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# Solution- Relations and Functions Test-4 Class 12 Notes | EduRev

## Class 12 : Solution- Relations and Functions Test-4 Class 12 Notes | EduRev

``` Page 1

CBSE TEST PAPER-04
CLASS - XII MATHEMATICS (Relations and Functions)
Topic:- Relations and Functions
1. A function f: X  Y is said to be one – one and onto (bijective), if f is both one – one and
onto.
2. Let x 1, x 2 ?R
If f(x 1) = f(x 2)
4 4
1 2
2 2
1 1
1 2
x x
x x
x x
=
=
± = ±
Not one – one
4
1/4
1/4
( )
y x
x y
f y y
=
= ±
=
Not onto.
1/4
( ) f y y - =
3. f is one – one and onto, Ao that f is invertible with f
-1
= {(3,1) (2, 3) (1, 2)}
4. gof (x) = g[f(x)]
= g (8x
3
)
=
( )
1
3
3
8x
= 2x
5. (f. g) oh
(f. g) h (x)
f[h(x)]. g[h(x)]
foh. goh
6. 3 * 4 = 2 (3) + 4-3 = 7
7. (i)  Each triangle is similar to at well and thus (T1, T1) ? R
?  R is reflexive.
Page 2

CBSE TEST PAPER-04
CLASS - XII MATHEMATICS (Relations and Functions)
Topic:- Relations and Functions
1. A function f: X  Y is said to be one – one and onto (bijective), if f is both one – one and
onto.
2. Let x 1, x 2 ?R
If f(x 1) = f(x 2)
4 4
1 2
2 2
1 1
1 2
x x
x x
x x
=
=
± = ±
Not one – one
4
1/4
1/4
( )
y x
x y
f y y
=
= ±
=
Not onto.
1/4
( ) f y y - =
3. f is one – one and onto, Ao that f is invertible with f
-1
= {(3,1) (2, 3) (1, 2)}
4. gof (x) = g[f(x)]
= g (8x
3
)
=
( )
1
3
3
8x
= 2x
5. (f. g) oh
(f. g) h (x)
f[h(x)]. g[h(x)]
foh. goh
6. 3 * 4 = 2 (3) + 4-3 = 7
7. (i)  Each triangle is similar to at well and thus (T1, T1) ? R
?  R is reflexive.
(ii)  (T1, T2) ? R
? T 1 is similar to T 2
? T 2 is similar to T 1
(T 2, T 1) ? R
R is symmetric
(iii)  T1 is similar to T2 and T2 is similar to T3
? T1 is similar to T3
? (T 1, T 3) ? R
?  R is transitive.
Hence R is equivalence
(II) part T 1 = 3, 4, 5
T2 = 5, 12, 13
T3 = 6, 8, 10
3 4 5 1
6 8 10 2
= = = T 1 is relative to T 3.
8. (a)  a * b = 1
b * a = 1
for all a, b ? N also
(a * b) * c = 1 * c = 1
a * (b * c) = a * (1) = 1 for all, a, b, c R N
Hence R is both associative and commutative
(b)  a * b =
2
a b +
,  b * a =
2
b a +
Hence commutative.
(a * b) * c = *
2
a b
c
+ ? ?
? ?
? ?

=
2
2 4
a b a b c
c
+ + + ? ?
+ =
? ?
? ?
=
2
*( * ) *
2 2
a b
a
a b
a b c a
+ ? ?
+
? ?
+ ? ?
? ?
= =
? ?
? ?
2
4
a b c + +
=
* is not associative.
Page 3

CBSE TEST PAPER-04
CLASS - XII MATHEMATICS (Relations and Functions)
Topic:- Relations and Functions
1. A function f: X  Y is said to be one – one and onto (bijective), if f is both one – one and
onto.
2. Let x 1, x 2 ?R
If f(x 1) = f(x 2)
4 4
1 2
2 2
1 1
1 2
x x
x x
x x
=
=
± = ±
Not one – one
4
1/4
1/4
( )
y x
x y
f y y
=
= ±
=
Not onto.
1/4
( ) f y y - =
3. f is one – one and onto, Ao that f is invertible with f
-1
= {(3,1) (2, 3) (1, 2)}
4. gof (x) = g[f(x)]
= g (8x
3
)
=
( )
1
3
3
8x
= 2x
5. (f. g) oh
(f. g) h (x)
f[h(x)]. g[h(x)]
foh. goh
6. 3 * 4 = 2 (3) + 4-3 = 7
7. (i)  Each triangle is similar to at well and thus (T1, T1) ? R
?  R is reflexive.
(ii)  (T1, T2) ? R
? T 1 is similar to T 2
? T 2 is similar to T 1
(T 2, T 1) ? R
R is symmetric
(iii)  T1 is similar to T2 and T2 is similar to T3
? T1 is similar to T3
? (T 1, T 3) ? R
?  R is transitive.
Hence R is equivalence
(II) part T 1 = 3, 4, 5
T2 = 5, 12, 13
T3 = 6, 8, 10
3 4 5 1
6 8 10 2
= = = T 1 is relative to T 3.
8. (a)  a * b = 1
b * a = 1
for all a, b ? N also
(a * b) * c = 1 * c = 1
a * (b * c) = a * (1) = 1 for all, a, b, c R N
Hence R is both associative and commutative
(b)  a * b =
2
a b +
,  b * a =
2
b a +
Hence commutative.
(a * b) * c = *
2
a b
c
+ ? ?
? ?
? ?

=
2
2 4
a b a b c
c
+ + + ? ?
+ =
? ?
? ?
=
2
*( * ) *
2 2
a b
a
a b
a b c a
+ ? ?
+
? ?
+ ? ?
? ?
= =
? ?
? ?
2
4
a b c + +
=
* is not associative.
9. Let (a1 b1) and (a2, b2) ? A × B
(i)  f(a 1 b 1) = f(a 2, b 2)
b 1 = b 2 and a 1 = a 2
(a 1 b 1) = (a 2, b 2)
Then f(a 1 b 1) = f(a 2, b 2)
(a 1 b 1) = (a 2, b 2) for all
(a 1 b 1) = (a 2, b 2) ? A × B
(ii)  f is injective,
Let (b, a) be an arbitrary
Element of B × A. then b ? B and a ? A
? (a, b) ) ? (A × B)
Thus for all (b, a) ? B × A their exists (a, b) ) ? (A × B)
Hence that
f(a, b) = (b, a)
So f: A × B  B × A
Is an onto function.
Hence bijective
```
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