Sample QuestionsRELATIONS AND FUNCTIONS questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
The range of the function $\text{f(x)}=^{7-\text{x}}\text{P}_{\text{x}-3}$ is:
- A
$\{1, 2, 3, 4, 5\}$
- B
$\{1, 2, 3, 4, 5, 6\}$
- C
$\{1, 2, 3, 4\}$
- ✓
$\{1, 2, 3\}$
Answer: D.
View full solution →If $f : R \rightarrow R, g : R \rightarrow R$ and $h : R \rightarrow R$ are such that $f(x) = x^2$, $\text{g(x)}=\tan\text{x}$ and $\text{h(x)}=\log\text{x},$ then the value of $(go(foh)) (x),$ if $x = 1$ will be:
Answer: D.
View full solution →On the power set P of a non-empty set A, we define an operation $\triangle \text{ by }\text{X}\triangle\text{Y}=(\text{X}\cap\text{Y})∪(\text{X}∩\text{Y})\text{X}\triangle\text{Y}=\text{X}∩\text{Y}∪\text{X}∩\text{Y}$
Then which are of the following statements is true about $\triangle$
- A
Commutative and associative without an identity.
- B
Commutative but not associative with an identity.
- C
Associative but not commutative without an identity.
- ✓
Associative and commutative with an identity.
Answer: D.
View full solution →The relation S defined on the set R of all real number by the rule aSb iff a ≥ b is:
- A
- ✓
Reflexive, transitive but not symmetric.
- C
Symmetric, transitive but not reflexive.
- D
Neither transitive nor reflexive but symmetric.
Answer: B.
View full solution →$S$ is a relation over the set $R$ of all real numbers and it is given by $(\text{a, b})\in\text{S}\Leftrightarrow\text{ab}\geq0.$ Then, $S$ is:
- A
Symmetric and transitive only.
- B
Reflexive and symmetric only.
- C
- ✓
Answer: D.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
If $A = \{1, 2, 3\}, B = \{4,5, 6, 7\}$ and $f = \{(1, 4), (2,5), (3, 6)\}$ is a function from $A$ to $B.$
Assertion: $f(x)$ is a one $-$ one function.
Reason: $f(x)$ is an onto function.
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- ✓
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
Answer: C.
View full solution →Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: $A$ relation $R = \{(1, 1), (1, 2), (2, 2), (2, 3), (3, 3)\}$ defined on the set $A = \{1, 2, 3\}$ is symmetri.
Reason: $A$ relation $R$ on the set $A$ is symmetric $(\text{a},\text{b})\in\text{R} \Rightarrow(\text{b},\text{a})\in\text{R}.$
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- ✓
$A$ is false but $R$ is true.
Answer: D.
View full solution →Directions : In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as :
Consider the set $A = \{1, 3, 5\}$.
Assertion : The number of reflexive relations on set $A$ is $2^9$.
Reason : A relation is said to be reflexive if $xRx, \forall\ \text{x}\in\text{A}.$
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- ✓
$A$ is false and $R$ is true.
Answer: D.
View full solution →Directions : In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion :$ A = \{1, 2, 3\}, B = \{4, 5, 6, 7\}, f = \{(1, 4), (2, 5), (3, 6)\}$ is a function from $A$ to $B$.Then $f$ is one $-$ one.
Reason : A function $f$ is one $-$ one if distinct elements of $A$ have distinct images in $B$.
- ✓
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both A and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
Answer: A.
View full solution →Directions : In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as :
Assertion : If $n(A) = p$ and $n(B) = q$ then the number of relations from $A$ to $B$ is $2^{pq}$.
Reason : A relation from $A$ to $B$ is a subset of $A \times B.$
- ✓
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
- B
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
- C
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
Answer: A.
View full solution →If R = {(x, y) : x + 2y = 8} is a relation on N, write the range of R.
The binary operation * : R x R $\rightarrow$ R is defined as a * b = 2a + b. Find (2 * 3) * 4.
View full solution →Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.
If $f : R → R$ be defined by $f(x) = (3 - x^3)^{1/3}$, then find fof(x).
View full solution →Let * be a binary operation on N given by a * b = HCF (a, b), a, b $\in$ N. Write the value of 22 * 4.
View full solution →Examine whether the operation * defined on R by $\text{a}^*\text{b}=\text{ab}+1$ is (i) a binary or not. (ii) if a binary operation, is it associative or not?
View full solution →Let $\text{f(x)=}\begin{cases}1+\text{x,}&0\leq\text{x}\leq2\\3-\text{x,}&2<\text{x}\leq3\end{cases}.$ Find fof.
View full solution →If $f : R → R$ is defined by $f(x) = x^2$, find $f^{-1}(-25)$.
View full solution →The binary operation *: R × R → R is defined as a * b = 2a + b. Find (2 * 3) * 4.
Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}
If the function f: R$\rightarrow$R be given by $f(x) = x^2 + 2$ and g: R$\rightarrow$R be given by g(x) $ = \frac{\text{x}}{\text{x} - 1 },\text{x}\neq1 ,$ find fog and gof and hence find fog (2) and gof (–3).
View full solution →Consider$\text{f}:\text{R}_{+}\rightarrow[4,\infty)$given by $f (x) = x^2 + 4$. Show that f is invertible with the inverse $f^{–1}$ of f given by $f^{–1}$ (y) =$\sqrt{\text{y} - 4 },$ where $R_+$ is the set of all non-negative real numbers.
View full solution →Find the value of k, for which $\text{f}(\text{x}) = $ $ \begin{matrix} \frac{\sqrt{1 + \text{kx}} - \sqrt{1 - \text{kx}}}{\text{x}} , \text{if} - 1\leq\text{x} < 0\\ \frac{2\text{x} + 1}{\text{x} - 1} , \text{ if}0\leq\text{x}< 1 \end{matrix} $ is continuous at x = 0.
View full solution →Let $f : R \rightarrow R$ be defined as $f(x) = 10x + 7.$ Find the function $g : R \rightarrow R$ such that $gof = fog = I_R.$
View full solution →Show that the relation S in the set A = {x $\in $ Z : 0 < x < 12} given by S = {(a, b): a, b $\in $ Z, | a – b | is divisible by 4} is an equivalence relation. Find the set of all elements related to 1.
View full solution →Show that the binary operation $\ast \text{ on A = R - {-1}}$ defined as a $\text{a} \ast \text{b} = \text{a + b + ab}$ for all $\text{a, b}\in \text{A}$ is communicative and associative on A. Also find the identity element of $\ast$ in A and prove that every element of a is invertible.
View full solution →Determine whether the relation R defined on the set $\Re$ of all real numbers as R =$(\text{a,b) : a, b} \in \Re$ and $\text{a - b} + \sqrt{3} \in \text{S},$where S is the set of all irrational numbers, is reflexive, symmetric and transitive.
View full solution →$\text{Let A = Q} \times \text{Q}$ and let $*$ be a binary operation on A defined by$\text{(a, b)} $*$ \text{(c, d) = (ac, b + ad)} \text{ for (a, b), (c, d)} \in \text{A}.$ Determine, whether $*$ is commutative and associative. Then, with respect to $*$ on A.
- Find the identity element in A.
- Find the invertible elements of A.
View full solution →Consider $\text{f : R} - \left\{-\frac{4}{3}\right\} \rightarrow \text{R} - \left\{\frac{4}{3}\right\} \text{given by f(x)} = \frac{\text{4x + 3}}{\text{3x + 4}}.$ Show that f is bijective. Find the inverse of f and hence find $f ^{–1}(0)$ and x such that $f ^{–1}(x) = 2$.
View full solution →Show that the binary operation $\ast \text{ on A = R - {-1}}$ defined as a $\text{a} \ast \text{b} = \text{a + b + ab}$ for all $\text{a, b}\in \text{A}$ is communicative and associative on A. Also find the identity element of $\ast$ in A and prove that every element of a is invertible.
View full solution →A relation R on a set A is said to be an equivalence relation on A iff it is:
- Reflexive i.e., $(\text{a, a})\in\ \text{R} \ \forall \ \text{a}\in\text{A}.$
- Symmetric i.e., $(\text{a, b})\in\ \text{R} \Rightarrow \text{(b, a) } \in\text{R}\ \forall \ \text{a, b}\in\text{A}.$
- Transitive i.e., $(\text{a, b})\in\ \text{R} \ \text{and}\ \text{(b, c) } \in\text{R}\Rightarrow\text{(a, c)}\in\text{R}\ \forall \ \text{a, b, c}\in\text{A}.$
Based on the above information, answer the following questions.
- If the relation R = {(1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)} defined on the set A = {1, 2, 3}, then R is:
- Reflexive
- Symmetric
- Transitive
- Equivalence
- If the relation R = {(1, 2), (2, 1), (1, 3), (3, 1)} defined on the set A = {1, 2, 3}, then R is:
- Reflexive
- Symmetric
- Transitive
- Equivalence
- If the relation R on the set N of all natural numbers defined as R = {(x, y): y = x + 5 and x < 4}, then R is:
- Reflexive
- Symmetric
- Transitive
- Equivalence
- If the relation R on the set A = {1, 2, 3, ........., 13, 14} defined as R = {(x, y): 3x - y = O}, then R is:
- Reflexive
- Symmetric
- Transitive
- Equivalence
- If the relation R on the set A = {I, 2, 3} defined as R = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}, then R is:
- Reflexive only
- Symmetric only
- Transitive only
- Equivalence
View full solution →Consider the mapping $f: A \rightarrow B$ is defined by $f(x) = x - $1 such that $f$ is a bijection.
Based on the above information, answer the following questions.
- Domain of $f$ is:
- $R - \{2\}$
- $R$
- $R -\{1, 2\}$
- $R - \{0\}$
- Range of $f$ is:
- $R$
- $R - \{2\}$
- $R -\{0\}$
- $R - \{1, 2\}$
- If $g: R - \{2\} \rightarrow R - \{1\}$ is defined by $g(x) = 2f(x) - 1,$ then $g(x)$ in terms of $x$ is:
- $\frac{\text{x}+2}{\text{x}}$
- $\frac{\text{x}+1}{\text{x}-2}$
- $\frac{\text{x}-2}{\text{x}}$
- $\frac{\text{x}}{\text{x}-2}$
- The function g defined above, is:
- One-one
- Many-one
- into
- None of these
- A function $f(x)$ is said to be one-one iff.
- $f(x_1) = f(x_2) \Rightarrow -x_1= x_2$
- $f(-x_1) = f(-x_2) \Rightarrow -x_1 = x_2$
- $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$
- None of these
View full solution →A relation R in a set A is called ____________, if $(a_1, B_2)$ $\in\text{R}$ implies $(a_2, a_1) \in\text{R}$ for all $a_1, a_2$ $\in\text{A}.$
View full solution →Fill in the blank.
Let f = {(1, 2), (3, 5), (4, 1) and g = {(2, 3), (5, 1), (1, 3)}. Then gof = ______ and fog = ______.
Fill in the blank.
Let the relation R be defined in N by aRb if 2a + 3b = 30. Then R = ______.
Fill in the blank.
Let the relation R be defined on the set $A = \{1, 2, 3, 4, 5\}$ by $R = \{(a, b): |a^2 – b^2| < 8$. Then R is given by _______.
View full solution →Fill in the blank.
Let f : R → R be defined by $\text{f}(\text{x})=\frac{\text{x}}{\sqrt{1+\text{x}^2}}.$ Then (fofof)(x) = _______.
View full solution →State True or False for the statements.
The relation R on the set A = {1, 2, 3} defined as R = {(1, 1), (1, 2), (2, 1), (3, 3)} is reflexive, symmetric and transitive.
State True or False for the statements.
An integer m is said to be related to another integer n if m is a integral multiple of n. This relation in Z is reflexive, symmetric and transitive.
State True or False for the statements.
Let A = {0, 1} and N be the set of natural numbers. Then the mapping f : N → A defined by $\text{f}(2\text{n}-1)=0,\ \text{f}(2\text{n})=1,\ \forall\ \text{n}\in\text{N},$ is onto.
View full solution →State True or False for the statements.
Let f : R → R be the function defined by $\text{f}(\text{x})=\sin(3\text{x}+2)\ \forall\ \text{x}\in\text{R}.$ Then f is invertible.
View full solution →State True or False for the statements.
The composition of functions is associative.