Question

[ JEE MAINS - 2014 ]
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✨ Irrelevant answers will be reported
✨ Answer with explanation
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Answer 1

Answer :

D) Y

Note :

★ Set : A well defined collection of distinct objects is called a set .

★ Cardinal number / Cardinality : The number of elements/members/objects in a finite set is called cardinal no. / Cardinality .

→ Cardinal no. of a finite set A is denoted by n(A) .

★ Method of representing a set :

a). Roster / Tabular / Listed form

b). Set Builder form

★ Roster form :

→ All elements are listed .

→ Elements are separated by commas .

→ Elements are enclosed within braces { } .

→ The order of writing elements doesn't matter .

→ The elements are not repeated

★ Set builder form :

→ The common properties of elements are written .

→ The elements is described using symbols like x , y , z (mostly x) .

→ Whole description of the elements are enclosed within braces { } .

★ Union of two sets : The union of two sets A and B is the set of all those elements which are either in A or in B or in both .

→ This set is denoted by A U B .

★ Intersection of two sets : The intersection of two sets A and B is the set of all those elements which are in common in both A and B .

→ This set is denoted by A ∩ B .

★ Difference of sets : The difference of two sets A and B in the order ( also called relative complement of B in A ) is the set of all those elements of A which are not the elements of B .

→ It is denoted by (A - B) .

Solution :

• Given : X = { 4ⁿ - 3n - 1 : n € N }

Y = { 9(n - 1) : n € N }

• To find : X U Y = ?

Here ,

X = { 4ⁿ - 3n - 1 : n € N }

For n = 1 , 2 , 3 , 4 , . . .

x = 0 , 9 , 54 , 243 , . . .

→ X = { 0 , 9 , 54 , 243 , . . . }

Clearly ,

The set X contains some non-negative multiples of 9 . ---------(1)

[ Also , any number of the form 4ⁿ - 3n - 1 is always divisible by 9 which can be proved by Principle of mathematical induction . Thus , any number of the form 4ⁿ - 3n - 1 is a non-negative multiple of 9 for n € N . ]

Also ,

We have , Y = { 9(n - 1) : n € N }

For n = 1 , 2 , 3 , 4 , . . .

y = 0 , 9 , 18 , 27 , . . .

→ Y = { 0 , 9 , 18 , 27 , . . . }

Clearly ,

The set Y contains all the non-negative multiples of 9 . ---------(2)

[ Also , it is clear that , 9(n - 1) is a multiple of 9 as 9 is a factor of 9(n - 1) . ]

Now ,

From statements (1) and (2) , we can conclude that , X ⊂ Y .

Thus ,

X U Y = Y

[ °•° If X ⊂ Y , then X U Y = Y ]

Hence ,

Required answer is option D .

Answer 2

Answer:

△ABC is right angled at A. AD is perpendicular to BC. If AB = 5cm, BC = 13cm and AC = 12cm. Find the area of \triangle△ ABC. Also find the Length of AD.

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