MCQ 11 Mark
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.
AnswerCorrect option: C. $A$ is true but $R$ is false.
Given$, A= \{1, 2, 3\}, B = \{4, 5, 6, 7\}$
and $f : A \rightarrow B$ is defined as $f = \{(1, 4), (2, 5), (3, 6)\}$
i.e., $f(1) = 4, f(2) = 5$ and $f(3) = 6.$
It can be seen that the images of distinct elements of $A$ under $f$ are distinct.
So$, f$ is one $-$ one.
So$, A$ is true.
Range of $f = \{4, 5, 6\}.$
$Co -$ domain $= \{4, 5, 6, 7\}.$
Since $co -$ domain $\neq$ range$, f(x)$ is not an onto function.
Hence $R$ is false.
View full question & answer→MCQ 21 Mark
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.
AnswerCorrect option: D. $A$ is false but $R$ is true.
View full question & answer→MCQ 31 Mark
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.
AnswerCorrect option: D. $A$ is false and $R$ is true.
By definition, a relation in $A$ is said to be reflexive if $xRx, \forall\ \text{x}\in\text{A}.$
So $R$ is true.
The number of reflexive relations on a set containing $n$ elements is $2^{n2-n}.]$
Here $n = 3.$
The number of reflexive relations on a set $A = 2^{9-3} = 2^6$.
Hence $A$ is false.
View full question & answer→MCQ 41 Mark
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 51 Mark
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 61 Mark
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 $\text{f w.r.t}$. 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.
- B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
- C
Assertion is correct but Reason is incorrect.
- D
Both Assertion and Reason are incorrect.
AnswerCorrect option: A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
View full question & answer→MCQ 71 Mark
Directions : In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R)$. Mark the correct choice as :
Let $W$ be the set of words in the English dictionary. $A$ relation $R$ is defined on $W$ as $R =$ $(\text{x},\text{y})\in\text{W}\times\text{W}$such that $x$ and $y$ have at least one letter in common
Assertion : $R$ is reflexive.
Reason : $R$ is symmetric.
- 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.
AnswerCorrect option: B. Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
For any word $x$ and $x$ have atleast one $($all$)$ letter in common
$\therefore(\text{x},\text{x})\in\text{R},\forall\ \text{x}\in\text{W}$
$\therefore R$ is reflexive
Let $(\text{x},\text{y})\in\text{R},\text{x},\text{y}\in\text{W}$
$\Rightarrow x$ and $y$ have atleast one letter in common
$\Rightarrow y$ and $x$ have atleast one letter in common
$\Rightarrow(\text{y},\text{x})\in\text{R}$
$\therefore R$ is symmetric
Hence $A $ is true, $R$ is true; $R$ is not a correct explanation for $A.$
View full question & answer→MCQ 81 Mark
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.
AnswerCorrect option: B. Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
View full question & answer→MCQ 91 Mark
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 $A.$
- 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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 101 Mark
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 reflexive.
Reason : A relation $R$ on the set $A$ is reflexive if $(\text{a},\text{a})\in\text{R},\forall\ \text{a}\in\text{A}.$
- ✓
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.
- D
$A$ is false but $R$ is true.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 111 Mark
Directions : In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as :
Assertion : The range of the function $\frac{\text{x}^{2}}{1+\text{x}^{2}}$ is $(0, 1).$
Reason : If $\text{f}(\text{x})\leq\text{g}(\text{x})$ then the range of $\frac{\text{f}(\text{x})}{\text{g}(\text{x})},\text{g}(\text{x})\neq0$ is $(0, 1).$
- 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.
AnswerCorrect option: C. $A$ is true but $R$ is false.
View full question & answer→MCQ 121 Mark
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 $R = \{(1,1), (1, 3), (3, 1), (3, 3), (3, 5)\}$ defined on the set $A = \{1, 3, 5\}$ is reflexive.
Reason : A relation $R$ on the set $A$ is transitive if $(\text{a},\text{b})\in\text{R}$ and $(\text{b},\text{c})\in\text{R}$
$\Rightarrow(\text{a},\text{c})\in\text{R}).$
- 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.
AnswerCorrect option: D. $A$ is false but $R$ is true.
View full question & answer→MCQ 131 Mark
Directions : In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as :
Assertion : The function $f : R \rightarrow R, \text{f}(\text{x})=\mid\text{x}\ \mid$ is not one $-$ one.
Reason : The function $\text{f}(\text{x})=\mid\text{x}\ \mid$ is not onto.
- 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.
AnswerCorrect option: B. Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
View full question & answer→MCQ 141 Mark
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 $X = \{0, 1, 2\}$ and the function defined by $f(x) = x^2 - 2$ is surjection then $Y = \{-2, -1, 0\}$.
Reason : If $f : X \rightarrow Y$ is surjective if $f(X) = Y$.
- ✓
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 151 Mark
Directions : In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as :
Let $R$ be the relation in the set of integers $Z$ given by $R = (a, b) : 2$ divides ${a - b}.$
Assertion : $R$ is a reflexive relation.
Reason : $A$ relation is said to be reflexive if $xRx, \forall\ \text{x}\in\text{Z}.$
- ✓
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
By definition, a relation in $Z$ is said to be reflexive if $xRx, \forall\ \text{x}\in\text{Z}.$
So $R$ is true.
$a - a = 0$
$\Rightarrow 2$ divides $a - a$
$\Rightarrow aRa.$
Hence $R$ is reflexive and $A$ is true. $R$ is the correct explanation for $A.$
View full question & answer→MCQ 161 Mark
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 function $f : A \rightarrow B,$ cannot be an onto function if $n(A) < n(B).$
Reason : A function $f$ is onto if every element of co $-$ domain has at least one pre $-$ image in the domain.
- ✓
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→MCQ 171 Mark
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 $R = \{(1, 1), (1, 3), (1.5), (3, 1), (3, 3), (3,5\}$ defined on the set $A = \{1, 3, 5\}$ is transitive.
Reason : A relation $R$ on the set $A$ is symmetric $(\text{a},\text{b})\in\text{R}$ and $(\text{a},\text{c})\in\text{R}$
$\Rightarrow(\text{a},\text{c})\in\text{R}).$
- 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 $A.$
- ✓
$A$ is true but $R$ is false.
- D
$A$ is false but $R$ is true.
AnswerCorrect option: C. $A$ is true but $R$ is false.
View full question & answer→MCQ 181 Mark
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 function $f : R \rightarrow R$ defined as
$\text{f}(\text{x})=\frac{\text{x}}{\text{x}^{2}+1}.$
Assertion : $f(x)$ is not one $-$ one.
Reason : $f(x)$ is not onto.
- 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.
AnswerCorrect option: B. Both $A$ and $R$ are true but $R$ is not the correct explanation of $A.$
View full question & answer→MCQ 191 Mark
Directions : In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as :
Assertion : Domain and Range of a relation $R = \{(x, y) : x -2y = 0\} $ defined on the set $A = \{1, 2, 3, 4\}$ are respectively $\{1, 2, 3, 4\}$ and $\{2, 4, 6, 8\}.$
Reason : Domain and Range of a relation $R$ are respectively the sets $\text{a}:\text{a}\in\text{A}$ and $(\text{a},\text{b})\in\text{R}.$ and $\text{b}:\text{b}\in\text{A}$ and $(\text{a},\text{b})\in\text{R}.$
- 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.
AnswerCorrect option: D. $A$ is false but $R$ is true.
View full question & answer→MCQ 201 Mark
Directions : In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as :
Assertion : Consider the function $f : R \rightarrow R$ defined by $f(x) = x^3$. Then $f$ is one $-$ one.
Reason : Every polynomial function is one $-$ one.
- 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.
AnswerCorrect option: C. $A$ is true but $R$ is false.
View full question & answer→MCQ 211 Mark
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 $n(A) = m,$ then the number of reflexive relations on $A$ is $m.$
Reason: A relation $R$ on the set $A$ is reflexive if $(\text{a},\text{a})\in\text{R},$ $\forall\ \text{a}\in\text{A}.$
- 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 but $R$ is true.
AnswerCorrect option: D. $A$ is false but $R$ is true.
View full question & answer→MCQ 221 Mark
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 function $f : R > R$ defined as $f(x) = x^3.$
Assertion: $f(x)$ is a one $-$ one function.
Reason: $f(x)$ is a one $-$ one function if co $-$ domain $=$ range.
- 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 and $R$ is true.
AnswerCorrect option: C. $A$ is true but $R$ is false.
$f(x)$ is a one $-$ one function if
$f(x_1) = f(x_2) > x_1 = x_2,$
Hence $R$ is false.
Let $f(x_1) = f(x_2)$ for some $\text{x}_{1},\text{x}_{2}\in\text{R}$
$\Rightarrow (x_1)^3= (x_2)^3$
$\Rightarrow x_1 = x_2$
Hence $f(x)$ is one $-$ one.
Hence $A$ is true.
View full question & answer→MCQ 231 Mark
Directions: In the following questions, a statement of assertion $(A)$ is followed by a statement of reason $(R).$ Mark the correct choice as:
Assertion: Consider the function $f : R \rightarrow R$ defined by $\text{f}(\text{x})=\frac{\text{x}}{\text{x}^{2}+1}.$ Then $f$ is one $-$ one.
Reason: $\text{f}(4)=\frac{4}{17}$ and $\text{f}\big(\frac{1}{4}\big)=\frac{4}{17}.$
- 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.
AnswerCorrect option: D. $A$ is false but $R$ is true.
View full question & answer→MCQ 241 Mark
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 $n(A) = m$ and $n(B) = n.$ The number of one $-$ one functions from $A$ onto $B$ is $n_{pm}$, if $\text{n}\geq\text{m}.$
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.
AnswerCorrect option: A. Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
View full question & answer→