3. Suppose set \( A \) consists of first 250 natural numbers that are multiples of 3 and set \( B \) consists of first 200 even natural numbers. How many elements does \( A \cup B \) have? (a) 324 (b) 364 (c) 384 (d) 400

We have A a set of natural numbers which are multiples of 2 M B = set of natural numbers which are multiples of 3 C = set of natural...

Correct Answer - B According to the question `2^m-2^m= 112 ` `rArr 2^n (2^(m-n)-1)= 2^4 xx 7` Comparing exponets on both sides ,we get `2^n-2^4 and 2^(m-n)-1= 7 ` Now ,`2^n=...

Correct Answer - C `B=B cap (A cup B)` `=B cap (A cup C)" " (therefore A cup B= A cup C)` `=(B cap A)cup (B cap C ) ` `=...

Correct Answer - A `(a)` Writing `((200),(r ))=((200),(200-r))`, we have `((100),(0))((200),(50))+((100),(1))((200),(49))+((100),(2))((200),(48))+......+((100),(50))((200),(0))` `=` Coefficient of `x^(50)` in the expansion of `(1+x)^(100)(x+1)^(200)` `=` Coefficient of `x^(50)` in the binomial expansion of `(1+x)^(300)` `=((300),(50))`

Given : n(X)=6  n(y)=5  n(z)=4     Also,the elements are distinct  Therefore,these three are disjoint sets ==>n(X∩Z) =0 -------- (1) Now,her S=(X-Y)∪Z=X∪Z [Because X∩Z=∅==>X-Z=X] ==>n(S)=n(X∪Z) ==>n(S)=n(X)+n(Z)-n(X∩Z) ==>n(S)=n(X)+n(Z)-0 [From (1)] ==>n(S)=6+4=10 Therefore, Number of proper subsets of S=2n(S)-1                ...

Correct Answer - `mu = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12---------, 99}`

Correct Answer - (i),(iv),(v),(vi)

Correct Answer - (i) `{1,2,3,4,5,6,7,8}` (ii) `{4,5,6,7,8,9,10,11}` (iii) ` {1,2,3,4,5,7,8,9,10,11}` ` (iv) `{4,5,6,7,8,10,11,12,13,14}` (v) `{1,2,3,4,5,6,7,8,9,10,11}` (vi) `{7,8}` (vii) ` {4,5,10,11,12,13,14}` (viii) `{4,5,7,8}` (ix) ` {7,8,9,10,11}`

Correct Answer - (i) `{1,(1)/(2).(1)/(3),(1)/(4),(1)/(6),(1)/(7)/(1)/(8)} ` (ii) `{(1)/(2),(1)/(4),(1)/(6)}` (iii) {1,(1)/(3) ,(1)/(3),(1)/(5),(1)/(7)}` (iv) `{(1)/(8)}`

Correct Answer - ` (i) 1/2 (ii) 3/4 `