Download FAS, Mathematics Assignment - Class XII CH

FAS, Mathematics Assignment - Class XII
CH-1/ RELATIONS AND FUNCTIONS
VERY SHORT ANSWER QUESTIONS (1 MARK)
1. Let A be the set of all 46 students of XII in a school. Let f: A → N be a function defined
by f(x) = Roll number of the student XII. Show that ‘f’is one-one but not onto.
2. Let f: Z → Z: f(x) = 2x. Find g: Z → Z: g o f = IZ.
Ans: y/2 where 2x=y
3. Let ‘f’ be a real valued function defined by f(x) = 4x + 3. Find the real valued function
‘g’ such that gof = fog = IR.
Ans: (x-3)/2
3 1/3
4. If f: R ⟶R: f(x) = (3 – x ) , show that (f o f) (x) = x.
5. Show that the relation R:{1, 2, 3} → {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3), (1, 2),
(2, 3)} is reflexive but neither symmetric nor transitive.
6. If f(x) = x3 and g(x)= x2, then find fog and gof. If fog = gof?
7. Show that the function f: R → R: f(x) = x2 is neither one-one nor onto.
8. If the binary operation * on the set of integers Z, is defined by a*b = a+3b2, then find the
value of 2*4.
Ans: 50
9. Let A and B be two finite sets having m and n elements respectively. How many
mappings are possible from A to B?
Ans: nm
10. If f: R→R defined by f(x) = (3x+5)/2 is an invertible function, then find f-1.
Ans: f-1(x) = (2x-5)/3
11. Let * be the binary operation on N defined by a*b = a+b+10 for all a, b є N and if
3*p=15 then find p.
Ans: p = 2
SHORT/LONG ANSWER QUESTIONS(4 /6MARKS)
12. Let N be the set of all natural numbers and let R be a relation on N ×N, defined by (a, b)
R (c, d) ⇔ad = bc. Show that R is an equivalence relation.
13. Let N be the set of all natural numbers and let R be a relation in N, defined by R = {(a,
b): a is a factor of b}. Then, show that R is reflexive and transitive but not symmetric.
14. Let S be the set of all real numbers and let R be a relation in S, defined by R = {(a, b): a
≤ b3}. Show that R satisfies none of reflexivity, symmetry and transitivity.
15. Show that the function f : R → R : f(x) = 3 – 4x is one-one onto and hence bijective
16. Let f be the greatest integer function and g be the absolute value function, find the value
of gof(5/3) –fog(-5/3)
Ans: 0
17. Let * be a binary operation of N, given by a * b = 1.c.m (a,b), a, b ∈ N. Find: (i) 2 * 4 (ii)
Is (N, *) associative?
18. If f(x) = e2x and g(x) = log √x > 0, find (i) fog (ii) gof (iii) f + g (iv) fg.
−1, 𝑖𝑓 𝑥 < 0
19. Show that the signum function f: R → R, defined by f(X) = { 0, 𝑖𝑓 𝑥 = 0 is neither
1, 𝑖𝑓 𝑥 > 0
one-one nor onto.
FAS, Mathematics Assignment - Class XII
20. For any relation R in a set A, we can define the inverse relation aR−1b if and only if
b R a.Prove that R is symmetric if and only if R = R−1. Show that R is an equivalence
relation in A.
21. Let f, g : R → R be two functions such that
Find f(x)
and g(x).
22. If f, g : R → Rare defined respectively by Find
i) fog
ii) gof
iii) fof
iv) gog
23. Show that the relation are in the set a = {x : x є w, x ≤10}given by
is an equivalence relation , find the elements related to
3.
24. 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.
5𝑥+3
25. If (𝑥) = 4𝑥−5 . show that f(f(x)) is an identity function.
26. Let A =NxN and Let * be a binary operation on A defined by (a,b)*(c,d) = (ad+bc,bd) for
all (a,b),(c,d) ∈ NxN.Determine if * is commutative, associative. Does there exist any
identity element?
27. Consider f, g : N → N and h : N → Rdefined as f(x) = 2x, g(y)= 3y+4, and h(z)= Sin z
for all x,y,z ε N Show that ho(gof) = (hog)of
28. Let * be a binary operation on Q defined by a*b = 3ab/5. Show that * is commutative as
well as associative. Also find its identity element,if it exists.
Ans: e=5/3
29. Consider f:{1, 2, 3} → {p, q, r} and g:{p, q, r} → {bat, ball, pitch} defined as f(1) = p,
f(2) = q, f(3) = (r), g(p) = but, g(q) = ball, g(r) = pitch. Show that ‘f’, ‘g’ and ‘gof ‘are
invertible. Find, f −1g−1and (gof) −1and show that (gof)−1= f−1og −1.
30. Define a binary operation * on the set {0, 1, 2, 3, 4, 5}as a*b = (ab) mod 6,where (ab)
mod 6 is the remainder obtained on dividing ab by 6 . Show that 1 is the identity element
for * and 1 and 5 are the only invertible elements with 1– 1 = 1 and 5 – 1 = 5.
31. Let S = { 1, – 1 , i , – i} be the set of fourth root of unity. Form the composition table for
multiplication on S. Show that multiplication on S satisfies the closure property,
associative law and commutative law. What is the identity element in S? Write down the
inverse of each element of S.
32. Let A be the set of all numbers except 1 and ‘o’ be the mapping from A× A to A defined
by aob = a+b-ab far all a, b € A. Prove that
i)
O is a binary operation on A
ii)
The given operation is commutative
iii)
The given operation is associative
iv)
0(zero) is the identity element
𝑎
v)
For each a ∈A, -1−𝑎 is inverse of a under ‘o’