CBSE TEST PAPER-03
CLASS - XII MATHEMATICS (Relations and Functions)
[ANSWERS]
Topic:- Relations and Functions
1.
2.
the function f is one – one, for
f(x1) = f(x2)
2x1 = 2x2
x1 = x2
is x1, x2 ∈ R
f(x1) = f(x2)
3 – 4x1 = 3 – 4x2
x1 = x 2
Hence one – one
Y = 3 – 4x
 3− 4 
x=

 4 
 3− 4 
 3− 4 
f
 = 3 − 4

 4 
 4 
=y
Hence onto also.
3.
f is one – one and onto, so that f is invertible with inverse f-1 = {(1, 1) (2, 2) (3, 3)}
4.
fog (x) = f(g x)
= f{|5x – 2|)
= |5x – 2|
5.
f = {(1, a) (2, b) (3, c)}
f-1 = { (a, 1) (b, 2) (c, 3)}
(f -1) -1 = {(1, a) (2, b) (3, c)}
Hence (f-1)-1 = f.
6.
(fog) (x) = f[g(x)]
= f(x – 7)
=x–7+7
=x
(fog) (7) = (7)
7.
(i) (x, x) ∈ R, as x and x have the same no of pages for all x ∈ R ∴ R is reflexive.
(ii) (x, y) R
x and y have the same no. of pages
y and x have the same no. of pages
⇒ (y, x) ∈ R
⇒ (x, y) = (y, x) R is symmetric.
(iii) if (x, y) ∈ R, (y, y) ∈ R
(x, z) ∈ R
∴ R is transitive.
8.
(i)
a * b = a – b + ab
b * a = b – a + ab
a*b ≠ b*a
(ii) a * (b * c) = a * (b – c + bc)
= a – (b – c + bc) + a. (b – c + bc)
= a – b + c – bc + ab – ac + abc
(a * b) * c = (a – b + ab) * c
= [ (a – b + ab) – c ] + ( a – b + ab)
= a- b + ab – c + ac – bc + abc
a * (b * c) ≠ (a * b) * c.
9.
(i)
gof (x) = g[f(x)]
= g (2x + 1)
= (2x + 1)2 – 2
(ii) fog (x) = f (fx)
= f (2x + 1)
= 2(2x + 1) + 1
= 4x + 2 + 1 = 4x + 3
10.
Let x1 x2 ∈ A
Such that f(x1) = f(x2)
x1 − 2 x2 − 2
=
x1 − 3 x2 − 3
x1 = x2
f is one – one
y
x−2
=x
x −3
1
2y − 2
x=
y −1
 3y − 2 
f
= y
 y −1 
Hence onto