1.Algebra Booster

Quadratic Equations and Expressions 2.15
4. The set of values of p for which the roots of the equation
3x 2 + 2x + p(p – 1) = 0 are of opposite sign is:
(a) (– , 0) (b) (0, 1)
(c) 1, (d) (0, )
5. The value of p, for which both the roots of the equation
4x 2 – 20px + (25p 2 + 15p – 66) = 0 are less than 2, lies in
Ê4 ˆ
(a) Á ,2 ˜
(b) (2, )
Ë5
¯
Ê -4ˆ
(c) Á-1, ˜
(d) (– , –1)
Ë 5 ¯
6. The number of values of k for which the equation x 2 –3x
+ k = 0 has two distinct roots lying in the interval (0, 1)
is:
(a) three
(b) two
(c) infinitely many
(d) no value of k satisfies the requirement
7. If a, b be the roots of (x – a)(x – b) + c = 0, c π 0, the
roots of (ab – c)x 2 + (a + b)x + 1 = 0 are
1 1 1 1 1 1 1 1
(a) , (b) - , (c) , - (d) ,
a b a b a b
- a
- b
8. The quadratic equation with real co-efficients one of
whose complex roots has the real part 12 and modulus
13 is
(a) x 2 – 12x + 13 = 0 (b) x 2 – 24x + 13 = 0
(c) x 2 – 24x + 169 = 0 (d) x 2 – 24x – 169 = 0
9. The root of the equation
x 6 6 6 ..... is
(a) 3 (b) 2 (c) –3 (d) 2
10. If x 2 + mx + 1 = 0 and (a – b)x 2 + (b – c)x + (c – a) =
0 have both roots common, which of the following is
true?
(a) m = –1 (b) m = 2
(c) m = –2
(d) a, b, c are in AP
11. If the quadratic equation
p(q – r)x 2 + q(r – p)x + r(p – q) = 0
has equal roots, 2 is equal to
q
1 1
(a) + (b) p + r
p r
1
(c) r
p + (d) 1
p +
r
12. If x, y, z are real and unequal, the expression x 2 + 4y 2 +
9z 2 – 6yz – 3zx – 2xy is always
(a) non-negative (b) non-positive
(c) zero
(d) none of these
13. Both the roots of the equation:
(x – 1)(x – 2) + (x – 2)(x – 3) + (x – 3)(x – 1) = 0 are
(a) complex numbers (b) always real
(c) integers
(d) rational numbers
14. The number of real roots of the equation
(x – 3) 2 + (x – 5) 2 + (x – 7) 2 = 0 is
(a) 2 (b) 0
(c) 4
(d) infinitely many
15. The equation x 2 + 5|x| + 4 = 0 has
(a) 4 real roots
(b) 2 real roots
(c) no real root
(d) 6 real roots
16. The roots of x 2 – 8|x| + 12 = 0
(a) do not form a progression
(b) form an AP with 0 sum
(c) form an AP with non-zero sum
(d) form a GP
17. If a, b, g, d be the roots of 16x 4 + 4x 2 + 1 = 0, the value
of a 4 + b 4 + g 4 + d 4 is
(a) –1/8 (b) 0
(c) 1/8 (d) 1
18. If a, b, g, d be the roots of x 4 + x 2 + 1 = 0, the value of
(a + b)(a + g)(a + d)(b + g)(b + d)(g + d)
(a) –1 (b) 0
(c) 1
(d) none of these
19. The value of a for which the equation
(a 2 + 4a + 3)x 2 + (a 2 – a – 2)x + (a + 1) = 0 in x has more
than two roots
(a) 1 (b) –1
(c) 2
(d) no value of a
20. The number of values of the triplet (a, b, c) for which
a cos 2x + b sin 2 x + c = 0 is satisfied by all real x is
(a) 2 (b) 4
(c) 6
(d) infinite
21. If the roots of ax 2 + bx + c = 0 be the Reciprocals of
those of lx 2 + mx + n = 0, where all the co-efficients are
non-zero real numbers, then
(a)
a b c
= = (b)
l m n
a b c
= =
n m l
(c) al = bm = cn (d) none of these
22. If the equations x 2 + ax + 12 = 0; x 2 + bx + 15 = 0;
x 2 +(a + b)x + 36 = 0 have a common positive root, the
values of a and b, respectively, are
(a) –7, –8 (b) –8, –7
(c) –5, –3
(d) no value of a exist
23. If x is any real number, the value of the expression
x
f()
x =
x
2
2
- 5x+
4
+ 5x+
4
(a) lies between –9 and –1/9
(b) do not lies between –9 and –1/9
(c) may be any real number
(d) is always positive