Download Class XI th – Sets

Class XIth – Sets
Q 1.
If A = {0,1,2,3,4,5,6, 7,8,9,10}, then insert the appropriate symbol  or  in each of the
following blank spaces:
(i) 4.....A
(ii) -4......A
(iii) 12 ....A
(iv) 9 ....A
(v) 0.....A
(vi) -2.....A
1. (i)  (ii)  (iii)  (iv)  (v)  (vi) 
Q 2.
Describe each of the following sets in Roster form
(i) {x :x is a positive integer and a divisor of 9} (ii) {x : x  Z and | x | ≤2}


(iii) {x :x is a letter of the word 'PROPORTION'} (iv)  x :
n

and 1  n  3, where n  N 
n 1

2
1 2 3 

 2 5 10 
2. (i) {1, 3, 9} (ii) {-2, -1, 0, 1, 2} (iii) {P, R, O, T, I, N} (iv)  , ,
Q 3.
Match each of the set on the left described in the roster form with the same set on the
right described in the set-builder form
(i) {P, R, I, N, C, A, L}
(ii) {0}
(a) {x : x is a positive integer and is a divisor of 18}
(b) {x :x is an integer and x2 - 9 = 0}
(iii) {1,2,3,6,9,18}
(c) {x : x is an integer and x + 1 = 1}
(iv) {-3,3}
(d) {x : x is a letter of the word 'PRINCIPAL'.}
Q4.
1 2 3 4 5 6 7 8 9 
 in the set-builder form.
 2 3 4 5 6 7 8 9 10 
Write the set  , , , , , , , ,


4.  x : x 
Q 5.
Describe the following sets in set-builder form:
(i) A = {1, 2,3,4,5, 6};
(ii) B = {1,1/2, 1/3, 1/4, 1/5,...};
(iii) C = {0,3, 6, 9,12,...};
(iv) D = {10,11,12,13,14,15};
(v) E = {0};
(vi) {1,4,9,16, ...,100}
(vii) {2,4,6,8,.....}
(viii) {5,25,125,625}
n

, n  N, n  9 
n 1

5. (i) {x : x  N, x < 7} (ii) {x:x = 1/n, x  N} (iii) {x: x = 3n, n  Z+} (iv) {x : x  N, 9 < x < 16} (v){x:x
= 0) (vi) {x2 :x  N, 1 ≤ x ≤ 10} (vii) {x : x = 2n, n  N} (viii) {5n : n  N, 1 ≤ n ≤ 4}
Q 6.
Are the following pairs of sets equal ? Give reasons.
(i) A = {2, 3}, B = {x : x is a solution of x2 + 5x + 6 = 0};
(ii) A = {x : x is a letter of the word "WOLF"}; B = {x : x is a letter of the word "FOLLOW"}.
6. (i) No (ii) Yes
Q 7.
Let A, B and C be three sets. If A  B and B  C, is it true that A  C? If not give an
example.
Q 8.
Prove that A   implies A = .
Q 9.
In each of the following, determine whether the statement is true or false. If it is true,
prove it. If it is false, give an example.
(i) If x  A and A  B, then x  B
(ii) If A  B and B  C, then A  C
(iii) If A  B and B  C, then A  C
(iv) If A  B and B  C, then A  C
(v) If x  A and A  B, then x  B
(vi) If A  B and x  B, then x  A
Ans 9. (i) False (ii) False (iii) True (iv) False (v) False (vi) True
Q 10. Which of the following statements are true ? Give reason to support your answer
(i) For any two sets A and B either A  B or B  A;
(ii) Every subset of an infinite set is infinite;
(iii) Every subset of a finite set is finite;
(iv) Every set has a proper subset;
(v) {a, b, a, b, a, b,...} is an infinite set;
(vi) {a, b, c} and {1,2,3} are equivalent sets;
(vii) A set can have infinitely many subsets.
Ans 10. (i) F, A = {1, 2,3}, B = {a, b} (ii) F, A = {1,2} is a finite subset of N. (iii) T (iv) F,  does not
have a proper subset (v) F, Given set = {a, b} (vi) T (vii) F
Q 11. If A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}, C = {7, 8, 9, 10,11} and D = {10, 11, 12,13,14},
find;
(i) A  B
(ii) A  C
(iii) B  C
(iv) B  D
(iv) A  B  C
(vi) A  B  D
(vii) B  C  D
(viii) A  (B C)
(ix) (A  B)  (B  C)
(x) (A  D)  (B  C).
Ans 11. (i) {1,2,3,4,5,6,7,8} (ii) {1,2,3,4,5,7,8,9,10,11} (iii) {4, 5, 6, 7, 8,9,10,11} (iv) {4, 5, 6, 7,
8,10,11,12,13,14} (v) {1,2,3,4,5,6,7,8,9,10,11} (vi) {1,2,3,4,5,6,7,8,10,11,12,13,14}
(vii) {4, 5, 6, 7, 8, 9,10,11,12,13,14} (viii) {4, 5} (ix) 0 (x) {4, 5,10, 11}
Q 12. Let A = {3, 6,12,15,18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8,10,12,14,16} and D =
{5,10,15, 20}. Find:
(i) A - B
(ii) A - C
(iii) A - D
(iv) B - A
(v) C - A
(vi) D - A
(vii) B - C
(viii) B – D
Ans 12. (i) {3,6,15,18,21} (ii) {3,15,18,21} (iii) {3,6,12,18,21} (iv) {4,8,16,20} (v) {2,4,8,10,14,16}
(vi) {5,10,20} (vii) {20} (viii) {4,8,12,16}
Q 13. Let A and B be sets, if A  X = B  X =  and A  X = B  X for some set X, prove that. A
= B.
Q 14. If a  N such that aN = {ax : x  N}. Describe the set 3N  7N.
Ans 14. 21N
Q 15. Suppose A1, A2,.......... A30 are thirty sets each with five elements and B1, B2,.,., Bn are n
30
sets each with three elements. Let
i 1
Ai 
n
B j  S . Assume that each element of S belongs to
j1
exactly ten of the Ai's and exactly 9 f Bj's. Find n.
Ans 15. n = 45
Q 16. If U = {2,3,5,7,9} is the universal set and A = {3, 7}, B = {2,5,7,9}, then prove that:
(i) (A  B)' = A'  B'
(ii) (A  B)' = A'  B'.
Q 17. For any two sets A and B, prove that :A  B =   A  B'.
Q 18. For any two sets A and B, prove that: A' - B' = B - A
Q 19. If A, B, C are three sets such that A  B, then prove that C - B  C – A
Q 20. In a group of 800 people, 550 can speak Hindi and 450 can speak English. How
many can speak both Hindi and English ?
Ans 20. 200
Q 21. There are 200 individuals with a skin disorder, 120 has been exposed to chemical
C1, 50 to chemical C2 and 30 to both the chemicals C1 and C2. Find the number of
individuals exposed to (i) chemical C1 or chemical C2 (ii) chemical C1 but not chemical
C2 (iii) chemical C2 but not chemical C1.
Q 22. Out of 500 car owners investigated, 400 owned Maruti car and 200 owned. Hyundai car;
50 owned both cars. Is this data correct ?
Ans 22. Incorrect
Q 23. If A, B and C are three sets and U is the universal set such that n(U) = 700, n(A) = 200,
Ans 23. 300
n(B) = 300 and n(A  B) = 100. Find n (A'  B').
Q 24. A survey shows that 63% of the Americans like cheese whereas 76% like apples. If x% of
the Americans like both cheese and apples, find the value of x.
Ans 24. 39 ≤ x ≤ 63
Q 25. A college awarded 38 medals in Football, 15 in Basketball and 20 to Cricket. If, these
medals went to a total of 58 men and only three men got medals in all the three sports, how
many received medals in exactly two of the three sports ?
Ans 25. 9
Class XI th – Relations and Functions
Assignment
Q 1.
Q 2.
Q 3.
Q 4.
If A = {1, 2, 3}, B = {3, 4} and C = {4, 5, 6}, find :
(i) A × (B  C)
(ii) A × (B  C)
(iii) (A × B)  (B × C)
Let R be the set of all real numbers. What does (R × R) represent ?
Express {(x, y) : x2 + y2 = 25, where x, y  W} as a set of ordered pairs.
Find the values of a and b, when:
(i) (a + 3, b - 2) = (5, 1)
(ii) (a + b, 2b - 3) = (4, -5)
(iii)
Q
Q
Q
Q
1 5 2
a
  1, b     , 
3  3 3
3
(iv) (a - 2, 2b + 1) = (b - 1, a + 2)
5.
6.
7.
8.
If A × B = {(-2, 3), (-2, 4), (0, 3), (0, 4), (3, 3), (3, 4)}, find A and B.
Let A = {2, 3} and B = {4, 5}. Find (A × B). How many subsets will (A × B) have?
If A  B, prove that (A × C)  (B × C).
Let A = {1, 2,3} and B = {2,4, 6}.
Show that R = {(1,2), (1,4), (3, 2), (3,4)} is a relation from A to B.
Find (i) dom (R), (ii) co-domain (R), (iii) range (R). Depict the above relation by an arrow diagram.
Q 9.
Let A = {1, 3} and B = {2,3,4}. Find the number of relations from A to B.
Q 10. Find the domain and range of each of the following relations:
(i) R = {(-1,1), (1,1), (2,4), (-2,4), (3, 9)} (ii)R =
 1 

 x,  : x is an int eger, 0  x  6 
 x 

(iii) R = {(x, y) : x and y are integers and xy = 4} (iv) R = {(x, y) : x, y  Z and x2 + y2 = 25}
Q11.
Let A = (1,2,3,5} and B = {4, 6,9}.
Let R = {(x, y): | x – y | is odd, x  A and y  B}. Write R in the roster form.
Q 12. Let A = {1, 2, 3,4, 6} and let R = {(a, b): a, b  A and a divides b}.
(i) Write R in the roster form.
(ii) Find dom (R) and range (R).
Q 13. Let A = {1,2,3,4,5} and B = {1, 2,3,4}.
Let R be the relation, 'is greater than' from A to B. Write R as a set of ordered pairs.
Find dom (R) and range (R).
Q 14. Let f(x) = x2 and g(x) = (3x + 2) be two real functions. Then, find :
(i) (f + g)(x)
(ii) (f – g)(x)
(iii) (fg)(x)
(iv)
f 
  (x)
g
x 2  2x  1
Q 15. Find the domain of the real function, f (x)  2
.
x  7x  12
Q 16. Find the domain and range of the real function, f(x) = x2.
Q 17. Find the domain and range of the real function f (x) 
1
.
(x  3)
Q 18. Find the domain and range of the real function f (x) 
9  x2 .
1
Q 19. Find the domain and range of the real function f (x)
.
(1  x 2 )
Q 20. Find the domain and range of the real function f(x) = - | x |.
Q 21. The function F(x) =
9x
 32 is formula to convert xCto Fahrenheit units. Find
5
(i) F(0),
(ii) F(-10)
(iii) the value of x when F(x) = 212.
Q 22. Let A = {-1, 1, 2, 3} and B = {1, 4, 9,16}.
Let f= {(x, y): x  A, y  B and y = x2}. ,
Show that f is a function from A to B. Find its domain and range.
Q 23. Let A = {2, 3, 5, 7) and B = {3, 5, 9,13,15}.
Let f = {(x, y): x  A, y  B and y = 2x - 1}.
Write f in the roster form. Show that f is a function from A to B. Find the domain and range of f.
Q 24. If f(x) = x 3, find the value of
{f (5)  f (1)}
.
(5  1)
Q 25. Find the domain of the real-valued function f(x) =
Q 26. Find the domain of the real function f(x) =
x 2  2x  3
.
x 2  5x  6
x2  4 .
Answer
1. (i) {(1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5), (3, 6)}.
(ii) {(1, 4), (2, 4), (3, 4)}
(iii) {(3, 4)}
3. {(0, 5), (3, 4), (4, 3), (5, 0)}.
4. (i) a = 2;b = 3 (ii) a = 5,b = -1 (iii) a = 2, b = 1 (iv) a = 3, b = 2
5. A = {-2, 0, 3} and B = {3, 4}
6. A × B = {(2, 4), (2, 5), (3, 4), (3, 5)}; number of subsets of (A × B) = 2 4 = 16
10. (i) dom (R) = {-2, -1, 1, 2, 3), range (R) = {1, 4, 9)
(ii) dom (R) = {1, 2, 3, 4, 5}, range (R) =
 1 1 1 1
1, , , , 
 2 3 4 5
(iii) dom (R) = {-4, -2, -1,1, 2,4} = range (R)
(iv) dom (R) = {-5, -4, -3, 0, 3,4, 5} = range (R)
11. R = {(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)}
12. (i) R = {(1,1), (1,2), (1,3), (1,4), (1,6), (2,2), (2,4), (2,6), (3,3), (3,6), (4,4), (6,6)}
(ii) dom (R) = {1, ,2 3, 4, 6} and range (R) = {1, 2, 3, 4, 6}
13. R = {(2, 1), (3, 1), (3,2), (4,2), (4, 3), (5, 1), (5,2), (5,3), (5, 4)}
14. (i) x  3x  2 , (ii) (x 2  3x  2) , (iii) 3x  2x
2
15.
16.
17.
19.
20.
21.
22.
23.
24.
25.
26.
3
2
(iv) 
2
3
R – {3, 4}
[0, ]
R = {0}18. [0, 3]
[1, ]
[-, 0]
(i) 32F
(ii) 14F,
(iii) 100F.
Dom (f) = {-1,1, 2, 3} and range (f) = {1, 4, 9}
f = {(2,3), (3,5), (5,9), (7,13)}, dom (f) = {2,3,5, 7}, range (f) = {3,5,9,13}.
31
R – {2, 3}
[-, -2] [2, ]
Class XIth – Complex Numbers and Quadratic Equations
Q 1.
Prove that :
(i) in + in+1 + in+2 + in+3 = 0
(ii) i107 + i112 + i117 + i122 = 0
4
 1
(iii) (1 + i)4 ×  1   = 16
 i
Ans 1. (i) 0,
Q 2.
(ii) 0,
(iii) 16.
Explain the fallacy :
1  (i  i)  1  1  (1)  (1)  1  1 .
Q 3.
Evaluate : 4 4  5 9  3 16
Q 4.
Find the modulus of 
Q 5.
If z1 = (1 – i) and z2 = (-2 + 4i), find Im 
Q 6.
1 i 
Find the least positive integral value of m for which 
 =1.
 1 i 
Q 8.
If (x + iy)3 = (u + iv) then show that
Ans 3. 11i
 1 i 1 i 


 1 i 1 i 
Ans 4. 2
 z1z 2 
.
 z1 
Ans 5. 2
m
u v
   = 4(x2 - y2).
x y
Ans 6. m = 4
Q 9.
If (a + ib) =
1 i
then prove that (a2 + b2) = 1.
1 i
Q 10. If (a + ib) (c + id)(e + if)(g + ih) = (A + iB) then show that
(a2 + b2)(c2 + d2)(e2 + f2)(g2 + h2) = (A2 + B2).
Q 11. If | z1 | = | z2 | = | z3 | = ….= | zn | = 1 then prove that
1 1 1
1
   .... 
= | z1 + z2 + z3 +…. zn|.
z1 z 2 z3
zn
Q 12. Represent each of the following complex numbers in the polar form :
(i) ( 3 + i)
(ii) (1 + i)


Ans 12. (i) 2  cos


 i sin 
6
6
(ii)



2  cos  i sin 
4
4

Q 13. Represent each of the following complex numbers in the polar form 1  3i
ANS 13.
   
    
2 cos    i sin    , 2,
3
 3 
  3 
Q 14. Represent each of the following complex numbers in the polar form
Ans 14.
(5  i)
(2  3i)




2  cos  i sin  , 2,
4
4
4

 3  2i 3  2i 
,

5 
 5
Q 15. Solve 25x2 – 30x + 11 = 0
Ans 15. 
Q 16. Solve 8x2 + 2x + 1 = 0
Ans 16. 
Q 17. 27x2 + 10x +1 = 0
Ans 17.
Q 18. 2x2 -
Ans 18. 
3x + 1 = 0
Q 19. Evaluate :
Q 20. Evaluate
5  12i
7  30 2
 1  7i 1  7i 
,

8
8


 5  2i 5  2i 
,


27 
 27

 3  5i 3  5i 

,

4
4




Ans 19. (2 + 3i) or (-2 – 3i)
Ans 20. (5  3 2i)