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MATHS Assertion (A) & Reason (B) MCQ

RELATIONS AND FUNCTIONS · Rajasthan Board (RBSE) STD 12 Science (Rajasthan - English Medium). 24 questions.

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Assertion (A) & Reason (B) MCQ

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24 questions · timed · auto-graded

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:
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
Correct 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.
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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: 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
Correct option: A.
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
Both $A$ and $R$ are true and $R$ is the correct explanation of $A.$
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Question 31 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 f w.r.t. to g is $\frac{\text{du}}{\text{dv}}=\frac{\frac{\text{du}}{\text{dx}}}{\frac{\text{dv}}{\text{dx}}}.$
  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
  2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
  3. Assertion is correct but Reason is incorrect.
  4. Both Assertion and Reason are incorrect.
Answer
  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
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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: $A, B$ 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
Correct option: B.
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A$.
Both $A$ and $R$ are true but $R$ is not the correct explanation of $A$.
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Question 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: 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
  1. Both A and R are true and R is the correct explanation of A. 
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
Answer
  1. Both A and R are true and R is the correct explanation of A.
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Question 61 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).
  1. Both A and R are true and R is the correct explanation of A. 
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
Answer
  1. A is true but R is false.
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Question 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:
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 $(\text{b},\text{c})\in\text{R}$
$\Rightarrow(\text{a},\text{c})\in\text{R}).$
  1. Both A and R are true and R is the correct explanation of A. 
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
Answer
  1. A is false but R is true.
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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 :
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.
Answer
Correct 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.
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Question 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: Consider the function f : R → 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 $\text{f}\big(\frac{1}{4}\big)=\frac{4}{17}.$
  1. Both A and R are true and R is the correct explanation of A. 
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
Answer
  1. A is false but R is true.
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Question 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 symmetri.
Reason: A relation R on the set A is symmetric $(\text{a},\text{b})\in\text{R}$
$\Rightarrow(\text{b},\text{a})\in\text{R}.$
  1. Both A and R are true and R is the correct explanation of A. 
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
Answer
  1. A is false but R is true.
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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: $A, B$ 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.
Answer
Correct option: A.
Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.
Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.. 
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Question 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 = {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.
  1. Both A and R are true and R is the correct explanation of A. 
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
Answer
  1. Both A and R are true and R is the correct explanation of A. 
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Question 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:
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.
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false and R is true.
Answer
  1. Both A and R are true but R is not the correct explanation of A.
Solution:
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 $(\text{x},\text{y})\in\text{R},\text{x},\text{y}\in\text{W}$
⇒ x and y have atleast one letter in common
⇒ 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.
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Question 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: 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 $(\text{a},\text{a})\in\text{R},\forall\ \text{a}\in\text{A}.$
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
Answer
  1. Both A and R are true and R is the correct explanation of A.
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Question 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, $\forall\ \text{x}\in\text{Z}.$
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false and R is true.
Answer
  1. Both A and R are true and R is the correct explanation of A.
Solution:
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
⇒ 2 divides a - a
⇒ aRa.
Hence R is reflexive and A is true. R is the correct explanation for A.
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Question 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 → 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.
  1. Both A and R are true and R is the correct explanation of A. 
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
Answer
  1. Both A and R are true and R is the correct explanation of A. 
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Question 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: 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 $(\text{a},\text{b})\in\text{R}.$}
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
Answer
  1. A is false but R is true.
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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:
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.
Answer
Correct option: C.
$A$ is true but $R$ is false.
$A$ is true but $R$ is false.
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Question 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: 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 $(\text{a},\text{a})\in\text{R},$ $\forall\ \text{a}\in\text{A}.$
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
Answer
  1. A is false but R is true.
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Question 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:
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.
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false and R is true.
Answer
  1. A is true but R is false.
Solution:
Given, A= {1, 2, 3}, B = {4, 5, 6, 7}
and f : A  → 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.
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Question 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:
Consider the function f : R → 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.
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false and R is true.
Answer
  1. Both A and R are true but R is not the correct explanation of A.
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Question 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:
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 $(\text{a},\text{c})\in\text{R}$
$\Rightarrow(\text{a},\text{c})\in\text{R}).$
  1. Both A and R are true and R is the correct explanation of A. 
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
Answer
  1. A is true but R is false.
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Question 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: The function f : R → 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.
  1. Both A and R are true and R is the correct explanation of A. 
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
Answer
  1. Both A and R are true but R is not the correct explanation of A.
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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: 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.
Answer
Correct option: A.
Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.
Both $A$ and $R$ are true and $R$ is the correct explanation of $A$. 
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