If A = {x $$\in$$ R : |x $$-$$ 2| > 1}, B = {x $$\in$$ R : $$\sqrt {{x^2} - 3} $$ > 1}, C = {x $$\in$$ R : |x $$-$$ 4| $$\ge$$ 2} and Z is the set of all integers, then the number of subsets of the set (A $$\cap$$ B $$\cap$$ C)c $$\cap$$ Z is ________________.
Your Input ________
Answer
Correct Answer is 256
Explanation
A = ($$-$$$$\infty$$, 1) $$\cup$$ (3, $$\infty$$)
B = ($$-$$$$\infty$$, $$-$$2) $$\cup$$ (2, $$\infty$$)
C = ($$-$$$$\infty$$, 2] $$\cup$$ [6, $$\infty$$)
So, A $$\cap$$ B $$\cap$$ C = ($$-$$$$\infty$$, $$-$$2) $$\cup$$ [6, $$\infty$$)
z $$\cap$$ (A $$\cap$$ B $$\cap$$ C)' = {$$-$$2, $$-$$1, 0, $$-$$1, 2, 3, 4, 5}
Hence, no. of its subsets = 28 = 256.
2
JEE Main 2021 (Online) 27th July Morning Shift
Numerical
Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S $$\to$$ S such that f(m . n) = f(m) . f(n) for every m, n $$\in$$ S and m . n $$\in$$ S is equal to _____________.
Your Input ________
Answer
Correct Answer is 490
Explanation
F(mn) = f(m) . f(n)
Put m = 1 f(n) = f(1) . f(n) $$\Rightarrow$$ f(1) = 1
Let A = {n $$\in$$ N | n2 $$\le$$ n + 10,000}, B = {3k + 1 | k$$\in$$ N} an dC = {2k | k$$\in$$N}, then the sum of all the elements of the set A $$\cap$$(B $$-$$ C) is equal to _____________.
Your Input ________
Answer
Correct Answer is 832
Explanation
B $$-$$ C $$ \equiv $$ {7, 13, 19, ......, 97, .......}