Class 11 & 12 Maths – Relations & Functions 50 TYPE Questions (Only Question Set) By Grand Master Bikram Sutradhar
Class 11 & 12 Maths – Relations & Functions 50 TYPE Questions (Only Question Set) By Grand Master Bikram Sutradhar
📘 INTEGRATION – 200 QUESTIONS Easy to Pro
Relations & Functions
Class 11 & 12 Maths – Relations & Functions
50 Grand Master Bikram Sutradhar TYPE Questions (Only Question Set)
By Grand Master Bikram Sutradhar
1. Let $R$ be a relation on $\mathbb{Z}$ defined by
$$
R=\{(a,b): a^3-b^3 \text{ is divisible by } (a-b)\}.
$$
Examine whether $R$ is reflexive, symmetric, and transitive.
2. Determine whether the relation
$$
R=\{(a,b): a^2+b^2=2ab\}
$$
on $\mathbb{R}$ is an equivalence relation.
3. Let
$$
R=\{(a,b): a-b\in 3\mathbb{Z}\}
$$
on $\mathbb{Z}$. Find all distinct equivalence classes.
4. For the relation
$$
R=\{(x,y): x^2=y^2\}
$$
on $\mathbb{R}$, find the quotient set.
5. Let
$$
f:\mathbb{R}\to\mathbb{R}, \quad f(x)=x^3+ax^2+bx.
$$
Find conditions on $a,b$ for $f$ to be one-one.
6. Show that
$$
R=\{(a,b): a-b \text{ is rational}\}
$$
on $\mathbb{R}$ is an equivalence relation and find the equivalence class of $\sqrt{2}$.
7. Let $A=\{1,2,3,4\}$. Find the number of equivalence relations on $A$.
8. If
$$
f(x)=x+\frac{1}{x}, \quad x\neq 0,
$$
determine whether $f$ is invertible.
9. Prove that
$$
f(x)=x^3-3x
$$
is not one-one on $\mathbb{R}$.
10. Find the domain on which
$$
f(x)=x^4-4x^2
$$
is invertible and hence find $f^{-1}$.
11. Let $f:\mathbb{R}\to\mathbb{R}$ satisfy
$$
f(x+y)=f(x)f(y), \quad f(0)=1.
$$
Show that $f(x)\neq 0$ for all $x$.
12. If
$$
f(x)=\ln\left(\frac{1+x}{1-x}\right),
$$
find the largest interval on which $f$ is invertible.
13. Let $R$ be a relation on $\mathbb{N}$ defined by
$$
aRb \iff \gcd(a,b)=1.
$$
Examine whether $R$ is transitive.
14. Find the number of onto functions from a set of 5 elements to a set of 3 elements.
15. Let
$$
f(x)=|x-1|+|x+1|.
$$
Examine injectivity and surjectivity.
16. Show that the binary operation
$$
a*b=a+b+ab
$$
on $\mathbb{R}\setminus\{-1\}$ is associative and find its identity element.
17. If
$$
f(x)=\frac{ax+b}{cx+d}
$$
is invertible, find the condition on $a,b,c,d$.
18. Let
$$
f:\mathbb{R}^+\to\mathbb{R}, \quad f(x)=\ln x.
$$
Prove that $f$ is bijective.
19. Let
$$
f(x)=x^2
$$
with domain $[0,\infty)$. Determine whether $f$ is invertible and find $f^{-1}$.
20. If
$$
f(x)=x^5+x,
$$
prove that $f$ is bijective.
21. Determine whether the relation
$$
R=\{(a,b): a=b^2\}
$$
on $\mathbb{R}$ is symmetric.
22. Let $A=\{1,2,3\}$. Find all equivalence relations on $A$.
23. If
$$
f(x)=\sqrt{x+1}+\sqrt{x-1},
$$
find the domain and test invertibility.
24. Let
$$
g(x)=\frac{x-1}{x+1}.
$$
Find $g^{-1}(x)$ and verify $g(g^{-1}(x))=x$.
25. Show that
$$
f(x)=\tan x
$$
is invertible on $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$.
26. Let $R$ be a relation on $\mathbb{Z}$ defined by
$$
aRb \iff a-b \text{ is divisible by } 5.
$$
Find the quotient set.
27. Prove that
$$
f(x)=x+\sin x
$$
is one-one on $\mathbb{R}$.
28. Let $f:\mathbb{R}\to\mathbb{R}$ be such that
$$
f(x)+f(1-x)=x^2.
$$
Find $f(x)$.
29. If $f:\mathbb{R}\to\mathbb{R}$ satisfies
$$
f(f(x))=x^2,
$$
examine whether $f$ is invertible.
30. Determine whether the relation
$$
R=\{(a,b): a|b\}
$$
on $\mathbb{N}$ is transitive.
31. Let
$$
f(x)=\frac{x^2-1}{x-1}.
$$
Find the largest domain for which $f$ is invertible.
32. Show that
$$
f(x)=x^3+x
$$
is one-one using calculus.
33. If
$$
f(x)=x^2+ax+b
$$
is invertible on $[0,\infty)$, find conditions on $a,b$.
34. Let
$$
R=\{(a,b): a+b \text{ is even}\}
$$
on $\mathbb{Z}$. Find all equivalence classes.
35. Find the number of bijections from a set of 4 elements to itself.
36. Let
$$
f(x)=e^{2x}, \quad g(x)=\ln x.
$$
Find $(g\circ f)(x)$ and examine invertibility.
37. If
$$
f(x)=\frac{1}{x}
$$
with suitable domain and codomain, examine bijectivity.
38. Show that the relation
$$
R=\{(a,b): |a-b|<2\}
$$
on $\mathbb{R}$ is not transitive.
39. Let
$$
f(x)=x|x|.
$$
Examine whether $f$ is one-one and onto.
40. If
$$
f(x)=x^3-x,
$$
find intervals on which $f$ is invertible.
41. Let $R$ be a relation on $\mathbb{R}$ defined by
$$
aRb \iff a^2+b^2=1.
$$
Is $R$ reflexive?
42. If
$$
f(x)=\ln(\ln x),
$$
find domain, range, and test invertibility.
43. Prove that
$$
f(x)=x+\frac{1}{x}
$$
is not one-one on $\mathbb{R}\setminus\{0\}$.
44. Let
$$
f(x)=x^3-3x+1.
$$
Determine whether $f$ is bijective.
45. If
$$
f(x)=\frac{x}{|x|}
$$
for $x\neq 0$, examine injectivity and surjectivity.
46. Let $A$ be a finite set with $n$ elements. Find the number of equivalence relations on $A$.
47. If $f:\mathbb{R}\to\mathbb{R}$ is continuous and one-one, prove that $f$ has an inverse.
48. Show that
$$
f(x)=x^7
$$
is bijective using properties of odd functions.
49. Let
$$
f(x)=\sqrt[3]{x+1}.
$$
Find $f^{-1}(x)$.
50. If
$$
f(x)=ax+b \quad \text{and} \quad g(x)=cx+d
$$
are invertible, find the condition under which $(f\circ g)$ is also invertible.
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