All algebra questions

ALGEBRA
1) Sets
2) Relation
3) Complex Numbers
4) Quadratic Equations
5) Permutation
6) Combination
7) Binomial Theorem
8) Sequence and Series
9) Matrices
10) Determinants
11) System of Equations
12) Probability
13) Conditional Probability
14) Baye’s Theorem
15) Probability Distribution
16) Statistics
17) Logarithms
18) Mathematical Induction

, 2 MATHEMATICS  FOR N.D.A AND N.A



ALGEBRA
Set
12. Let A = {1, 2}, B = {{1}, {2}}, C = {{1}}, {1, 2}. Then
1. Write the set builder form of A = {–1, 1) which of the following relation is true?
a. A = {x : x is an integer} a. A = B b. B  C c. A  C d. A  C
b. A = {x : x is a root of the equation x2 + 1 = 0}
c. A = {x : x is a real number} 13. Two finite sets A and B have m and n elements
d. A = {x : x is a root of the equation x2 = 1} respectively. If the total number of subsets of A is 112
more than the total number of subsets of B, then the value
2. Which of the following set is an empty set? of m is
a. {x|x is a real number and x2 – 1 = 0} a. 7 b. 9 c. 10 d. 12
b. {x|x is a real number and x2 + 3 = 0}
c. {x|x is a real number and x2 – 9 = 0} 14. If the set A contains 5 elements, then the number of
d. {x|x is a real number and x2 = x + 2} elements in the power set P(A) is equal to
3. Which of the following set is empty? a. 32 b. 25 c. 16 d. 8
a. {x  R|x2 + x + 1 = 0}
b. {x  R|x2 = 9 and 2x = 6} 15. A set contains n elements. The power set contains
c. {x  R|x + 4 = 4} d. {x  R|2x + 1 = 3} a. n elements b. 2n elements
c. 2
n elements d. None of these
4. The set A = {x|x is a real number and x2 = 16 and 2x = 6} 16. If n(P) = 8, n(Q) = 10 and n(R) = 5 (‘n’ denotes carinality)
is equal to for three disjoint sets P, Q, R then n(P  Q  R) =
a. {4} b. {3} a. 23 b. 20 c. 18 d. 15
c.  d. None of these
17. If A and B are finite sets and A  B , then
5. The set A = {x : |2x + 3| < 7} is equal to the set a. n(A  B)  n(B) b. n(A  B)  n(B)
a. D = {x : 0 < x + 5 < 7} b. B = {x : –3 < x < 7} c. n(A  B)   d. n(A  B)  n(A)
c. E = {x : –7 < x < 7} d. C = {x : –13 < 2x < 4}
18. If X = {–2, –1, 0, 1, 2, 3, 4, 5, 6, 7, 8} and A{x:|x – 2|  3,
6. If X = {4n – 3n – 1|n  N} and Y = {9(n – 1)|n  N}, then x is an integer}, then X – A =
a. X  Y b. Y  X a. {–2, 6, 7, 8}
c. X = Y d. None of these b. {–2, –1, 1, 2, 3, 4, 5, 6}
c. {–1, 0, 1, 2, 3, 4, 5, 7, 8}
7. If A = {5n – 4n – 1 : n  N} and B = {16(n – 1) : n  N}, d. {–2, –1, 2, 3, 6, 7, 8}
then
19. The set (A | B)  (B | A) is equal to
a. A = B b. A  B  
a. [A | (A  B)]  [B | (A  B)]
c. A  B d. B  A
b. (A  B) | (A  B)
8. If a set A has 4 elements, then the total number of proper c. A | (A  B) d. (A  B) | (A  B)
subset of set A, is 20. Set A and B have 2 and 6 elements respectively. What can
a. 16 b. 14 c. 15 d. 17 be the minimum number of elements in A  B ?
a. 18 b. 9 c. 6 d. 3
9. The number of proper subsets of a set having n + 1
elements is 21. Let X = {1, 2, 3, ...., 10} and A = {1, 2, 3, 4, 5}. Then the
a. 2n + 1 b. 2n + 1 – 1 number of subsets B of X such that A – B = {4} is
n + 1 d. 2n – 2
c. 2 –2 a. 25 b. 24 c. 25 – 1 d. 1
10. Let A = {1, 2, {a, b}, 3, 4} which among the following 22. The shaded region in the figure represents
statements is incorrect?
a. {a, b}  A b. {a, b}  A U
c. {{a, b}}  A d. {1, 2}  A A B

11. The number of subsets of A = {2, 4, 6, 8} without empty
set is .......... a. A  B b. A  B
a. 14 b. 16 c. 15 d. 12 c. B – A d. None of these

,Algebra 3
31. If n(A) = 8 and n(A  B) = 2 then n((A  B)  A) is
23. If sets A and B are defined as
equal to
A = {(x, y) : y = 1/x, x  0, x  R}
a. 2 b. 4 c. 6 d. 8
B = {(x, y) : y = –x, x  R}, then
a. A  B  A b. A  B  B
32. 25 people for programme A, 50 people for programme B, 10
c. A  B   d. None of these people for both. So number of employee employed for
only A is
24. Let Z denote the set of all integers and a. 15 b. 20 c. 35 d. 40
A = {(a, b) : a2 + 3b2 = 28, a, b  Z} and
B = {(a, b) : a > b, a, b  Z}. Then the number of elements 33. A and B are subsets of universal set U such that n(U) =
in A  B is 800, n(A) = 300, n(B) = 400 & n(A  B) = 100. The
a. 2 b. 3 c. 4 d. 6 number of elements in the set A c  Bc is
a. 100 b. 200 c. 300 d. 400
25. For any two sets A and B, A – (A – B) equals
a. B b. A – B c. A  B d. A c  Bc 34. If aN = {ax : x  N} and bN  cN  dN , where b, c  N
are relatively prime, then
26. In a certain town, 25% of the families own a phone and a. b = cd b. c = bd
15% own a car, 65% families own neither a phone nor a car c. d = bc d. None of these
and 2000 families own both a car and a phone. Consider
the following three statements: 35. If S is a set with 10 elements and
1. 5% families own both a car and a phone A = {(x, y) : x, y  S, x  y}, then the number of elements
2. 35% families own either a car or a phone in A is
3. 40,000 families live in the town a. 100 b. 90 c. 50 d. 45
Then
a. only 1 and 2 are correct b. only 1 and 3 are correct Relation
c. only 2 and 3 are correct d. all 1, 2 and 3 are correct
36. The caretsian product A  A has 9 elements among which
27. In a class of 80 students numbered 1 to 80, all odd two elements are found (–1, 0) and (0, 1), then set A?
numbered students opt for Cricket, students whose a. {1, 0} b. {1, –1, 0} c. {0, –1} d. {1, –1}
numbers are divisible by 5 opt for Football and those
whose numbers are divisible by 7 opt for Hockey. The 37. For non-empty sets A and B, if A  B , then
number of students who do not opt any of the three (A  B)  (B  A) equals
games, is a. A  B b. A  A
a. 13 b. 24 c. 28 d. 52 c. B  B d. None of these

28. There is a group of 265 persons who like either singing or 38. If A and B have n elements in common, then the number
dancing or painting. In this group 200 like singing, 110 like of elements common to A  B and B  A is
dancing and 55 like painting. if 60 persons like both a. 0 b. n c. 2n d. n 2
singing and dancing, 30 like both singing and painting and
10 like all three activities, then the number of persons who 39. If n(A) denotes the number of elements to set A and if
like only dancing and painting is n(A) = 4, n(B) = 5 and n(A  B) = 3, then
a. 10 b. 20 c. 30 d. 40 n{(A  B)  (B  A)} =
a. 8 b. 9 c. 10 d. 11
29. A survey shoes that 63% of the Americans like cheese
whereas 76% like apples. If x% of the Americans like both 40. Let A and B be finite sets such that n(A) = 3. If the total
cheese and apples, then the value of x is number of relations that can be defined from A to B is 4096,
a. 39  x  63 b. 63 c. 39 d. 139  x then n(B) =
a. 5 b. 4 c. 6 d. 8
30. There are 100 students in a class. In an examination, 50 of
41. If R is a relation on a finite set having n elements, then the
them failed in Mathematics, 45 failed in Physics, 40 failed
number of relations on A is
in Biology and 32 failed in exactly two of three subjects. 2
n
Only one student passed in all the subjects. Then the a. 2n b. 2 c. n 2 d. n n
sumber of students failing in all the three subjects
a. is 12 b. is 4 42. Let A = {1, 2, 3, 4) and R be the relation on A defined by
c. is 2 {(a, b) : a, b  A, a  b is an even number}, then find the
d. cannot determined from the given information range of R
a. {1, 2, 3, 4} b. {2, 4} c. {2, 3, 4} d. {1, 2, 4}

, 4 MATHEMATICS  FOR N.D.A AND N.A


43. Let the number of elements of the sets A and B be p and 53. On R, the set of real numbers, a relation  is defined as
q respectively. Then the number of relations from the set ‘ab if and only if 1 + ab > 0. Then
A to the set B is a.  is an equivalence relation
a. 2p + q b. 2pq c. p + q d. pq b.  is reflexive and transitive but not symmetric
c.  is reflexive and symmetric but not transitive
44. Let S = {(a, b) : b = |a – 1|, a  Z and |a| < 3} where Z d.  is symmetric
denotes the set of integers. Then the range of S is
a. {1, 2, 3} b. {–1, 2, 3, 1} 54. Let R be a reflexive relation on a finite set A having n
c. {0, 1, 2, 3} d. {–1, –2, –3, –4} elements and let there be m ordered pairs in R then
a. m  n b. m  n
45. The relation R defined on set A = {x : |x| < 3, x  I} by c. m = n d. None of these
R = {(x, y) : y = |x|} is
a. {(–2, 2), (–1, 1), (0, 0), (1, 1), (2, 2)} 55. Let R be a relation defined on the set Z of all integers and
b. {(–2, –2), (–2, 2), (–1, 1), (0, 0), (1, –1), (1, 2), (2, –1), xRy when x + 2y is divisible by 3. Then
(2, –2)} a. R is not transitive b. R is symmetric only
c. {(0, 0), (1, 1), (2, 2)} d. None of these c. R is an equivalence relation
d. R is not an equivalence relation
46. If n(A) = 5 and n(B) = 7, then the number of relations on
A  B is 56. The number of equivalence relations on the set {1, 2, 3}
a. 235 b. 249 c. 225 d. 270 containing (1, 2) and (2, 1) is
a. 3 b. 1
47. Let A = {x, y, z} and B = {a, b, c, d}. Which one of the c. 2 d. None of these
following is not a relation from A to B? 57.
a. {(x, a), (x, c)} b. {(y, c), (y, d)}
c. {(z, a), (z, d)} d. {(z, b), (y, b), (a, d)} 58. Let X = {a, b, c, d, e} and R = {(a, a), (b, b), (c, c), (a, b),
(b, a)}. Then the relation R on X is
48. R is a relation on N given by R = {(x, y)(4x + 3y) = 20}. a. reflexive and symmetric
Which of the following belongs to R? b. not reflexive but symmetric only
a. (3, 4) b. (2, 4) c. (–4, 12) d. (5, 0) c. symmetric and transitive, but not reflexive
d. reflexive but not transitive
49. A set A has 5 elements. Then the maximum number of
relations on A (inclusing empty relation) is 59. Let r be a relation over the set N  N and it is defined by
a. 5 b. 25 c. 225 d. 25 (a, b)r (c, d)  a + d = b + c. Then r is
a. reflexive only b. symmetric only
50. On the set R of real numbers we define xRy if and only if c. transitive only d. an equivalence relation
xy  0. Then the relation P is
a. reflexive but not symmetric 60. A relation  on the set of real number R is defined as
b. symmetric but not reflexive follows: “x  y if and only if xy > 0”. Then which os the
c. transitive but not reflexive following is/are true?
d. reflexive and symmetric but not transitive a.  is reflexive and symmetric
b.  is symmetric but not reflexive
51. On R, the relation  be defined by ‘xy holds if and only c.  is symmetric and transitive
if x – y is zero or irrational’. Then d.  is an equivalence relation
a.  is reflexive and transitive but not symmetric
b.  is reflexive and symmetric but not transitive 61. Let R be the relation on the set R of all real numbers
c.  is symmetric and transitive but not reflexive defined by aRb iff |a – b|  1. Then R is
d.  is equivalence relation a. reflexive b. transitive
c. anti-symmetric d. None of these
52. On set A = {1, 2, 3}, relations R and S are given by
R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} 62. If A = {1, 2, 3, 4}, then which one of the following is
S = {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)}. Then reflexive?
a. R  S is an equivalence relation a. {(1, 1), (2, 3), (3, 3)} b. {(1, 1), (2, 2), (3, 3)(4, 4)}
b. R  S is reflexive and transitive but not symmetric c. {(1, 2), (2, 1), (3, 2)(2, 3)} d. {(1, 2), (1, 3), (1, 4)}
63. Let S be the set of all real numbers. A relation R has been
c. R  S is reflexive and symmetric but not transitive
defined on S by aRb  |a – b|  1, then R is
d. R  S is symmetric and transitive but not reflexive
a. symmetric and transitive but not reflexive
b. reflexive and transitive but not symmetric