Which of the following sequences are in G.P.? (vii) 16, 4, 1, 1/4,..........

Which of the following sequences are in G.P.?

(vii)  16, 4, 1, 1/4,..........

Solution :

First term (a)  = 16, common ratio

r  = t2/t1   =  4/16 r  = 1/4

r  = t2/t1  r  = 1/4

Since the common ratios are same, it is geometric sequence.

Question 3: Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive. Chapter 1 - Relations And Functions

Chapter 1 - Relations And FunctionsNCERT Solutions for Class 12 Science Math

Question 3:

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} asR = {(a, b): b = a + 1} is reflexive, symmetric or transitive.

ANSWER:

Let A = {1, 2, 3, 4, 5, 6}.

A relation R is defined on set A as:

R = {(a, b): b = a + 1}

∴R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)}

We can find (a, a) ∉ R, where a ∈ A.

For instance, 

(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) ∉ R

∴R is not reflexive.

It can be observed that (1, 2) ∈ R, but (2, 1) ∉ R.

∴R is not symmetric.

Now, (1, 2), (2, 3) ∈ R

But, 

(1, 3) ∉ R

∴R is not transitive 

Hence, R is neither reflexive, nor symmetric, nor transitive.

Let f : A -> B be a function defined by f (x) = (x/2) - 1 where A = {2, 4, 6, 10, 12}, B = {0, 1, 2, 4, 5, 9} .

Let f : A -> B be a function defined by f (x) =  (x/2) - 1 where A = {2, 4, 6, 10, 12}, B = {0, 1, 2, 4, 5, 9} .

Represent f by

(i) set of ordered pairs; (ii) a table; (iii) an arrow diagram; (iv) a graph

Solution :

(i) set of ordered pairs

f (x) =  (x/2) - 1

Set of ordered pairs  = {(2, 0) (4, 1) (6, 2) (10, 4) (12, 5)}

(ii) a table

(iii) an arrow diagram;

(iv) a graph

For A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7},show that (i) A U (B ∩ C) = (A U B) ∩ (A U C) (ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

(11) For A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7},show that

(i) A U (B ∩ C) = (A U B) ∩ (A U C)

(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

(iii) Verify using Venn diagrams

Solution:

(i) A U (B ∩ C) = (A U B) ∩ (A U C)

L.H.S

A U (B ∩ C)

(B ∩ C) = {-1,-2,3,4,5,6} ∩ {-1,2,3,4,5,7}

       = {-1,3,4,5}

A U (B ∩ C) = {-3,-1,0,4,6,8,10} U {-1,3,4,5}

            = {-3,-1,0,3,4,5,6,8,10}  --- (1)

R.H.S

(A U B) ∩ (A U C)

(A U B) = {-3,-1,0,4,6,8,10} U {-1,-2,3,4,5,6}

        = {-3,-2,-1,0,3,4,5,6,8,10}

(A U C) = {-3,-1,0,4,6,8,10} U {-1,2,3,4,5,7}

       = {-3,-1,0,2,3,4,5,6,7,8,10}

(AUB)∩(AUC)={-3,-2,-1,0,3,4,5,6,8,10} ∩ {-3,-1,0,2,3,4,5,6,7,8,10}

             = {-3,-1,0,3,4,5,6,8,10}  --- (2)

(1) = (2)

A U (B ∩ C) = (A U B) ∩ (A U C)

Hence proved

---

(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

A = {-3,-1,0,4,6,8,10} B = {-1,-2,3,4,5,6} and C = {-1,2,3,4,5,7}

L.H.S

(B U C) = {-1,-2,3,4,5,6} U {-1,2,3,4,5,7}

       = {-2,-1,2,3,4,5,6,7}

A ∩ (B U C) = {-3,-1,0,4,6,8,10} ∩ {-2,-1,2,3,4,5,6,7}

            = {-1,4,6} ---- (1)

R.H.S

(A ∩ B) = {-3,-1,0,4,6,8,10} ∩ {-1,-2,3,4,5,6}

       = {-1,4,6}

(A ∩ C) = {-3,-1,0,4,6,8,10} ∩ {-1,2,3,4,5,7}

       = {-1,4}

(A ∩ B) U (A ∩ C) = {-1,4,6} U {-1,4}

                    = {-1,4,6} ---- (2)

(1) = (2)

A ∩ (B U C) = (A ∩ B) U (A ∩ C)

Hence proved

For A = {x|x is a prime factor of 42}, B ={x|5 < x ≤ 12, x ∈ N} and C = {1,4,5,6} verify A U (B U C) = (A U B) U C.

Question 7

For A = {x|x is a prime factor of 42}, B ={x|5 < x ≤ 12, x ∈ N} and C = {1,4,5,6} verify A U (B U C) = (A U B) U C.

Solution:

A = {x|x is a prime factor of 42}

A = {2,3,7}

B ={x|5 < x ≤ 12, x ∈ N}

B = {6,7,8,9,10,11,12}

C = {1,4,5,6}

L.H.S

A U (B U C)

(B U C) = {6,7,8,9,10,11,12} U {1,4,5,6}

       = {1,4,5,6,7,8,9,10,11,12}

A U (B U C) = {2,3,7} U {1,4,5,6,7,8,9,10,11,12}

            = {1,2,3,4,5,6,7,8,9,10,11,12}  --- (1)

R.H.S

(A U B) U C

(A U B) = {2,3,7} U {6,7,8,9,10,11,12}

       = {2,3,6,7,8,9,10,11,12}

(A U B) U C = {2,3,6,7,8,9,10,11,12} U {1,4,5,6}

             = {1,2,3,5,6,7,8,9,10,11,12}  ----(2)

(1) = (2)

Verify the commutative property of set intersection for A = {l,m,n,o,2,3,4,7} and B = {2,5,3,-2,m,n,o,p}

Question 6

Verify the commutative property of set intersection for A = {l,m,n,o,2,3,4,7} and B = {2,5,3,-2,m,n,o,p}

Solution:

commutative property of set intersection

A ⋂ B = B ⋂ A

A ⋂ B = {l,m,n,o,2,3,4,7} ⋂ {2,5,3,-2,m,n,o,p}

     = {m,n,o}  --- (1)

B ⋂ A = {2,5,3,-2,m,n,o,p} ⋂ {l,m,n,o,2,3,4,7}

     = {m,n,o}  --- (2)

(1) = (2)

If A = {4, 6, 7, 8, 9} , B = {2, 4, 6} and C = {1, 2, 3, 4, 5, 6}, then find (i) A U (B ∩ C) (ii) A ∩ (B U C) (iii) A \ (C \ B)

Question 4 :

If A = {4, 6, 7, 8, 9} , B = {2, 4, 6} and C = {1, 2, 3, 4, 5, 6}, then find

(i) A U (B ∩ C) (ii) A ∩ (B U C)    (iii) A \ (C \ B)

Solution :

(i) A U (B ∩ C)

(B ∩ C) = {2, 4, 6} ∩ {1, 2, 3, 4, 5, 6}

 = {2, 4, 6}

A U (B ∩ C) = {4, 6, 7, 8, 9} U {2, 4, 6}

 = {2, 4, 6, 7, 8, 9}

(ii) A ∩ (B U C)

(B U C) = {2 ,4, 6} U {1, 2, 3, 4, 5, 6}

       = {1, 2, 3, 4, 5, 6}

A ∩ (B U C) = {4, 6, 7, 8, 9} ∩ {1, 2, 3, 4, 5, 6}

            = {4, 6}

(iii) A \ (C \ B)

C \ B = {1, 2, 3, 4, 5, 6} \ {2, 4, 6}

    = {1, 3, 5}

A \ (C \ B) = {4, 6, 7, 8, 9} \ {1, 3, 5}

           = {4, 6, 7, 8, 9}