Assertion (A) & Reason (B) MCQ

MCQ 11 Mark

The function $f$ be given by $f(x)=2 x^{3}-6 x^{2}+6 x+5$.
Assertion (A): $x=1$ is not a point of local maxima.
Reason (R): $x=1$ is not a point of local minima.

A
Both A and R are true and R is the correct explanation of A .
✓
$A$ is true but $R$ is false.
C
Both A and R are true but R is not the correct explanation of A .
D
A is false but $R$ is true.

Answer

Correct option: B.

$A$ is true but $R$ is false.

(B) Both A and R are true but R is not the correct explanation of A .
Explanation: We have,
$f(x)=2 x^{3}-6 x^{2}+6 x+5$
$\Rightarrow f^{\prime}(x)=6 x^{2}-12 x+6=6(x-1)^{2}$
and $f^{\prime \prime}(x)=12(x-1)$
Now, $f^{\prime}(x)=0$ gives $x=1$.
Also, $\mathrm{f}^{\prime \prime}(1)=0$.
Therefore, the second derivative test fails in this case.
So, we shall go back to the first derivative test.
Using first derivatives test, we get $\mathrm{x}=1$ is neither a point of local maxima nor a point of local minima and so it is a point of inflexion.

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MCQ 21 Mark

Assertion (A): If $A=\left[\begin{array}{ccc}2 & 3 & -1 \\ 1 & 4 & 2\end{array}\right]$ and $B=\left[\begin{array}{ll}2 & 3 \\ 4 & 5 \\ 2 & 1\end{array}\right]$, then $A B$ and $B A$ both are defined.
Reason ( $\mathbf{R}$ ): For the two matrices $A$ and $B$, the product $A B$ is defined, if number of columns in $A$ is equal to the number of rows 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 .

(A) Both A and R are true and R is the correct explanation of A .
Explanation: The given matrices are $A=\left[\begin{array}{ccc}2 & 3 & -1 \\ 1 & 4 & 2\end{array}\right]$ and $B=\left[\begin{array}{ll}2 & 3 \\ 4 & 5 \\ 2 & 1\end{array}\right]$
Order of $\mathrm{A}=2 \times 3$; Order of $\mathrm{B}=3 \times 2$
Since, number of columns in $A$ is equal to the number of rows in $B$.
$\Rightarrow \mathrm{AB}$ is defined.
Also, number of columns in B is equal to the number of rows in A .
$\therefore$ The product BA is also defined.

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Applied Maths Assertion (A) & Reason (B) MCQ

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