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.
Choose the correct answer from the given four options. Let $f : [2, \infty ) \rightarrow R$ be the function defined by $f(x) = x^2 – 4x + 5,$ then the range of $f$ is:
- A
$\text{R}$
- ✓
$[1,\infty)$
- C
$[4,\infty)$
- D
$[5,\infty)$
Answer: B.
View full solution →Let R be the relation “is congruent to” on the set of all triangles in a plane is:
- Reflexive.
- Symmetric.
- Symmetric and reflexive.
- Equivalence.
The function $\text{f}:\Big[\frac{-1}{2},\frac{1}{2},\frac{1}{2}\Big]\rightarrow\ \Big[\frac{-\pi}{2},\frac{\pi}{2}\Big],$ defined by $\text{f(x)}=\sin^{-1}(3\text{x}-4\text{x}^3),$ is:
- Bijection.
- Injection but not a surjection.
- Surjection but not an injection.
- Neither an injection nor a surjection.
View full solution →The mapping $f : N \rightarrow N$ is given by $f(n) = 1 + n^2, n \in N$ when $N$ is the set of natural numbers is$:$
Answer: C.
View full solution →If N be the set of all-natural numbers, consider f : N → N such that f(x) = 2x, ∀ x ∈ N, then f is:
- One-one onto.
- One-one into.
- Many-one onto.
- None of these.
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 $xR, \forall\ \text{x}\in\text{A}.$
- 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: 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 $A\ x\ B$.
- ✓
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, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: $\text{u}=\text{f}(\cot\text{x})\&\text{f}(1)=\sqrt2$ and $\text{g}(\sqrt{2})=2$ then $\Big(\frac{\text{du}}{\text{dv}}\Big)_{\text{x}=\frac{\text{x}}{4}}=1.$
Reason: If u = f(x), v = g(x) then derivative of f w.r.t. to g is $\frac{\text{du}}{\text{dv}}=\frac{\frac{\text{du}}{\text{dx}}}{\frac{\text{dv}}{\text{dx}}}.$
- Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
- Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
- Assertion is correct but Reason is incorrect.
- Both Assertion and Reason are incorrect.
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: $A, B$ are two sets such that $n(A) = p$ and $n(B) = q,$ The number of functions from $A$ onto $B$ is $q^p ..$
Reason: Every function is a relation.
- A
Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.
- ✓
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: B.
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: n(A) =5, n(B) =5 and f : A B is one - one then f is bijection.
Reason: If n(A) = n(B) then every one - one function from A to B is onto
- Both A and R are true and R is the correct explanation of A.
- Both A and R are true but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
- Both A and R are fals.
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 $f : R \rightarrow R$ 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 →Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
Let * be a binary operation on the set Q of rational numbers as follows:
a * b = a + ab
If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x| – x, $\forall\ \text{x}\in\text{R}.$ Then find fog and gof. Hence find fog(-3), fog(5) and gof(-2).
View full solution →The following defines a relation on N:
$\text{x}>\text{y, x, y}\in\text{N}$
Determine which of the above relations are reflexive, symmetric and transitive.
View full solution →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 go $f = f\ o\ g = I_R.$
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 →Let $\text{A = Q} \times \text{Q}$ 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 $\text{f : R} - \left\{-\frac{4}{3}\right\} \rightarrow \text{R} - \left\{\frac{4}{3}\right\}$ 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. If $\text{f}(\text{x})=[4-(\text{x}-7)^3],$ then $f^{-1}(x) =........$
View full solution →State True or False for the statements.
Every relation which is symmetric and transitive is also reflexive.
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.
Every function is invertible.
State True or False for the statements.
The composition of functions is associative.
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.