Assertion (A) & Reason (B) MCQ

MCQ 11 Mark

Let $f(x)=x^4-2 x^2+5$ be defined on $[-2,2]$.
Assertion (A): The range of f(x) is [2, 13].
Reason (R): The greatest value of f is attained at x = 2.

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.

Answer

Correct option: D.

A is false but R is true.

(d) A is false but R is true.
Explanation:$f(x)=x^4-2 x^2+5 \Rightarrow f^{\prime}(x)=4 x^3-4 x$
$\Rightarrow f^{\prime}(x)=4 x(x-1)(x+1)$
$\begin{array}{l}\Rightarrow f^{\prime}(x)=4 x\left(x^2-1\right. \\ \Rightarrow f^{\prime}(x)=4 x(x-1)(x+1)\end{array}$
For critical points,$f ^{\prime}( x )=0 \Rightarrow x =0,-1,1$
Now, $f(-2)=(-2)^4-2(-2)^2+5=16-8+5=13$
$\begin{array}{l}f(2)=2^4-2(2)^2+5=16-8+5=13 \\ f(-1)=(-1)^4-2(-1)^2+5=1-2+5=4 \\ f(0)=0-2 \times 0+5=5 \\ f(1)=14-2(1)^2+5=4\end{array}$
So, the range of f is [4, 13]
$\therefore$ Assertion is false.
Also, f attains it maximum value at $x =-2$ and $x =2$
$\therefore$Reason is true.

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MCQ 21 Mark

Assertion (A): If $A=\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]$ and $I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, then the value of k such that $A ^2= kA -2 I$, is -1.
Reason (R): If A and B are square matrices of same order, then (A + B)(A + B) is equal to $A^2+A B+B A+B^2$

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.

Answer

Correct option: D.

A is false but R is true.

(d) A is false but R is true.
Explanation: Assertion: Given,$A^2=k A-2 I$
$\Rightarrow AA = kA -2 I$
$\begin{array}{l}\Rightarrow\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]=k\left[\begin{array}{ll}3 & -2 \\ 4 & -2\end{array}\right]-2\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right] \\ \Rightarrow\left[\begin{array}{cc}9-8 & -6+4 \\ 12-8 & -8+4\end{array}\right]=\left[\begin{array}{ll}3 k & -2 k \\ 4 k & -2 k\end{array}\right]-\left[\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right] \\ \Rightarrow\left[\begin{array}{cc}1 & -2 \\ 4 & -4\end{array}\right]=\left[\begin{array}{cc}3 k-2 & -2 k \\ 4 k & -2 k-2\end{array}\right]\end{array}$
By definition of equality of matrix, the given matrices are equal and their corresponding elements are equal.
Now, comparing the corresponding elements, we get
$3 k-2=1 \Rightarrow k=1$
$\Rightarrow-2 k =-2= k =1$
$\Rightarrow 4 k =4 \Rightarrow k =1$
$\Rightarrow-4=-2 A-2 \Rightarrow k =1$
Hence, k = 1
Reason: We have,
(A + B)(A + B) = A(A + B) + B(A + B)
$= A ^2+ AB + BA + B ^2$

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

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