CPP–QEE

1. Find the coefficients of (i) x99 (ii) x98 in the polynomial expression (x – 1) (x – 2) … (x – 100) .

2. If the quadratic equations x2 + ax + b = 0 and x2 + bx + a = 0 (a  b) have a common root, then the numerical
value of a + b is ……….

3. If a  b  c  d, then the roots of the equation (x – a) (x – c) + 2(x – b) (x – d) = 0 are real and district

4. If n 1, n2, …. np are p positive integers, whose sum is an even number, then the number of odd integers among
them is odd.

5. If P(x) = ax2 + bx + c and Q(x) = –ax2 + dx + c, where ac  0, then P(x) Q(x) = 0 has at least two real roots.

6. If x, y and z are real and different and u = x2 + 4y2 + 9z2 – 6yz – 3zx – 2xy, then u is always
(A) non negative (B) zero
(C) non positive (D) none of these

7. If (x2 + px + 1) is a factor of (ax3 + bx + c), then

(A) a2 + c2 = -ab (B) a2 =- c2 = - ab
(C) a2 – c2 = ab (D) none of these

8. If p, q, r are any real numbers, then

1
(A) max (p, q)  max (p, q, r) (B) min (p, q) = (p  q  | p  q |)
2
(C) max (p, q)  min (p, q, r) (D) none of these

9. The largest interval for which x12 – x9 + x4 – x + 1  0 is

(A) –4  x  0 (B) 0  x  1
(C) –100  x  100 (D) –  x  

2 2
10. The equation x   1 has
x 1 x 1
(A) no root (B) one root
(C) two equal roots (D) infinitely many roots

11. If a2 + b2 + c2 = 1, then ab + bc + ca lies in the interval

1 
(A)  , 2  (B) [–1, 2]
2 
 1   1
(C)   ,1 (D)  1, 
 2   2

12. If  and  are the roots of x2 + px + q = 0 and 4, 4 are the roots of x2 – rx + s = 0, then the equation
x2 – 4qx + 2q2 – r = 0 has always
(A) two real roots (B) two positive roots
(C) two negative roots (D) one positive and one negative root

13. Let a, b, c be real numbers, a  0. If  is a root of a2x2 + bx + c = 0.  is the root of a2x2 – bx – c = 0 and 0   
, then the equation a2x2 + 2bx + 2c = 0 has a root  that always satisfies
  
(A)   (B)    
2 2
(C)  =  (D)     

14. Let ,  be the roots of the equation (x - a) (x – b) = c, c  0. Then the roots of the equation
(x - ) (x - ) + c = 0 are
(A) a, c (B) b, c
(C) a, b (D) a + c, b + c

1
15. If the roots of the equation x2 – 2ax + a2 + a – 3 = 0 are real less than 3, then
(A) a  2 (B) 2  a  3
(C) 3  a  4 (D) a  4

16. If  and  (  ) are the roots of the equation x2 + bx + c = 0, where c  0  b, then

(A) 0     (B)   0    ||
(C)     0 (D)   0  ||  

17. If b  a, then the equation (x – a) (x – b) – 1 = 0 has

(A) both roots in (a, b) (B) both roots in (–, a)
(C) both roots in (b, +) (D) one root in (–, a) and the other in (b, +)

18. For all ‘x’, x2 + 2ax + 10 – 3a  0, then the interval in which ‘a’ lies is
(A) a  –5 (B) –5  a  2
(C) a  5 (D) 2  a  5

19. If ,  are the roots of ax2 + bx + c = 0, (a  0) and  + ,  +  are the roots Ax2 + Bx + C = 0, (A  0) for
b 2  4ac B2  4AC
some constant , then prove that 2

a A2

20. Let a, b, c be real numbers with a  0 and let ,  be the roots of the equation ax2 + bx + c = 0. Express the roots
of a3x2 + abcx + c3 = 0 in terms of , .

21. If x2 + (a – b) x + (1 – a – b) = 0 where a, b  R, then find the values of a for which equation has unequal real
roots for all values of b.

22. Let a & b be the roots of the equation x2 – 10cx – 11d = 0 and those of x2 – 10ax – 11b= 0 are c, d then the value
of a + b + c + d when a  b  c  d, is

1
23 Let a, b, c, p, q be real numbers. Suppose ,  are the roots of the equation x2 + 2px + q = 0 and , are the

roots of the equation ax2 + 2bx + c = 0, where 2  {-1, 0, 1}
STATEMENT–1: (p2 – q) (b2 – ac)  0
and
STATEMENT–2: b  pa or c  qa
(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

Write-up (24-25)
af() < 0 is the necessary and sufficient condition for a particular no.  to lie between the roots of a quadratic equation
f(x) = 0, where f(x) = ax2 + bx + c. Again if f(1).(2) < 0 then exactly one of the roots will lie between 1 and 2 . So,

24. If a(a + b + c) < 0 < (a + b + c)c then

(A) one root is less than 0, the other is greater than 1 (B) exactly one of the roots lies in (0, 1)
(C) both the roots lie in (0, 1) (D) atleas one of the roots lies in (0, 1)

25. If (a + b + c)c < 0 < a(a + b + c) then

(A) one root is less than 0, then other is greater than 1 (B) one root less than 0 other lies in (0, 1)
(C) both the roots lie in (0, 1) (D) no root lies in (0, 1)

26. Let the line x - 8y + k = 0, kI meets the rectangular hyperbola xy = 1 at the points whose abscissae are integers
then the number of each lines is
(A) 4 (B) 5
(C) 8 (D) 10

2
27. Number of points with integral coordinates lying on the circle x2 + y2 = 2008 is
(A) 0 (B) 4
(C) 8 (D) 16

28. If the equation ax2 – bx + 12 = 0 where a and b are +ve integers not exceeding 10, has roots both greater than 2
then the number of ordered pair (a, b) is
(A) 0 (B) 1
(C) 3 (D) 5

29. Let f(x) = ax2 + bx + c, a  0, a, b, cR. f(x) = 0 has two real and distinct roots  and . If f(x + k) + f(x) = 0
(k, R) has exactly one root between  and  then
(A) f( + k) f( + k) < 2 (B) k does not depend upon the value of 
(C) k can take any real value (D) 0<|a k|<  where  is discriminant of f(x)

30. STATEMENT I: The range of values of p (p > 1) for which the expression
p 2  3p  2
px2 – (p – 1) px – 2  0 for atleast one –ve value of x, is (2, 5)  (6, )
p  11p  30
STATEMENT II: If ac < 0 then the equation ax2 + bx + c = 0 (a, b, cR) will have a negative solution.

31. If the roots of x2 - ax + b = 0 are real and differ by a quantity which is less than c(c > 0), prove that b lies
between (1/4) (a2 - c2) and (1/4)a2.

32. Find m for which the inequality m22x - 4.2x + 3m + 1 < 0 is satisfied for atleast one real x.

33. Find the values of 'a' for which the equation x4 + (1 - 2a)x2 + a2 - 1 = 0
(i) has no solution (ii) has one solution
(iii) has two solutions (iv) has three solutions
(v) has four distinct real solution

x 2  14x  9
34. If x is real, then the value of the expression f lies between
x 2  2x  3
(A) -3 and 3 (B) -4 and 5
(C) -4 and 4 (D) -5 and 4

35. Consider the equation x2 + x - n = 0, where n is an integer lying between 1 to 100. Total number of different
values of n so that the equation has integral roots, is
(A) 6 (B) 4
(C) 9 (D) none of these

d e f
36. If a, b, c are in G.P. then the equation ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have a common root if , ,
a b c
are in
(A) A.P. (B) G.P.
(C) H.P. (D) none of these

37. If the expression (mx - 1 + 1/x) is non-negative for all positive real x, then the minimum value of m must be
(A) -1/2 (B) 0
(C) 1/4 (D) 1/2

38. The number of quadratic equations which are unchanged by squaring their roots is
(A) 2 (B) 4
(C) 6 (D) none of these

39. If the equations ax2 + bx + c = 0 and x3 + 3x2 + 3x + 2 = 0 have two common roots, then
(A) a = b  c (B) a = -b= c
(C) a = b = c (D) none of these

3
40. The equation ax2 + bx + a = 0 and x3 - 2x2 + 2x - 1 = 0 have two roots in common. Then a + b must be equal to
(A) 1 (B) -1
(C) 0 (D) none of these

3
41. Let p(x) = 0 be a polynomial equation of least possible degree, with rational coefficients, having 7  3 49 as
one of its roots. Then the product of all the roots of p(x) = 0 is
(A) 7 (B) 49
(C) 56 (D) 63

42. If both the roots of the equation x2 - 2ax + a2 + a - 3 = 0 are less than 3, then
(A) a < 2 (B) 2  a  3
(C) 3 < a  4 (D) a > 4

43. If ,  are the roots of the equation 4x2 - 16x +  = 0, R such that 1 <  < 2 and 2 <  < 3, then find the
number of integral values of .

44. Find the least value of (6x2 - 22x + 21)/ (5x2 - 18x + 17) x R.

45. Show that 2x2 + 2xy + y2 - 2x + 2y + 2 is never smaller than -3; x, yR.

 7
46. Find all the values of the parameters c for which is inequality has at least one solution 1 + log2  2x 2  2x   9
 2
 log2 (cx2 + c).

47. Find the value of 'b' for which the equation 2log1/25 (bx + 28) = - log5 (12 - 4x - x2) has
(i) only one solution (ii) two different solutions (iii) no solution.

48. In a triangle PQR, R = /2. If tan (P/2) & tan (Q/2) are the roots of the equation ax2 + bx + c = 0 (a  0) then
(A) a + b = c (B) b + c = a
(C) a + c = b (D) b = c

49. The number of integer values of m, for which the x-coordinate of the point of intersection of the lines 3x + 4y =
9 and y = mx + 1 is also an integer, is

50. Let f(x) = ax2 + bx + c, a  0 and  = b2 - 4ac. If  + , 2 + 2 and 3 + 3 are i G.P., then
(A)   0 (B) b = 0
(C) c = 0 (D) bc  0

51. If x = 2 + 22/3 + 21/3, then the value of x3 - 6x2 + 6x is

(A) 3 (B) 2
(C) 1 (D) none of these

52. If a, b, c are odd integers, then ax 2  bx  c  0 can’t have rational roots. Prove it.

53. For what k  I , the quadratic equation x 2  kx  24  0 has integral roots.

54. For what values of k  I, the quadratic equation  x  k  x  10   89  0 has integral roots.

55. Find the integral value of a for which the equation x 2   a  1 x  a  1  0 has integral roots

56. Prove that 2x 4  1402  y 4 has no integral solution.

4
57. Prove that the equation x 2  2px  2q  0 cannot have rational roots if p and q are odd integers
58. If a & c are odd primes, b  N , b  1  ac and roots are rational then show that one of the roots of the equation
ax 2  bx  c  0 is always –1.

59. Let a, b, c  N , a  1. if p be prime number and ax 2  bx  c  p has two distinct integral solution then prove
that ax 2  bx  c  2p has no integral solution

60. Consider a polynomial function, f  x  = a n x n  a n 1 x n 1  .....  a1x  a 0 , a i  I .

If f  0  and f 1 are odd integers, then show that f  x   0 can’t have integral solution.

1/ n
61. 
Let f  x   x 2  px  q  p, q  R  and g  x   1  x n  , n  N . If f  x   x has no real solution then prove that
f  f  x    g  g  x    0 will also have no real solution.

62. If  and  are the roots of the equation ax 2  bx  c  0 and 4 ,  4 are the roots of the equation
x 2  mx  n  0, then prove that the roots of the equation a 2 x 2  4acx  2c 2   a 2 m  0 are always real and
opposite in sign  ,  are real & distinct  .

63. If ax 2  bx  6  0 does not have 2 distinct real roots, find the minimum possible values of a  b , 3a  b and
a  2b .

64. Let x 2   m  3 x  m  0  m  R  be a quadratic equation, find ‘m’ for which roots are
(i) opposite in sign (ii) equal in magnitude but opposite sign
(iii) 2 lies between roots. (iv) exactly one root in (1, 2).
(v) both roots between –2 and 3. (vi) atleast one root +ve.
(vii) one root < 4, other > 7. (viii) both roots > 3.
(ix) both roots < –1. (x) atleast one root > 4.

65. If b 2  2ac, prove that ax 3  bx 2  cx  d  0 has exactly one real root.

66. Find out minimum non-negative real values of a, b and c, given that the equation x 4  ax 3  bx 2  cx  1  0
has real roots.

67. Let f  x   Ax 2  Bx  C, where A, B, C  R. Prove that if f  x  is an integer whenever x in an integer, then

2A, A  B, C are all integer. Again prove that if 2A, A + B, C are all integers, f(x) is an integer whenever x is an
integer.

68. Find the value of k if the product of two of the four roots of the equation x 4  18x 3  kx 2  200x  1984  0 is –
32.

69. a, b, c are distinct real numbers, such that ax 2  bx  c  0, bx 2  cx  a  0 and cx 2  ax  b  0 .

a 2  b2  c2
Prove that, 1  4
ab  bc  ca

70. If , ,  are two roots of 2x 3  ax  b  0 , find the values of

1 1 1
(i)   (ii)  3    3   3   
2 2 2 

5
 (iii) 3   3   3 (iv) 1   1   1   
3 3 3

(v) 33  3 3   3 3

 a  x  b  x  2
71. Prove that minimum value of ,  a  c, b  c, x  c  is  a  c  b  c  and it is
c  x
at x   a  c  b  c   c .

72. For sin 2 x  a sin x  1  0 to have no real solution find ‘a’.

73. For x 2   a  3 x  4  0 to have real solution, find range of values of a.

74. For e2sin x  aesin x  1  0 to have no real solution find ‘a’

75. 
Find all the possible values of the parameter ‘a’ so that the function f  x   x 3  3  7  a  x 2  3 9  a 2 x  2, 

assume local maximum value at some x  R .

76. Prove that any real  the quadric equation  x  a  x  c     x  b  x  d   0 ,  a  b  c  d  has real roots.

p 2  3p  2
77. Find range of p (p > 1) for which px 2   p  1 px   0 for atleast one –ve x.
 P  5 P  6 

78. If  is a root of ax 2  bx  c  0 ,  is a root of ax 2  bx  c  0 then prove that there will be a root of equation
a 2
x  bx  c  0 lying between  and  .
2

9
79. Find a for which 4 t   a  4  2 t  a  0  t  (1,2)
4

12x 3
80. Prove that  1 for x  R and equality holds when x  .
4x 2  9 2

81. Find all x  R for each of which 3  x  a  x 2 has at least one negative solution.

82. (i) Find ‘a’ if x 2  ax  a  3  0 for atleast one x  R

(ii) Find ‘a’ if x 2  ax  a  3  0 for atleast one x  R

83. Find the parameter a  R such that 4 x  a.2 x  a  3  0, for at least one x  R.

84. Find the value of a for which ax 2   a  3 x  1  0 for at least one positive real x.

3
85. Consider, f  x   x 2  x  and g  x   x 2  ax  1, then find the values of ‘a’ for which g  f  x    0 will have
4
no real solution.

86. Let a, b, c be real, if ax 2  bx  c  0 has two real roots  and 

c b
(i) where   1 and   1, then show that 1    0.
a a
c b
(ii) where 2  ,   2, then show that 4   2  0.
a a

6
 1 sec4   3 tan 2 
87. Prove that,  1
3 sec4   tan 2 

88. Solve x 2  4  x 2  9  8

ax 2  4x  5
89. The expression is less than 6 for all real x. Find the greatest integral value of ‘a’.
x 2  2x  4

90.    
When x100 is divided by x 2  3x  2, the remainder is 2 N 1  1 x  2 2 N1 where N is a numerical quantity,
then find N.

n
91. If r
r 1
2

 5r  7  r  2 !   3128  3128 ! 18, then find the value of n.

p m n
92. If m, n, p are positive integers such that m n .n p .p m  3mnp, find the value of
m + n + p.

93. Find the least value of n which 3 + 6 + 9 + …….. to n terms exceeds 1000.

94. If x 3  ax 2  11x  6 and x 3  bx 2  14x  8 have a common factor of the form x 2  px  q, then find (a+b).

95. If a + b + c = 0 , then fine the value of


7 a 2  b 2  c2  a 3
 b3  c3 
7 7 7
a b c

96. Let the number of integral roots of the equation x8  24x 7  18x 5  39x 2  1155  0 , be n then find
2009n  2010.

97. If p is a prime and both roots of the equation x 2  px  444p  0 are integers, find p.

7
 Mathematics (Quadratic Equation & Expression)
1. (i) –5050 (ii) 12582075 2. 1
3. T 4. F
5. T 6. (A)
7. (D) 8. (B)
9. (B) 10. (A)
11. (C) 12. (A)
13. (D) 14. (C)
15. (A) 16. (B)
17. (D) 18. (B)
20. 2 , 2 21. a>1
22. 1210 23. (A)
24. A 25. B
26. (A) 27. (A)
28. (B) 29. (A, B, D)
30. (B) 32. (–, 1)
33. (i) a  (–, –1)  (5/4, + ) (ii) a = –1
(iii) a  (–1, 1)  {5/4} (iv) a = 1
(v) a  [1, 5/4)
34. (D) 35. (C)
36. (A) 37. (C)
38. (B) 39. (C)
40. (C) 41. (C)
42. (A) 43. 3
44. 1 46. (0, 8]
 14 
47. (i) (–, –14)  {4}   ,   (ii) (4, 14/3) (iii) [–14, 4)
3 
48. (A) 49. (A)
50. (C) 51. (B)
77. p   2,5    6,   89. 5
90. 99 91. 3125
92. 6 93. 26
94. 13 95. 12
96. 2010 97. 37