Assertion (A) & Reason (B) MCQ

MCQ 11 Mark

Assertion $(A):$ The function $f : R ^* \rightarrow R ^*$ defined by $f(x)=\frac{1}{x}$ is one$-$one and onto, where $R*$ is the set of all non$-$zero real numbers.
Reason $(R):$ The function $g : N \rightarrow R ^*$ defined by $f(x)=\frac{1}{x}$ is one$-$one and onto.

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.

Assertion: It is given that$f: R^* \rightarrow R^*$ is defined by
$f(x)=\frac{1}{x}$
For one$-$one, $f(x)=f(y)$
$\Rightarrow \frac{1}{x}=\frac{1}{y}$
$\Rightarrow x=y$
Therefore, $f$ is one$-$one.
For onto, it is clear that for $y \in R^*$,
there exists $x=\frac{1}{y} \in R^* ($exists as $y \neq 0 )$ such that $f(x)=\frac{1}{\left(\frac{1}{y}\right)}=y$
Therefore, $f$ is onto.
Thus, the given function $( f )$ is one$-$one and onto.
Reason: Now, consider function $g : N \rightarrow R ^*$ defined by $g(x)=\frac{1}{x}$.
We have, $g\left(x_1\right)=g\left(x_2\right) \Rightarrow \frac{1}{x_1}=\frac{1}{x_2}$

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MCQ 21 Mark

Assertion $(A):f(x)=2 x^3-9 x^2+12 x-3$ is increasing outside the interval $(1, 2).$
Reason $(R): f ^{\prime}( x )<0$ for $x \in(1,2)$

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 $A$

We have, $f(x)=2 x^3-9 x^2+12 x-3$$\Rightarrow f^{\prime}(x)=6 x^2-18 x+12$
For increasing function, $f^{\prime}(x) \geq 0$
$\therefore 6\left(x^2-3 x+2\right) \geq 0$
$\Rightarrow 6(x-2)(x-1) \geq 0$
$\Rightarrow x \leq 1$ and $x \geq 2 $
$\therefore f(x)$ is increasing outside the interval $(1, 2),$ therefore it is a true statement.
Reason: Now, $f ^{\prime}( x )<0$
$\Rightarrow 6( x -2)( x -1)<0$
$\Rightarrow 1< x <2$
$\therefore$ Assertion and Reason are both true but Reason is not the correct explanation of Assertion.

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Maths Assertion (A) & Reason (B) MCQ

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