Assertion (A) & Reason (B) MCQ

MCQ 11 Mark

Assertion (A): If $A=\{x \in z: 0 \leq x \leq 12\}$ and R is the relation in A given by $R =\{( a , b ): a = b$.Then, the set of all elements related to 1 is {1, 2}.
Reason (R): If $R _1$ and $R _2$ are equivalence relation in a set A , then $R_1 \cap R_2$ is an equivalence relation.

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.

Answer

(d) A is false but R is true.
Explanation: Assertion: The elements that are related to 1 will be those elements from set A which are equal to 1. Hence, the set of elements related to 1 is {1}.
Reason: Since , $R _1$ and $R _2$ are equivalence relations, therefore $( a , a ) \in R_1,(a, a) \in R_2, \forall a \in A$.
This implies that $(a, a) \in R_1 \cap R_2, \forall a$.
Hence, $R_1 \cap R_2$ is reflexive.
Further, $( a , b ) \in R_1 \cap R_2 \Rightarrow( a , b ) \in R_1$ and $( a , b ) \in R _2$ and $( b , a ) \in R _2$
$\Rightarrow(b, a) \in R_1 \cap R_2$
Hence, $R _1 \cap R _2$ is symmetric.
Similarly, $(a, b) \in R_1 \cap R_2$ and $(b, c) \in R_1 \cap R_2$
$\Rightarrow(a, c) \in R_1 \text { and }(a, c) \in R_2 \Rightarrow(a, c) \in R_1 \cap R_2$
This implies that $R_1 \cap R_2$ is transitive.
Hence, $R _1 \cap R _2$ is an equivalence relation.

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MCQ 21 Mark

Assertion (A): If x is real, then the minimum value of $x ^2-8 x +17$ is 1
Reason (R): If f"(x) > 0 at a critical point, then the value of the function at the critical point will be the minimum value of the 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
C
A is true but R is false.
D
A is false but R is true.

Answer

(a) Both A and R are true and R is the correct explanation of A.
Explanation: Let $f(x)=x^2-8 x+17$
$\therefore f^{\prime}(x)=2 x-8$
So, $f^{\prime}(x)=0$, gives $x=4$
Here $x=4$ is the critical number
Now, $f ^{\prime \prime}( x )=2;0, \forall x$
So, $x=4$ is the point of local minima.
$\therefore$ Minimum value of $f(x)$ at $x=4$,
$f(4)=4 \times 4-8 \times 4+17=1$
Hence, we can say that both Assertion and Reason are true and Reason is the correct explanation of the Assertion.

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Maths Assertion (A) & Reason (B) MCQ

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