QUADRATIC EQUATIONS

DAY-1
2 2
1. If the difference of the roots of x – px + q = 0 is unity, then prove that p – 4q = 1 and
2 2 2
p + 4q = (1 + 2q) .

2 2 b2  ac
2. If ,  are roots of ax + 2bx + c = 0 and  + ,  +  are roots of Ax + 2Bx + C = 0, then =
B2  AC
2 2
a  A a
(A)   (B)   C)   (D)none of these
 A a  A
2 2
3. If ,  are roots of ax + bx + c = 0 and ,  are the roots of px + qx + r = 0 and D1, D2 be the
respective discriminants of these equations. If , ,  and  are in A.P, then D1 : D2 = ___
2 2
(A) a :p (B) a:p (C) p:a (D) none of these

4. Match the following

Column– I Column– II
(P) Average of two positive distinct roots > its product (1) Two real roots
4 4
(Q) (x + 3) + (x + 5) = 16 (2) Sum of real roots =–1
2 2
(R) (x + x – 2)(x + x – 3) = 12 (3) True always
2
(S) x –1>0 (4) Not always true
P Q R S
(A) 1 2 4 3
(B) 2 3 1 4
(C) 3 1 2 4
(D) 2 1 4 3

Comprehension -1
2
Let ,  are roots of x – p(x + 1) – c = 0 (c not equal to 1)
5. The value of (1 + )(1 + ) is
(A) 1 – c (B) 1 + c (C) c (D) –c

 2  2  1 2  2  1
6. The value of  is
 2  2  c 2  2  c
(A) 1 (B) 2 (C) 3 (D) none of these

7. If p, q is roots of (x – )(x – ) = c, then roots of (x – p)(x – q) + c = 0 will be

(A) ,  (B) –,  (C) , – (D) –, –

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DAY-2

1. If p, q, r, s are real and pr > 4(q + s), then show that at least one of the equations
2 2
x + px + q = 0 and x + rx + s = 0 has real roots.

a2 b2 c2
2. Equation   =m - n2 x (a, b, c, m, n  R) has necessarily
x  x  x  
(A) all the roots real (B) all the roots imaginary
(C) two real and two imaginary roots (D) two rational and two irrational roots.
2
3. The equation x + nx + m = 0, n, m  I, can not have
(A) integral roots (B) non-integral rational roots
(B) irrational roots (D) complex roots

n
4. The number of real roots of the quadratic equation  x  k 
k 1
2
 0 (n > 1) is

(A) 1 (B) 2 (C) n (D) 0

2 2
5. For a  0, the roots of the equation x – 2a|x – a| – 3a = 0.
(A) a  6 1 (B) a  
6 1   
(C) a 1  2 ,a 1  6  
(D) a 1  2 
DAY-3

1. If a, b, and c are odd integers, then the roots of ax 2  bx  c  0 , if real, cannot be

(A) integers (B) rational numbers (C) irrational (D) equal

9
log2 x 2  log2 x  5
2. The equation x 2  2 2 has
(A) at least one real root (B) exactly three real roots
(C) exactly one irrational root (D) no complex root
2
3. Let ,  are roots of px + qx + r = 0 are real and opposite in sign then roots of the equation
2 2
(x – ) + (x – ) = 0 are
(A) irrational conjugate of each other (B) opposite in sign
(C) equal (D) none of these

 
4. If , ,  be disting real numbers lying  0,  then the number of real roots of
 2
1 1 1
   0.
x  sin  x  sin  x  sin 

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5. The set of value(s) of k  R for which

Column I Column II
2
(A) Kx – (k + 1)x + 2k – 1 = 0 has no real roots p. {–2, 1}
2 2
(B) x – 2(4k – 1)x + 15k – 2k – 7 > 0 for each x q.  1
 ,  7   (1,  )
 
2 2 2
(C) Sum of the roots of x + (2 – k – k )x – k = 0 is zero r. {–4}
2 2
(D) The roots of x + (2k – 1)x + k + 2 = 0 are in the ratio 1 : 2 s. (2, 4)

6. Match the following

Column– I Column– II
(P) f(x) = x(x + 1), f(x + 4) = 4(x) + 4 (1) Only one real solution
(Q) 2x = |x| + 1 (2) Only two real solutions
2
(R) x |x| = 8 (3) x  [7, 9]
(S) x– x 1= 5 (4) No solution
P Q R S
(A) 1 2 3 4
(B) 2 3 1 5
(C) 3 7 6 4
(D) 4 1 2 3

DAY-4

x 2  2x  a
1. If x  R, then can take all real values, then find the interval in which a belongs.
x 2  4x  3a

 a  x b  x 
2. If x < –c, then find the maximum value of , (c < a, c < b)
cx

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2 2 2 2
3. If a + b +c + d = 1, then the maximum value of ab + bc + cd +da is
(A) zero (B) One (C) Two (D) None of these

4. If x, y, z are real variables such that x + y + z = 6, xy + yz + zx = 7 then range of x is

 6  2 15 6  2 15 
(A)  ,  (B) [3- 15,3  15
 3 3 
(C)   15, 15  (D) None of these
 

5. The set of value (s) of k  R for which

Column– I Column– II
(x – 1)(x – 3) + k(x – 2)(x – 4) = 0
(P) (1) (–5, –1)
(k  R), has real roots for k
x 1
Range of the function 2
(Q) x k 1 (2) 
Does not contain any value in the interval [–1,
1] for k 
   
The equation  x   0,   ,
(R)   2  (3) (–, )
sec x + cosec x = k has real roots if k 

2
The equation x + 2(k – 1)x + k + 5 = 0 has 2 2, 
(S) (4) 
roots positive and distinct, if k 
P Q R S
(A) 1 2 4 3
(B) 2 3 1 4
(C) 3 2 4 1
(D) 2 1 4 3

DAY-5
2
1. Find values of k for which x + kx + 2 < 0 for atleast one negative x.

mx 2  3x  4
2. If the inequality < 5 is satisfied for all x  R, then
x 2  2x  2
71
(A) 1 < m < 5 (B) –1 < m < 5 (C) 1 < m < 6 (D) m <
24
2
3. If c > 0 and 4a + c < 2b, then ax – bx + c = 0 has a root in the interval
(A) (0, 2) (B) (2, 4) (C) (0, 1) (D) (–2, 0)
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2
4. If c + 1 < b, then roots of x – bx + c = 0
(A) are real and equal
(B) are real and distinct
(C) are Imaginary
(D) nothing can be said about the nature of roots as given condition is not sufficient.
2
5. Consider the graph of f(x) = ax +bx +c in the adjacent figure.
f(x
We can conclude that
(A) c > 0 (B) a < 0
(C) b < 0 (D) a + b+ c < 0

x
2 2
6. If ax + bx + c = 0 is a quadratic equation and if 4ac > b and a + c > b for real numbers a, b and c,
then which of the following is true?
(A) a > 0 (B) c > 0 (C) a + b + c > 0 (D) 4a + c > 2b
2
7. For which of the following graphs of the quadratic expression y = ax + bx + c, the product of a, b, c
is negative.
y y

(A) (B)
x x

y
y

(C) (D) x
x

2
8. The values of k for which the equation x + 2(k – 1)x + k + 5 = 0 has atleast one positive root, are
(A) [4, ) (B) (–, –1]  [4, ) (C) [–4, –1] (D) (–, –1]
2 bc
9. If the equation cx +bx – 2a =0 has no real roots and a < then
2
c b c  2b
(A) a c < 0 (B) a < 0 (C) a (D) a
2 8

DAY-6
3 2 3 2
1. Find the common roots of the equations x – 2x – x + 2 = 0 and x + 6x + 11x + 6 = 0.

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2. If a, b, c are in G.P., then the equations ax 2  2bx  c  0 and dx 2  2ex  f  0 have a common root
if d/a, e/b, f/c are in
(A) G.P (B) A.P (C) H.P (D) None of these
2
3. The values of the parameter a for which the quadratic equations (1 – 2a)x – 6ax – 1 = 0 and
2
ax – x + 1 = 0 have at least one root in common are
1 1 2 2 1 2
(A) 0, (B) , (C) (D) 0, ,
2 2 9 9 2 9
2 2
4. If ax + 2bx + c = 0, a1x + 2b1x + c1 = 0 have a common root, then the roots of the equation
2 2 2
(b – ac)x + (2bb1 – ac1 – a1c)x + (b1 – a1c1) = 0 are
(A) different (B) equal (C) zero (D) none of these
2 2
5. If ax + 2bx + c = 0 and px + 2qx + r = 0 have one and only one root in common and a, b, c being
rational, then
2 2
(A) b – ac and q – pr are both perfect squares
2 2
(B) b – ac is a perfect square and q – pr is not a perfect square
2 2
(C) q – pr is a perfect square and b – ac is not a perfect square
2 2
(D) both b – ac and q – pr are not perfect square
2 2
6. If the equations ax +bx +1 =0 and ax +x +b =0 have exactly one common root then
(A) a – b =2 (B) a – b = 1 (C) a + b = 2 (D) a + b = –1
2 2
7. If x + px + 1 = 0 and (a – b)x + (b – c)x + (c – a) = 0 have both roots common, then which of the
following is true ?
(A) p = –2 (B) b, a, c are in A.P (C) 2a – 3b + c = 1 (D) 2a – 3b + c = 0
2 2
8. If ax + bx + c = 0 and cx + bx + a = 0 (a, b, c  R) have a common non-real root, then
(A) –2|a| < b < 2 |a| (B) – 2|c| < |b| < 2|c| (C) a =  c (D) a = c

Comprehension-1
2 2
Consider two equations: ax + bx + c = 0 and ax + bx + c = 0.
2 2
Let  be a common root then a + b + c = 0 and a + b + c = 0
2  1
We have  
bc  bc ca  ca ab  ab
bc  bc ca  ca
We get  = and  
ca  ca ab  ab
 bc  bc   ca  ca 
Eliminating , 
 ca  ca   ab  ab 
2
2 a b b c c a
 (bc – bc) (ab – ab) = (ca – ca) or  
a b b c c a
This is the required condition for all root of two quadratic equations to be common.
a b c
If ab – ab = 0 then   .
a b c
2 2
9. Find the value of k, so that the equations x – x – 12 = 0 and kx + 10x + 3 = 0 may have one root in
common.
(A) 3 (B) 10 (C) 1 (D) 0
2 2
10. If equations ax + bx + c = 0 and x + 2x + 3 = 0 have a common root, then a : b : c =
(A) 2 : 1 : 3 (B) 3 : 2 : 1 (C) 1 : 2 : 3 (D) none of these

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2 2
11. If the equation x + cx + ab = 0 and x + bx + ca = 0 have a common root, then a + b + c is equal to
(A) 1 (B) 0 (C) –1 (D) none of these

DAY-7
2
1. Let x – (m – 3)x + m = 0, (m  R) be a quadratic equation. Find the value of m for which
(i) one root is smaller than 2, the other root is greater than 2

(ii) both the roots are greater than 2

(iii) both the roots are smaller than 2

(iv) exactly one root lies in the interval (1, 2)

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(v) both the roots lie in the interval (1, 2)

(vi) atleast one root lies in the interval (1, 2)

(vii) one root is greater than 2, the other root is smaller than 1

(viii) atleast one root is greater than 2

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2 2
2. Find the values of a for which the inequality x + ax + a + 6a < 0 is satisfied for all x  (1, 2).

3. Prove that for any value of a the inequality

2 2
(a + 4)x + (a + 2)x – 4 < 2 is true for at least one negative x.

x x
4. Find the values of a for which the inequality is satisfied 25 + (a + 2)5 – (a + 3) < 0 for atleast one x.

DAY- 8
x x
1. If all the real solutions of the equation 4 – (a – 3)2 + (a – 4) = 0 are non positive, then
(A) 4 < a  5 (B) 0 < a < 4 (C) a > 4 (D) a < 3
2 2 2
2. The value(s) of a for which one of the roots of (a + a + 1)x + (a – 1)x + a is greater than 3 and the
other less than 3 is
(A) R (B)  (C) (0, ) (D) (–, 1)

3. Which of the following is correct for the quadratic equation x 2  2  a  1 x  a  5  0

(A) The equation has positive roots, if a   5, 1

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(B) the equation has roots of opposite sign, if a   , 5 

(C) the equation has negative roots, if a  [4, ) (D) None of these

4. If roots of the equation a x 2  bx  c  0(a  0) are ,  such that   2 and   2 then

b c b c
(A) b2  4ac  0 (B)c > 0 (C)4-2  0 (D) 4  2  0
a a a a

2
5. If the equation ax + bx + c = 0 (a < 0) has two roots  and  such that  < -3 and  > 3, then
(A) 9a + 3|b| + c > 0 (B) c > 0 (C) 4a + 2|b| + c > 0 (D) none of these
2
6. The equation x – 4x + a sin  = 0 has real roots
 
(A) for all values of a (B) for all values of ‘a’ provided  
4 4
(C) for all values of a  4 provided     2 (D) for all a provided |a|  4
2
7. Consider the quadratic equation x + 2(a + 1)x + 9a – 5 = 0
Column– I Column– II
(P) a>7 (1) Both the roots are negative
(Q) a<0 (2) Roots are of opposite sign
(R) 2<a<5 (3) Roots are imaginary
(4) At least one root is negative
P Q R
(A) 1 2 3
(B) 2 3 1
(C) 3 4 3
(D) 2 1 4
2
8. For the Q.E x – (m – 3)x + m = 0. The value of ‘m’ match the following
Column– I Column– II
The smallest value of  for which
2
(P) ( – 2)x + 4(2x + 1) +  > 0 (1) 4
(where   I) for all real value of x is
The number of real solutions of the equation
(Q)
    (2) 2
x x/2
2x / 2  2  1  5  2 2 is
Let [.] denotes greatest integer in x. Then in [0, 3] the
(R) 2 (3) 5
number of solutions of the equation x – 3x + [x] = 0 is
 x2  x 
The number of solutions of 3 + 3 = 2cos2 
x –x
(S)  4  (4) 1
 
is
2 2
If x – ax – 21 = 0 and x – 3ax + 35 = 0 have one
(T) (5) 6
common root, then value of a such that a > 0 equals
Number of equations of the form
2
(U) ax + bx + c = 0 for all a, b, c  {1, 2, 3} having real (6) 0
roots
(7) 3
P Q R S T U
(A) 1 2 4 3 5 6
(B) 2 3 1 4 6 5
(C) 3 4 2 4 1 1
(D) 2 1 4 3 7 6

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Comprehension-3
Consider the quadratic equation ax
2
 bx  c  0 , a, b, c  N , which has two distinct real roots belonging to
the interval (1, 2).

9. The least value of a is

(A) 4 (B) 6 (C) 7 (D) 5

10. The least value of b is

(A) 10 (B) 11 (C) 13 (D) 15

11. The least value of c is

(A) 4 (B) 6 (C) 7 (D) 5

Comprehension-4
x x
Consider the inequality 9 – a3 – a + 3  0, where ‘a’ is a real parameter.

12. The given inequality has atleast one negative solution for a 
(A) (–, 2) (B) (3, ) (C) (–2, ) (D) (2, 3)

13. The given inequality has atleast one positive solution for a 
(A) (–, –2) (B) [3, ) (C) (2, ) (D) [–2, )

14. The given inequality has atleast one real solution for a 
(A) (–, 3) (B) [2, ) (C) (3, ) (D) [–2, )

DAY-11
4 3
1. The equation x +1=x has
(A) no real root (B) one real root (C) two real roots (D)all four real roots
3 2 2 2 2
2. If , ,  are the roots of the equation x + p0x + p1x + p2 = 0, then (1 –  )(1 –  ) (1 –  ) is equal to
(A) (1  p1 )  ( p0  p2 ) (B) (1  p1 )  ( p0  p2 )
2 2 2 2

(C) (1  p1 )  ( p0  p2 )
2 2
(D) None of these

3 3 3 3 3
3. The number of real solution of the equation (x + 4) + (x + 3) + (x + 2) + (x + 1) + (x – 5) + 180 = 0
is
(A) none (B) one (C) two (D)three

4. If b2  2ac then number of real roots of ax 3  bx 2  cx  d  0 is

(A) 3 (B) 2 (C) 1 (D) 0

x     
3
3
5.  3x  4  3 x3  3x  4  4  2 4x 3  3x  2 has
(A) All its solutions are real but not all positive
(B) Only 3 of its solutions are real
(C) 3 are positive 2 negative all other are imaginary
(D) Only one real solution

x x x
6. If a, b, c, d are unequal positive numbers, then the roots of equation   xd0
x a x b x c
are necessarily
(A) all real (B) all imaginary (C) two real and two imaginary
(D) at least two real
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   
2
7. If the roots of the equation  a  1 x 2  x  1   a  1 x 4  x 2  1  0 are real and distinct, then a
lies in the interval
(A) (–2, 2) (B) (–, –2)  (2, ) (C) (2, ) (D) None of these

If p, q, r  R and are distinct, the equation  x  p    x  q   x  r   0 has

5 5 5
8.
(A) four imaginary and one real root (B) two imaginary and three real roots
(C) all the roots real (D) None of these

9. If equation x5  10x 2  x  5  0 has one root as , then

(A) [] = –3 (where [.] denotes the greatest integer function)
(B) number of roots between –2 and –1 is three
(C) number of real roots is 3
(D) equation has at least one positive root
3 2
10. If a, b, c, d  R and f(x) = ax + bx – cx + d has local extrema at two points of opposite signs and
2
ab > 0, then roots of equation ax + bx + c = 0
(A) are necessarily negative (B) have necessarily negative real parts
(C) have necessarily positive real parts (D) are necessarily positive

11. If a, b, c  R and 3b2  8ac  0 then the equation ax 4  bx 3  cx 2  5x  7 =0 has

(A) all real roots (B) all imaginary roots
(C) exactly two real and two imaginary roots (D) None of these
3
12. The set of values of ‘a’ for which the equation x – 3x + a = 0 has three distinct real roots, is
(A) (–, ) (B) (–2, 2)
(C) (–1, 1) (D) none of these

13. The product of two roots of the equation x 4  18x 3  kx 2  200x  1984 is –32 then k =
(A) 16 (B) 56 (C) 76 (D) 86

14. Consider the polynomial equation 1 + 2x + 3x 2  4x 3  0 , If s be the sum of distinct real roots then s
lies in the interval
 1   3  3 1  1
(A)   ,0  (B)  11,   (C)   4 , 2  (D)  0, 
 4   4    4

15. x 6  12x 5  ax 4  bx 3  cx 2  dx  64  0 has all positive roots then

(A) a = 60 (B) b = –160 (C) c = 240 (D) d = –192

MISCELLANEOUS

  1
1. If the roots of the equation, ax 2  bx  c  0 , are the form and then the value of
 1 
 a  b  c 2 is
(A) 2b2  ac (B) b2  2ac (C) b2  4ac (D) 4b2  2ac

 3c 
2. It the equation ax 2  2bx  3c  0 has non real roots and     a  b  . Then c is always
 4 
(A) < 0 (B) > 0 (C)  0 (D) None of these

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3. If ,  are the roots of x 2  3x  a  0 , a  R and  < 1 < , then

 9  9
(A) a  (–, 2) (B) a   ,  (C)  2,  (D) None of these
 4  4

3
4. x – [x] = 3 then x =
3 3
(A) 2 (B) 3 (C) 4 (D) 6

5 If s  Z and the equation (x – a)(x – 10) + 1 = 0 has integral roots, then the values of a are
(A) 10, 8 (B) 12, 10 (C) 12, 8 (D) None of these

 1

If the equation (3x)  27  3  15  x  4  0 has equal roots , then p =
 p 2
6
 
 
(A) 0 (B) 2 (C) –1/2 (D) None of these

7 The integer k for which the inequality x 2  24k  1x  15k 2  2k  7  0 is valid for any x, is
(A) 2 (B) 3 (C) 4 (D) none of these

8 The roots of the equation

a  b  x2 15

 a b x2 15
 2a , where a2 – b = 1, are
(A)  2, 3 (B)  4 , 14 (C)  3,  5 (D)  6 , 20

If x, y and z are real numbers, then x  4 y  9 z 2  6 yz  3zx  2 xy is always

2 2
9.
(a) Positive (b) non-positive (c) zero (d) non-negative

10.  2
 2

If the equations k 6 x  3  rx  2 x  1  0 and 6k 2 x  1  px  4 x  2  0 have both roots
2
 2

common, then 2r - p = 0
(A) 2 (B) 1 (C) 0 (D) k

11. The equation x  2 x  3  0 and ax  bx  c  0 where

2 2
a1b1c  R have a common root then
a : b : c is
(A) 3 : 2 : 1 (B) 1 : 3 : 2 (C) 3 : 1 : 2 (D) 1 : 2 : 3

12. The number of positive integral roots of x 4  x 3  4 x 2  x  1  0 are

(A) 0 (B) 1 (C) 2 (D) 4

 
2
The nature of the roots of the equation x  6 x  9  x 3  4 x 2  9 x is
2
13.
(A) four real roots (B) no real roots
(C) two real and two complex roots (D) two multiple real roots

14. The value of ‘a’ for which the equation a 2

  
 4a  3 x 2  a2  a  2 x   a  1 a  0 has more
than two roots is
(A) 1 (B) 2 (C) -2 (D) –1

15. Let p(x) = 0 be a polynomial equation of least possible degree, with rational coefficients, having
3
7  3 49 as one of its roots. Then the product of all the roots of p(x) = 0 is

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(A) 7 (B) 49 (C) 56 (D) 63

16. If c > 0 and 4a + c < 2b, then ax 2  bx  c  0 has a root in the interval
(A) (0, 2) (B) (2, 4) (C) (0, 1) (D) (–2, 0)

17. If x3  ax  1  0 and x 4  ax 2  1  0 have common root then the exhaustive set of values of a is
(A) (–, –2) (B) [–2, ) (C) {–2} (D) [–2,2]
2
18. Let x + x + 1 is divisible by 3. If x is divided by 3, the remainder will be
(A) 2 (B) 1 (C) 0 (D) none of these
2
19. The least value of |a| for which tan and cot are the roots of the equation x + ax + b = 0 is
(A) 2 (B) 1 (C) ½ (D) 0

20. If a, b, c are three distinct positive real numbers then the number of real roots of
2
ax + 2b |x| – c = 0 is
(A) 4 (B) 2 (C) 0 (D) none of these
2 2
21. The equation formed by increasing each root of ax + bx + c = 0 by 1 is 2x + 8x + 2 = 0, then
(A) a + b = 0 (B) b + c = 0 (C) b = c (D) a = b

x 4  2x 2  2
22. Minimum value of the expression is
x2  1
(A) 1 (B) 2 (C) 3 (D) 4
6/5 3/4
23. The product of real roots of the equation |x| – 26 |x| – 27 = 0 is
(A) 310 (B) 312 (C) 312 / 5 (D) 321/ 5

24. If the equation ax 2  bx  c  0 is not altered when each of the coefficient is increased by the same
quantity then x 2  x  1 =
(A) 1 (B) 0 (C) 3 (D) 2
2
25. If a, b  R, a  0 and the quadratic equation ax –bx +1 = 0 has imaginary roots, then
a + b + 1 is
(A) positive (B) negative (C) zero (D) depends on the sign of b
2
26. The least integral value of k such that (k – 2)x + 8x +k + 4 is positive for all real values of x is
(A) 1 (B) 2 (C) 3 (D) 5

27. Let a, b, c be three distinct positive real numbers, then the number of real roots of
2
ax + 2b|x| + c = 0 is
(A) 0 (B) 1 (C) 2 (D) 4
2
28. If a, b, c are positive real numbers, then the roots of the equation ax + bx + c = 0
(A) are real and positive (B) are real and negative
(C) have negative real part (D) have positive real part.
2
29. If roots of ax +bx+c = 0, a 0, (a, b, c are real numbers), are imaginary and a +c < b, then
(A) 4a + c = 2b (B) 4a + c > 2b
(C) 4a + c < 2b (D) 4a + c < 2b, if a < 0 and 4a + c > 2b if a > 0

30. Suppose f(x) is a quadratic expression which is negative for all real x. If g(x) = f(x) + f (x) + f(x),
then for any real x
(A) g(x) < 0 (B) g(x) > 0 (C) g(x) = 0 (D) none of these
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31. The roots of the equation (x – b)(x – c) + (x – a)(x – c) + (x – a)(x – b) = 0 are always
(A) positive (B) negative (C) real (D) imaginary

32. Find the sum and product of roots of | x 2 | 3 | x | 10  0

(A) 0, –4 (B) –3 (C) 2, 4 (D) can’t be determined

33. The roots of the quadratic equation x 2  2  a  1 x  a2  6a  8  0 will be of opposite sign if ‘a’
belongs to
(A) (1, 4) (B) (2, 4) (C) (–2, –4) (D) (2, –4)

34 Statement – 1: If ,  are the roots of the equation x 2  2  a  3  x  9  0 , a  R and   6   then

3
a<
4
3
Statement – 2: If f(x) = x 2  2  a  3  x  9 , then f(6) < 0  a <
4
(A) Both A and R are true and R is the correct explanation of A
(B) Both A and R are true and R is not the correct explanation of A
(C) A is true but R is false
(D) A is false but R is true

35. The set of all values of k for which the equation x 2  2 k  1 x   k  5   0 has at least one non-
negative root is
(A) [1,  ) (B)[-1,1] (C) (-  ,-5] (D) None of these

36. For the quadratic equation 4x 2  2  a  c  1 x  ac  b  0 (a>b>c)

(A) both roots are greater than a (B) both roots are less than c
(C) both roots lie between c/2 and a/2 (D) Exactly one of the roots lies between c/2 and a/2

37. If roots of the equation x 2  2mx  m2  1  0 lie in the interval (–2, 4) , then
(A) –1 < m < 3 (B) 1 < m < 5 (C) 1 < m < 3 (D) –1 < m < 5
2
38. The values of a for which one root of equation. (a – 5)x – 2ax + a – 4 = 0 is smaller than 1 and the
other greater than 2 are
(A) 5 < a < 24 (B) 5 < a (C) a < 24 (D) 5 < a < 0

Number of roots of the equation sin x  cos x  x  2x  6 is

2
39
(A) 0 (B) 2
(C) 4 (D) an odd number

The set of all real numbers x for which x  | x  2 |  x  0 , is

2
40.
(A) (, 2)  ( 2, ) (B)  ,  2    2,  
(C) (, 1) (1, ) (D)  2,  
41. For a > 0, all the real roots of the equation x 2  3a | x  a | 7a2  0 are
(A) 4a,5a (B) -4a,5a (C)-4a,-5a (D)4a,-5a
2
42. The values of m for which (m – 1)x + mx – 11 > 0 for all positive x are
(A) R (B)  (C) (1, ) (D) (–, 1)

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2
43. If the equation x + 2(k + 1)x + 9k – 5 = 0 has only negative roots, then
(A) k  0 (B) k  0 (C) k  6 (D) k  6
2
44. The integral value(s) of a for which the equation x – 2(a – 1)x + (2a + 1) = 0 has both roots positive
is/are
(A) 3 (B) 4 (C) 1 (D) 5

1 1 2
45. If ,  be the roots of x 2  a  x  1  b  0 , then the value of   is
2
  a 2
  a ab
4 1
(A) (B) (C) 0 (D) None of these
ab ab

If  is a root of the equation x – x + x – x + x – x + x – x – 2 0 = 0 , then 

11 10 8 7 5 4 2 12
46.
(A) is equal to 61 (B) is greater than 61
(C) is less than 61 (D) cannot be estimated

47. The value of `a’ for which the equation x3  ax  1  0 and x 4  ax 2  1  0 have a common root is
(A) (B) –2 (C) 0 (D) None of these

1 2 3
48. Let f(x) = x3  x 2  100x  7 sin x, then equation   = 0 has
y  f 1 y  f  2  y  f  3 
(A) no real root (B) one real root (C) two real roots (D) more than two real roots
2 3
49. Let P(x) = 1 + a1x + a2x + a3x where a1, a2, a3  I and a1 + a2 + a3 is an even number, then
(A) P(–1) is an even number (B) P(–1) is an odd number
(C) P(x) = 0 has no integral solutions (D) P(x) = 0 has atleast one integral solution
3 2
50. The real value of c for which the equation, 3x – 6x + 3x + c = 0 has two distinct roots in [0, 1] lies in
the interval(s)
(A) (–, 2) (B) [0, 1] (C) [3, 4] (D) (–, )

51. y = f(x) is concave upward graph and y =g(x) be a function such that
f (x)g(x)  g(x)f(x)  x 4  2x2  10 then
(A) g(x) =0 has at least one root between roots of f(x) = 0
(B) g(x)=0 has no root between consecutive roots of f(x) = 0
(C) If  and  are consecutive roots of f(x) = 0 then  < 0
(D) If  and  are consecutive roots of f(x) = 0 then  > 0
2 2
52. If m be the number of integral solutions of the equation 2x -3xy-9y -11=0 and n be the number of
3
real solutions of the equation x -[x]-3=0; then m is equal to

Comprehension -1
Consider the equations (n > 2)
2
ax1 + bx1 + c = x2
2
ax2 + bx2 + c = x3
2
ax3 + bx3 + c = x4
..........................
2
ax n – 1 + bxn – 1 + c = xn
2
axn + bxn + c = x1
in the variables x1, x2, ......, xn
a, b, c be real numbers, a  0
Let us put x2 – x1 = X1, x3 – x2 = X2, ...... xn – xn – 1 = Xn – 1 and x1 – xn = Xn

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2
53. If (b – 1) – 4ac = 0, then the system has
(A) no solution (B) n solutions (C) (n – 1) solutions (D) one solution

Comprehension-2
If x  R , then the roots of the equation x 4  4 x 3  8 x 2  k  0 when
54. The values of k   0, 3 , are :
A) all real B) two real and two complex
C) no real root D) none of these

55. The values of k   3, 4 , are:

A) all real B) two real and two complex
C) no real D) coincident

56. The values of k   , 0  , are:

A) all real B) two real and two complex
C) no real C) repeated roots

Comprehension-3
P(2x) 56
Let P(x) be a polynomial such that P(1) = 1 and 8 for all real x for which both sides are
P(x  1) x7
defined.

57. The degree of P(x) is

(A) 2 (B) 3 (C) 4 (D) 5

58. The value of P(–1) is

2 4 5 11
(A) (B)  (C)  (D) 
7 21 21 21

59. The number of real roots of P(x) = 0 is

(A) 5 (B) 3 (C) 2 (D) none of these

ASSIGNMENT

   
x x
1 The equation 52 5 2  18 has
(A) only one solution (B) two solutions
(C) no solution (D) any number of solutions

x
3
   x  x  9 is
2
2. The solution set of 
5
 
(A) x > 0 (B) –2 < x < 2 (C) 0 < x < 2 (D) 
|x|
3. Total number of solutions of the equation 7 (|5 – |x||) = 1, is equal to,
(A) 12 (B) 2 (C) 3 (D) 4
2 2 2 2 3 3
4. In the quadratic equation ax + bx + c = 0, if  = b – 4ac and  + ,  +  ,  +  are in G.P.
2
where ,  are the roots of ax + bx + c = 0, then
(A)   0 (B) b = 0 (C) c = 0 (D)  = 0

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2
5. If one root is square of the other root of the equation x + px + q = 0, then the relation between p and
q is
3 2 3 2
(A) p – q(3p -1 ) + q = 0 (B) p – q(3p + 1) + q = 0
3 2 3 2
(C) p + q (3p – 1) + q = 0 (D) p + q(3p + 1) + q = 0

6. Both the roots of the equation (x – b)(x – a) + (x – a)(x – c) + (x – b)(x – c) = 0 are equal, then
(A) a = b = c (B) a2  bc (C) b2  ac (D) none of these

7. If the roots of the equation bx 2  cx  a  0 be imaginary then for all real values of x, then expression
3b2 x 2  6bcx  2c 2 is
(A) greater than 4ab (B) less than 4ab (C) greater than -4ab (D) less than –4ab

8.  
If ,  are real and  2 , 2 are the roots of the equation a2 x 2  x  1  a2  0 (a > 1) then 2 =

(A) a2 (B) 1 (C) 1  a2 (D) 1  a2

9.  
The polynomial ax 2  bx  c ax 2  dx  c ,ac  0 , has 
(A) four real zeroes (B) at least two real zeroes
(C) at most two real zeros (D) all the above.

10. If a, b, c, d are real and no two of them are simultaneously zero, then the equation
x 2

 ax  3b x2  cx  b  x 2

 dx  2b  0 has
(A) atleast two real roots (B) at least four real roots
(C) all roots real (D) atleast two imaginary roots

11. If y = tan x cot 3x, x  R then

1 1 1 1
(A)  y  1 (B)  y  1 (C) y3 (D) y  or y > 3
3 3 3 3

12. Assertion (A): The curve y = x 2  6x  9 touches the x-axis.

Reason (R): y = ax 2  bx  c touches the x-axis if   b2  4ac  0
(A) Both A and R are true and R is the correct explanation of A
(B) Both A and R are true and R is not the correct explanation of A
(C) A is true but R is false
(D) A is false but R is true

13. If ,  are the roots of x 2   a  2  x   a  1  0 where `a’ is a variable then the least value of
2  2 is
(A) 2 (B) 3 (C) 5 (D) 7

14.  
Let f(x) = 1  b2 x2  2bx  1 and m(b) be the minimum value of f(x). As b varies, the range of m(b)
is
 1 1 
(A) [0, 1] (B) 0,  (C)  ,1 (D) (0, 1]
 2 2 

15. The equation tan4 x  2 sec 2 x  a2  0 will have atleast one solution if
(A) |a|  4 (B) |a|  2 (C) | a | 3 (D) none of these

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2 2
16. The set of values of x for which the inequlities x – 3x – 10 < 0, 10x – x – 16 > 0 hold
simultaneously, is
(A) (–2, 5) (B) (2, 8) (C) (–2, 8) (D) (2, 5)

17. The solution set contained in R of the inequation 3 x  31 x  4  0 is

(A) (1, 3) (B) (0, 1) (C) (1, 2) (D) (0, 2)

3c
18. If the equation ax 2  2bx  3c  0 has non real roots and  a  b then c is always
4
(A) < 0 (B) > 0 (C)  0 (D) zero

xa
19. If x2  4cx  b2  0x  R , and a2  c 2  ab then range of f(x) = is
x  bx  c 2
2

(A)  ,0  (B)  0,  (C)  ,   (D) None of these

20. The common root of two equations p 2

 p  q  r 2 x2     
p  q  r 2  1 x  1  p2  0 and 
 
m2 x 2  r 2  m2 x  r 2  0 is
1 p2 p 2  r 2  m2
(A) 0 (B) (C) (D) none of these
r2 pq

21. The equation 22x   a  1 2x 1  a  0 has roots of opposite signs then exhaustive set of values of a
is
(A) a   1,0  (B)a < 0 (C) a   ,1/ 3  (D) a   0,1/ 3 

22. Let ,   R be the roots of the equation ax 2  bx  c  0 , then k  R lies between  and  if
(A) ak 2  bk  c  0 (B) a2 k 2  bak  ac  0
(C) a2 k 2  ba2 k  a2 c  0 (D) ak 2  bk  c  0

23. The number of solutions of the equation x 3  2x 2  5x  2 cos x  0 in [0, 2 ] is

(A) 0 (B) 1 (C) 2 (D) 3
2
24. Number of positive integers n for which n + 96 is a perfect square is
(A) 4 (B) 8 (C) 12 (D) infinite

If p(x) = ax + bx and q(x) = x + mx + n with p(1) = q(1), p(2) – q(2) = 1, and p(3) – q(3) = 4, then
2 2
25.
p(4) – q(4) is equal to
(A) 7 (B) 16 (C) 9 (D) none of these

   
2
26. If the two roots of the equation (c-1) x 2  x  1   c  1 x 4  x 2  1  0 are real and distinct and

1 x   1 
f(x) = , then f(f(x)) + f  f    =
1 x   x 
(A) –c (B) c (C) 2c (D) None of these
2
27. Let a, b, c  R such that 2a + 3b + 6c = 0. Then the quadratic equation ax + bx + c = 0 has
(A) at least one root in (0,1) (B) at least one root in (–1, 0)
(C) both roots in (1, 2) (D) imaginary roots

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SP-WB-MA(V)-20

28. If , ,  are such that       2 ,  2  2   2  6 , 3  3   3  8 then 5  5  5 =

(A) 10 (B) 12 (C) 18 (D) 32
2
29. If the roots of the equation (a – 3)x – 2ax + 5a = 0 are positive then a belongs to
 1 15   15  1 4   4 
(A)  ,  (B)  3,  (C)  ,  (D)  3, 
 3 4   4   3 15   15 
2 3 2
30. The real values of a for which the Quadratic equation 2x – (a + 8a – 1)x + a – 4a = 0 possesses
roots of opposite sign are given by
(A) a > 5 (B) 0 < a < 4 (C) a > 0 (D) a > 7

2 2
31. The all values of a for which the inequation 4 x  2(2a  1)2x  4a2  3 > 0 is satisfied for all x are
(A) (–, –1)  (0, ) (B) (–, –7)  (6, ) (C) (–, –11)  (9, ) (D) none of these

32. The equation  a  2  x 2   a  3  x  2a  1,a  2 has rational roots for

(A) all rational values of a except a = -2 (B) all real values of a except a = -2
1
(C) rational values of a > only (D) None of these
2

33. The integral values of m for which the roots of the equation mx 2   2m  1 x   m  2  =0 are rational
are given by the expression [where n is integer]
(A) n(n + 2) (B) n(n + 1) (C) n2 (D) None of these

2 2
34 If the roots of the equation x  2ax  a  a  3  0 are real and less than 3, then
(A) a < 2 (B) 2  a  3 (C) 3  a  4 (D) a  4

35. Statement 1: The number of values of a for which a  3a  2 x  a  5a  6 x  a  4  0 is  2

 2
 2
 2

an identity in x is one
Statement2: If ax  bx  c  0 is an identity in x then a =b =c =0
2

(A)Statement 1 is true; statement 2 is true; statement 2 is a correct

explanation for statement 1.
(B)Statement 1 is true; statement 2 is true; statement 2 is NOT a correct explanation for statement 1.
(C) Statement 1 is true; statement 2 is false
(D) Statement 1 is false; statement 2 is true

 
 x  3 | x | has
log  x 1  3 2 x  x 2
36. The equation x 1
(A) Unique (B) Two solutions (C) No solution (D) More than two

 x 2 10 x  24 
 
37. The number of solutions of the equation x  4  x 3   1 is
(A) 2 (B) 0 (C) infinitely many (D) 4

If the equations ax  3bx  3cx  d  0 and ax  2bx  c  0 have a common root then
3 2 2
38.
 bc  ad 
2


 ac  b  bd  c 
2 2

1 1
(A) (B) 4 (C) 9 (D)
4 9
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 SP-WB-MA(I)-21

17  1x 
2 1
39. The value of sum of the roots of the equation 10  25   50  is
x x
4 
1 3
(A) 0 (B) 1 (C) (D)
4 2

40. If ‘1’ lies between the roots of the equation ax 2  bx  sin   0, a  0 , then a + b is always less
than
3
(A) – 1 (B) 1 (C)  (D) none of these
2

41. The quadratic polynomials defined on real coefficients P(X) = a1x 2  2b1x  c1 ; Q(X) =
a2 x 2  2b2 x  c 2 ; where a1  0 and a2  0 and P(x) and Q(x) both take positive values  x  R .
g(x) = a1a2 x 2  b1b2 x  c1c 2 then
(A) g(x) takes + ve values only (B) g(x) takes –ve values only
(C) g(x) takes both +ve and –ve values (D) Nothing can be said about g(x).

The number of roots of quadratic equation ax 2  b | x | 5  0 if a, b,  R ; are

+
42.
(A) 2 (B) 4 (C) 0 (D) none of these

43. The greatest value of   0 for which both the equations 2x 2     1 x  8  0 and
x 2  8x    4  0 have real roots is
(A) 9 (B) 15 (C) 12 (D) 16

44. If x 2  4xy  4y 2  4x  cy  3 can be written as the product of two linear factors, then c =
(A) 2 (B) 4 (C) 6 (D) 8

45. If the equation x 2  bx  c  0 and x 2  bx  c  0 have real roots such that b  [–2, 2], then c
belongs to
(A) (0, 1) (B) [–1, 0) (C) (–1, 1) (D) [–1, 1]
x x
46. If both the roots of (2a – 4)9 – (2a – 3)3 + 1 = 0 are non-negative, then
5 5
(A) 0 < a < 2 (B) 2 < a < (C) a < (D) a > 3
2 4
2 2 2 2
47. If f(x) = x + 2bx + 2c and g(x) = –x – 2cx + b such that min f(x) > max g(x), then the relation
between b and c is
(A) no real value of b and c (B) 0 < c < b 2
(C) |c| < |b| 2 (D) |c| > |b| 2
2
48. If ax + bx + 1 = 0 does not have 2 distinct real roots then least value of 2a– b is ________
1 1
(A) (B)  (C) 1 (D) –1
2 2

49. The quadratic equation whose roots are the x and y intercepts of the line passing through (1, 1) and
making a triangle of area A with the axes, may be
(A) x 2  Ax  2A  0 (B) x 2  2Ax  2A  0

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SP-WB-MA(V)-22

(C) x 2  Ax  2A  0 (D) x 2  2Ax  2A  0

2
50. If x + 2ax + b  c,  x  R, then
2 2 2
(A) b – c  a (B) c – a  b (C) a – b  c (D) none of these

51. This set of value of k, for which roots of the equation (x – a) (x – c) + k(x – b) (x – d) = 0 are real
given by is (a > b > c > d).
(A) (–, d) (B) (a, ) (C) (–, ) (D) none of these
2 2
52. Roots of the quadratic equation (x  4x + 3) + (x  6x + 8) = 0,   R will be
(A) always real (B) real only when is positive
(C) real only when  is negative (D) always imaginary

1 x x x2
53. x 1 x x 2 =0 has roots x1, x 2 , x 3 , x 4 then |x1 x2 x3 x4| =
x2 x 1 x
(A) 0 (B) 1 (C) 4 (D) 16

54. Let S be the set of values of a for which  a  4  sec 4 x   a  3  sec 2 x  1  0 has real solutions.
Then S is
(A) R (B) (–, 3] (C) (4, ) (D) [3, 4)

55. The real value of c for which the equation, 3x3  6x 2  3x  c  0 has two distinct roots in [0, 1] lie in
the interval (s)
(A)  , 2  (B) [0,1] (C) [3, 4] (D)  ,  

2
56. If one root of x – x – k = 0 is square other then k equals
(A) 2  5 (B) 2  3 (C) 2  3 (D) 2  5

2 2
 x   x 
57. The equation,      a  a  1 has
 x  1  x  1
(A) four real roots if a > 2 (B) has two real roots if 1 < a < 2
(C) has no real roots if a < –1 (D) has four real roots for all a < –1

2
4    
58. The value of ‘x’ satisfying the equation x – 2  x sin  x   + 1 = 0
  2 
(A) 1 (B) –1 (C) 0 (D) no value of ‘x’

3 2
59. Let a, b, c be the roots of x – 9x + 11x – 1 = 0 and let s = a b c and
t = ab  bc  ca , then
2 4
(A) t = 11 + 2s (B) s = 125 + 36t + 8s
4 2 4 2
(C) s – 18s – 8s = –37 (D) s – 18s + 8s = 37
4 3 2
60. Let x – 6x + 26x – 46x + 65 = 0 have roots ak + ibk for k = 1, 2, 3, 4 where ak, bk are all integers,
then
  
(A) a12  b12 a22  b22 a32  b32 a24  b24  652  (B) a1 + a3 + a3 + a4 = 6
(C) |b1| + |b2| + |b3| + |b4| = 10 (D) |a1| + |a2| + |a3| + |a4| = 4

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 SP-WB-MA(I)-23

61. Let f(x) = ax 3  bx 2 + x + d has local extrema at x =  and x =  such that  < 0 and f    f     0
then
(A) f(x) = 0 has only one real root which is positive, If a f     0
(B) f(x) = 0 has only one real root which is negative, If a f    >0
(C) f(x) = 0 has only one real which is positive, If a f    > 0
(D) f(x) = 0 has only one negative real root, If a f    < 0

3 3
62. The equation x3  x is satisfied by
4 8
 5   7   17   23 
(A) x = cos   (B) x = cos   (C) x = cos   (D) x = cos  
 18   18   18   18 
3 2
63. The correct statements about the cubic equations 2x + x + 3x – 2 = 0 are
(A) it has exactly one positive root (B) it has either one or three negative roots
(C) it has a root between 0 and 1 (D) it has no imaginary roots
2 4 2
64. If b  4ac for the equation ax + bx + c = 0, then all roots of the equation will be real when
(A) b > 0, a < 0, c > 0 (B) b < 0, a > 0, c > 0
(C) b > 0, a > 0, c > 0 (D) b > 0, a < 0, c < 0
4 3 2
65. The equation 3x + 8x – 6x – 24x + r = 0 has no real root if r is equal to
(A) 18 (B) 20 (C) 23 (D) none of these
3 2
66. If the equation whose roots are the squares of the roots of the cubic x – ax + bx – 1 = 0 is identical
with this cubic, then the possible cases are
2
(A) a = b = 0 (B) a, b are roots of x + x + 3 = 0
(C) a = b = 3 (D) none of these
3 2
67. The equation |x| – 2|x| + 2|x| – 1 = 0 has
(A) only two real roots (B) rational real roots
(C) no imaginary roots (D) two rational roots of opposite sign

68. The equation | x |3 2 | x |2 2 | x | 1  0 has

(A) only two real roots (B) rational real roots
(C) no imaginary roots (D) two rational roots of opposite sign
2 2
69. The value of a for which one root of the equation (a – 5a + 3)x + (3a – 1)x + 2 = 0 is twice as large
as other, is

70. Find the solution of the following equations which have common roots:
4 3 2 4 3
2x – 2x + x + 3x – 6 = 0, 4x – 2x + 3x – 9 = 0.

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SP-WB-MA(V)-24

71. Match the following

Column– I Column– II
 x 2  3  2  x  3
If f(x) =  , then the quadratic equation
(P) 2x  5  3  x  4 (1)
2
x + 19x – 120 = 0
whose roots are lim f(x) and lim f(x) is
x 3 x 3

1 4 9 n2
If  = lim      and  =
n n3  1 n3  1 n3  1 n3  1
2
(Q) sin2x (2) 12x – 7x + 1 = 0
lim , then the quadratic equation whose roots are ,
x  0 sin8x
 is
2 4 2
If x – 3x + 2 is a factor of x – px + q, then the quadratic 2
(R) (3) x – 9x + 20 = 0
equation whose roots are p and q is
2
If one root of quadratic equation x + px + 12 = 0 is 4, while
2
the equation x + px + q = 0 has equal roots then the 2
(S) (4) x – 17x + 66 = 0
quadratic equation whose roots are [p] and [q] (where [.]
denotes the greatest integer) is
2
If one root of the quadratic equation x – ax + 6 = 0 is even
2 2
(T) prime while the one root of x + ax + 6 = 0 is smallest odd (5) x + 7x + 10
prime then the equation whose roots are  and  is
2
(6) x – 5x – 84 = 0
2
(7) 7x – 12x + 1 = 0
P Q R S T
(A) 1 2 3 4 5
(B) 2 3 1 5 7
(C) 3 7 6 4 5
(D) 4 2 3 6 1
2
72. For the Q.E x – (m – 3)x + m = 0. The value of ‘m’ match the following
Column– I Column– II
(P) The roots of one real and distinct (1) m  [9, )
(Q) The roots are equal (2) m  {3}
(R) Roots are of opp. Sign (3) m  (–, 1)  (9, )
(S) Opposite sign but equal in magnitude (4) m  (0, 5)
(T) Roots are both positive (5) m  (–, 0)  [9, )
(U) Roots are both negative (6) m  (-, 0)
(V) Atleast one root is positive (7) m  {1, 9}
P Q R S T U V
(A) 1 2 4 3 5 6 7
(B) 2 3 1 4 6 5 7
(C) 3 7 6 2 1 4 5
(D) 2 1 4 3 7 6 5

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 SP-WB-MA(I)-25

Comprehension -1
If both the roots  and  of a quadratic equation lie between numbers k1, k2, then
b
(i) D > 0 (ii) af (k1) > 0 (iii) af (k2) > 0 (iv) k1 <  < k2
2a
Y Y
3
f(x) = ax + bx + c  b D 
  , 
a>0  2a 4a 

X k1 k2
f(k1) f(k2) X
X   f(k1)  
k1 k2 X
a>0
3
 b D  f(x) = ax + bx + c
  2a ,  4a 
Y   Y

2
73. If the roots of the equation (a + 1)x – 3ax + 4a = 0 (a  –1) are greater than unity, then a belongs to
 16  7  16 7   16 
(A)   , 1 (B)  , 1 (C)  ,  (D)   ,1
 7  16   7 16   7 
2 2
74. If both the roots of equation 2x + ax + a – 5 = 0 are less than unity, then a belongs to
 13  1 40   40 1  13   40 1  13   13  1 40 
(A) 
 2
,     ,  (B)  

,  , 
 7   7 2   7 2   2 7 
 40 1  13   13  1 40   40 1  13   13  1 40 
(C)   ,    ,  (D)  ,    , 
 7 2 7   7 7 
   2  2   2
2
75. If 6 lies between the roots of the equation x + 2(a – 3)x + 9 = 0, then a belongs to
 3  3  3  3
(A)  ,  (B)  ,   (C)  ,  (D)  ,  
 4   4   4   4

Comprehension-2
Consider the equation x3  3x  K =0 answer the following questions.

76. The range of values of K for which above equation has only one real root.
(A)  , 2  U  2,   (B) (–2, 2) (C) (0, ) (D) (–, 0)

77. The range of values of K for which the above equation has no real root
(A) {–2, 2} (B) (–2, 2) (C) R (D) 

78. The range of values of K for which the above equation has 3-distinct real roots
(A)  , 2  (B) (2, ) (C) (–2,2) (D) 

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Saifabad  Narayanaguda  Dilsukhnagar  Kukatpally  Miyapur
Ph: 040–66777001 Ph: 040–66464102 Ph: 040–66757890 Ph: 040–48533211 Ph: 040–48565559
SP-WB-MA(V)-26

ANSWERS & KEY (QEE)

DAY-1

2. A 3. A 4. C 5. A
6. A 7. A

DAY-2

2. A 3. B 4. D 5. C

DAY-3

1. D 2. B 3. B 4. 2
5. A  q; B  s; C  p; D  r 6. D

DAY-4

 
2
1. (0, 1) 2. ac  bc 3. B

4. A 5. C

DAY-5

1. 2 2, 
  2. D 3. A 4. B

5. A, B, C 6. A, B, C, D 7. A, B, C 8. C, D
9. A, B, C, D

DAY-6

1. 1 2. B 3. C 4. B
5. A 6. D 7. A, B 8. A, B, D
9. A 10. C 11. B

DAY-7

1. (i) (10, ) (ii) [9, 10) (iii) (–, 1] (iv) (10, ) (v)  (vi) (10, )
(vii)  (viii) [9, )

7  3 5
2.  a  4  2 3 4. R – {–4}
2

DAY- 8

1. A 2. B 3. A, B, C 4. A, B, C, D
5. A, B, C 6. C, D 7. A 8. C

Hyderabad Centers
Saifabad  Narayanaguda  Dilsukhnagar  Kukatpally  Miyapur
Ph: 040–66777001 Ph: 040–66464102 Ph: 040–66757890 Ph: 040–48533211 Ph: 040–48565559
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9. D 10. B 11. B 12. D

13. C 14. B

DAY-11

1. A 2. A 3. B 4. C
5. D 6. A 7. B 8. A
9. A 10. B 11. C 12. B
13. D 14. C 15. A, B, C, D

MISCELLANEOUS

1. C 2. A 3. A 4. C
5. C 6. C 7. B 8. B
9. D 10. C 11. D 12. B
13. C 14. D 15. C 16. A
17. C 18. B 19. A 20. A
21. C 22. B 23. A 24. B
25. A 26. D 27. A 28. C
29. D 30. A 31. C 32. A
33. B 34. A 35. C 36. D
37. A 38. A 39. A 40. B
41. D 42. B 43. C 44. B
45. C 46. B 47. B 48. C
49. B 50. A, B, C, D 51. A, C 52. 2
53. D 54. A 55. B 56. B
57. B 58. C 59. B

ASSIGNMENT

1. B 2. D 3. D 4. D
5. A 6. A 7. C 8. B
9. B 10. A 11. D 12. A
13. C 14. D 15. C 16. D
17. B 18. A 19. C 20. D
21. D 22. B 23. A 24. A
25. C 26. B 27. A 28. D
29. B 30. B 31. A 32. A
33. A 34. A 35. A 36. C
37. A 38. B 39. A 40. A

Hyderabad Centers
Saifabad  Narayanaguda  Dilsukhnagar  Kukatpally  Miyapur
Ph: 040–66777001 Ph: 040–66464102 Ph: 040–66757890 Ph: 040–48533211 Ph: 040–48565559
SP-WB-MA(V)-28

41. D 42. C 43. C 44. D

45. D 46. B 47. D 48. B
49. C 50. A 51. C 52. A
53. B 54. A 55. A 56. A, D
57. A, B, D 58. A, B 59. A, B, C 60. A, B, C
61. A, B 62. A, B, C 63. A 64. B
65. B, C 66. A, C 67. A, B, D 68. A, B, D
2 3
69. 70.  71. D 72. C
3 2
73. A 74. C 75. B 76. A
77. D 78. C

Hyderabad Centers
Saifabad  Narayanaguda  Dilsukhnagar  Kukatpally  Miyapur
Ph: 040–66777001 Ph: 040–66464102 Ph: 040–66757890 Ph: 040–48533211 Ph: 040–48565559