KOTA ACADEMY --- KARAD
MATHEMATICS - TEST
TOPIC : QUADRATIC EQUATION
Q.1
Q.2
If 4x 3x 1/2 = 3x + 1/2 22x 1, then value of x is equal to(1) 5/2
(2) 2
(3) 3/2
If the roots of the equation ax2 + bx + c = 0 are of the form
is(1) 2b2 ac
(2) b2 2c
(4) 1
α
α +1
and
, then the value of (a + b + c)2
α −1
α
(3) b2 4ac
Q.3
The quadratic equation 12 cosec2 x + 5 cosec x 3 = 0 has(1) Infinitely many roots
(2) Exactly two roots
(3) Exactly four roots
(4) No root
Q.4
If ax3 + bx + c is divisible by x2 + bx + c, then a is a root of(1) cx2 bx 1 = 0
(2) ax2 bx 1 = 0
(3) bx2 ax 1 = 0
Q.5
(4) 4b2 2ac
(4) cx2 ax + b = 0
2
The equation x ( 3 /4 ) (log 2 x ) + log 2 x − 5 /4 = 2
has(1) Exactly two real roots
(3) One irrational root (4) None of these
(2) No real root
Q.6
If 0 < a < b < c < d, then the quadratic equation
ax2 + {1 a (b + c)) x + abc d = 0
has(1) Real and distinct roots out of which one lies between c and d.
(2) Real and distinct roots out of which one lies between a and b.
(3) Real and distinct roots out of which one lies between b and c.
(4) None real roots
Q.7
If α , β are the roots of the equation ax2 + bx + c = 0, then the equation ax2 bx (x 1) + c(x 1)2 = 0 has
roots(1)
Q.8
β
α
,
−
β
1
1−α
1−α 1−β
,
β
α
(3)
β
α
,
+
β
1
1+α
(4) None of these
Let the roots of the equation x3 + 3 l x2 + mx + n = 0 be in A.P., then 2 l 3 + n is(1) m l
Q.9
(2)
(2) n l
(3) m2
(4) None of these
If a, b, c, d ∈ R then the equation (x2 + ax 3b) (x2 cx + b) (x2 dx + 2b) = 0 has(1) 6 real roots
(2) 3 real roots
(3) 4 real roots
(4) At least 2 real roots
x
x2
+ |x| =
is equal toQ.10 Total number of real solutions of
x −1
|x − 1|
(1) 4
(2) 2
(3) 8
(4) x ∈ (1, ∞ ) ∪ {0}
KOTA ACADEMY --- KARAD
MATHEMATICS - TEST
Q.11 If the roots of x2 ax + b = 0 differ by unity then(1) b2 = 1 + 4a
(2) a2 = 1 + 4b
(3) b2 + 4a = 1
(4) a2 + 4b = 1
Q.12 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(1) (1, 16)
(2) (2, 32)
(3) (2, 16)
(4) (1, 32)
Q.13 If x2 x + a 3 < 0 for atleast one negative value of x, then complete set of values of 'a' is(1) ( ∞ , 4)
(2) ( ∞ , 2)
(3) ( ∞ , 3)
(4) ( ∞ , 1)
Q.14 Both roots of (a2 1) x2 + 2ax + 1 = 0 belong to the interval (0, 1) then exhaustive set of values of 'a' is(1) ( ∞ , 2) ∪ (0, ∞ )
(2) ( ∞ , 2)
(3) ( ∞ , 0) ∪
F 0, −1 + 5 I
GH 2 JK
(4)
F −2, −1 + 5 I
GH
2 JK
{0}
Q.15 If 22x + 2 a . 2x + 2 + 5 4a ≥ 0 has at least one real solution then complete set of values of 'a' is(1) ( ∞ , 1]
(2) ( 3, 1]
Q.16 The set of real values of x for which
(1) 1 < x < 2,
(3)
5
7
<x<
4
2
2(x − 3 )
>
2
2 x − 7x + 5
(4)
|x|log4
(4)
FG − 8 , − 1OP
H 7 Q
1
isx−2
(2) (1, 2) ∪
FG1, 5 IJ FG2, 7 IJ
H 2K H 4K
Q.17 The equation 2|x 2 −12| = e
(3) [1, ∞ )
FG 7 , 5 IJ
H 3 2K
LM1, 2 [∪] 7 , 5 LM
3 2N
N
has-
(1) No real solution
(2) Only two real solutions whose sum is zero
(3) Only two real solutions whose sum is not zero
(4) Four real solutions whose sum is zero.
Q.18 The set of all values of θ in [0, 2 π ] for which the equation (1 + sin θ ) x2 (2 cos θ ) x + sin θ = 0 has real
roots is(1)
(3)
FG 0, π IJ
H 3K
LM0, π OP
N 6Q
(2)
∪
LM 5π , 2πOP
N6 Q
(4)
LM π , πOP ∪ LM11π , 2πOP
N6 Q N 6 Q
LM π , 3π OP
N3 2 Q
KOTA ACADEMY --- KARAD
MATHEMATICS - TEST
Q.19 The value of m for which one of the roots of the equation (2m + 1) x2 mx + (m 2) = 0 is greater than unity
and the other smaller than unity lies in the interval-
1
(1) ∞ < m <
2
(2)
1
1
<m<
2
2
Q.20 The equation sin x = x2 + x + 1 has(1) One real solution
(3) More than one real solution
(3)
1
<m< ∞
2
(4) None of these
(2) No real solution
(4) Two positive solutions
Q.21 If sin α , sin β cos α are in G.P., then roots of the equation x2 + 2x cot β +1 = 0 are always(1) Equal
(2) Real
(3) Imaginary
(4) Positive
Q.22 If log|2 + cos 2x | (x2 3x + 3) ≤ 0 then x lies in the internal(1) ( ∞ , 1] ∪ [2, ∞ ]
(2) [1, 2]
(3) [1.5, 2]
(4) [1, 1.5]
Q.23 Curve y = ax2 + bx + c cuts x-axis at least at one point and curve y = cx2 bx + a cut y = 0 line almost at
one point if a, b, c > 0 then equation ax2 2bx + 4c = 0 has(1) Two real roots
(2) Equal roots
(3) No real roots
(4) None of these
Q.24 The value of a, for which exactly one root of the equation ea x2 e2a x + ea 1 = 0 lies between 1 and 2, are
given by-
F 5 − 17 I
F
I
GH 4 JK < a < l n GH 5 + 4 17 JK
F 5I
F 10 I
l n GH 4 JK < a < l n GH 3 JK
(1) l n
(2) 0 < a < 100
(3)
(4) None of these
Q.25 If α , β be the roots of x2 a (x 1) b = 0, then the value of
(1)
4
(a + b )
(2)
1
(a + b )
(3) 0
1
2
α − aα
(4)
+
1
2
β − aβ
+
2
isa−b
2
(a + b )
ANSWER KEY
Name : ..........................................................................................................
1
1
2
3
4
9
1
2
3
4
17
2
10
18
3
11
19
4
12
20
5
13
21
6
14
22
7
15
23
9
16
24
25
Roll No. : ..................................
1
2
3
4
MATHEMATICS - TEST
ANSWERS (QUADRATIC EQUATION)
Que.
Ans.
Que.
Ans.
1
3
16
2
2
3
17
4
3
4
18
3
4
1
19
2
5
3
20
2
6
1
21
2
7
3
22
4
8
1
23
2
9
4
24
1
10
4
25
3
11
2
12
2
13
3
14
1
15
1
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