 1)
48.
If the equation s in in4 x − (k + 2) sin2 x − (k + 3) = 0 has a solution then k must lie in the interval : (A) (− 4, − 2) (B) [− 3, 2) (C) (− 4, − 3) (D) [− 3, − 2]
49.
If bo both th root roots s of the the qu quad adra rati tic c 2 equation x + x + p = 0 exceed p where p ∈ R then p must lie in the interval : (A) (− ∞, 1) (B) (− ∞, − 2) (C) (− ∞, − 2) ∪ (0, 1/4) (D) (− 2, 1) ANSWER SHEET
1. B 7. A
2. C 8. A
3. C 4. B 5 . A 6. A 9 . A 10 . B 11 . B 1 2 . D
13. D 19. D 25. B 31. B 37. C 43. C 49. B
1 4. B 2 0. A 26. C 3 2. B 3 8. A 4 4. B
1 5. C 2 1. D 27 . A 33 . A 39 . B 4 5. D
16. D 22. C 28. A 34. C 40. D 46. B
17. A 23. B 29. D 35. D 41. D 47. D
1 8. D 24 . A 30 . B 36 . C 42 . B 4 8. D
