1.Algebra Booster

Quadratic Equations and Expressions 2.17
49. The total number of solutions of sin (px) = ln||x|| is
(a) 2 (b) 4 (c) 6 (d) 8
3
x + x+
2
50. The minimum value of y = , x> 0 is
x
(a) 3 (b) 4 (c) 5 (d) 2
51. If P(x) = ax 2 + bx + c and Q(x) = –ax 2 + dx + c, where
ac π 0 , P(x) ◊ Q(x) = 0 has
(a) exactly one root (b) at least two real root
(c) exactly three real roots (d) all four are real roots.
52. If a > 0, b > 0 and c > 0, both the roots of the equation
ax 2 + bx + c = 0
(a) are real and negative (b) have negative real parts
(c) are rational numbers (d) none
53. The real values of a for which the quadratic equation
2x 2 – (a 3 + 8a – 1)x – a 2 – 4a = 0 possesses roots of opposite
signs is given by
(a) a > 5 (b) 0 < a < 4
(c) a > 0 (d) a > 7
54. Suppose a, b and c are positive numbers such that
a + b + c = 1. The maximum value of ab + bc + ca is
(a) 1/3 (b) 1/4 (c) 1/2 (d) 2/3
55. The roots of (x – 1)(x – 3) + k(x – 2)(x – 4) = 0, k > 0 are
(a) real
(b) real and equal
(c) imaginary (d) one real and other
imaginary
56. The largest integral value of m for which the quadratic
expression y = x 2 – (2m + 6)x + 4m + 12 is always positive,
for every x in R, is
(a) –1 (b) –2 (c) 0 (d) 2
57. Let r 1
, r 2
and r 3
be the solutions of the equation x 3 – 2x 2
+ 4x + 5074 = 0, the value of (r 1
+ 2)(r 2
+ 2)(r 3
+ 2) is
(a) 5050 (b) 5066 (c) –5050 (d) –5066
58. The equation whose roots are sec 2 a and cosec 2 a can
be
(a) 2x 2 – x – 1 = 0 (b) x 2 – 3x + 3 = 0
(c) x 2 – 9x + 9 = 0 (d) None
59. Let a, b, c be the three roots of the equation
x 3 + x 2 – 333x – 1102 = 0, the value of a 3 + b 3 + c 3 is
(a) 2006 (b) 2005 (c) 2003 (d) 2002
60. The absolute term in the quadratic expression
Â
n
Ê 1 ˆÊ 1ˆ
Áx
- Áx
- ˜
k = 1Ë
k + 1
˜Ë ¯ k
¯
, is
(a) 1 (b) –1 (c) 0 (d) 1/2
61. The number of values of the parameter a Π[0, 2p], for
which the quadratic function
2 1
sin a◊ x + 2x◊ cos a + (cos a + sin a)
2
is the square of a linear function, is
(a) 2 (b) 3 (c) 4 (d) 1
62. The set of values of a for which the inequality
(x Р3a)(x Рa Р3) < 0 is satisfied for all x Π[1, 3] is
(a) (1/3, 3) (b) (0, 1/3)
(c) (–2, 0) (d) (–2, 3)
63. If a, b and g are the roots of the equation, x 3 – x – 1 = 0,
the value of 1 + a 1 + b 1 + g
+ +
1-a 1-b 1-g
is
(a) 0 (b) –1 (c) –7 (d) 1
64. For every x in R, the polynomial x 8 – x 5 + x 2 – x + 1 is
(a) positive
(b) never positive
(c) positive and negative (d) negative
65. If the roots of the cubic equation, x 3 + ax 2 + bx + c =
0 are three consecutive positive integers, the value of
2
a
is
b + 1
(a) 3 (b) 2 (c) 1 (d) 1/3
66. If both roots of (3a + 1)x 2 – (2a + 3b)x + 3 = 0 are infinite
, then
(a) a = , b = 0 (b) a = 0, b =
(c) a = 1/3, b = 2/9 (d) a = , b =
67. If tan a, tan b and tan g are the roots of the equation
3 2
x -( a + 1) x + ( b -a) x - b= 0,( b -aπ 1) ,
where a + b + g lies between 0 and p, the value of
(a + b + g) is
(a) p/4 (b) p/2 (c) 3p/4 (d) None
68. Three roots of the equation x 4 – px 3 + qx 2 – rx + s = 0
are tan A, tan B and tan C, where A, B, C are the angles
of a triangle. The fourth root of the biquadratic is
(a)
q - r
p - r
(b)
1 - q + s
1 + q - s
(c)
p + r
p + r
(d)
1 - q + s
1 + q - s
69. The number of real roots of (x + 3) 4 + (x + 5) 4 = 16 is
(a) 0 (b) 2 (c) 4 (d) None
70. If a, b be the roots of the equation x 2 – ax + b = 0 and
A n
= a n + b n , A n+1
– aA n
+ bA n–1
is equal to
(a) –a (b) b (c) 0 (d) a – b
71. If a, b, g are such that a + b + g = 2, a 2 + b 2 + g 2 = 6,
a 3 + b 3 + g 3 = 8, the value of a 4 + b 4 + g 4 is
(a) 5 (b) 18 (c) 12 (d) 36
72. The number of irrational solutions of the equation
2 2 2 2
x + x + 11 + x - x + 11 = 4 is
(a) 0 (b) 2 (c) 4 (d) 11
2x 1/ x Ê65ˆ
1/ x
73. The number of solutions of 10 + 25 = Á ¥ 50
Ë
˜
8 ¯
is
(a) 0 (b) 2 (c) 4 (d) infinite
74. The equation (2.4) x = (2.6) x – 1 has
(a) no solution (b) exactly one sol
(c) atleast 2 solution (d) infinite solution
75. If n ΠN, the number of real roots of
2 2n
x x
1 + x + + ... + = 0 is
2! (2 n)!
(a) n (b) 2 (c) 0 (d) none