Let A-3x2 1 6x- B-a-b-c- and C-x-22 5x2 2x 5x2 2x x-22 2x x-22 5x2- be three given matrices- where a-b-c and x- R- Given that tr-AB - tr-C - x- R- where tr-A denotes trace of A- Find the value of a-b-c

The correct option is C 7Given A=[3x^2 1 6x], B=[a,b,c] #160;Here, #160; AB=[ 3x^ 2 1 6x ] [a,b,c] ⇒ AB=[ 3ax^ 2 am ...

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Byju's Answer

Standard XII

Mathematics

One - One function

Let A=[3x2 ...

Question

Let $A = ⎡ ⎢ ⎣ \begin{matrix} 3 x^{2} 1 6 x \end{matrix} ⎤ ⎥ ⎦, B = [a, b, c]$ and $C = ⎡ ⎢ ⎢ ⎣ \begin{matrix} (x + 2)^{2} & 5 x^{2} & 2 x 5 x^{2} & 2 x & (x + 2)^{2} 2 x & (x + 2)^{2} & 5 x^{2} \end{matrix} ⎤ ⎥ ⎥ ⎦$ be three given matrices, where $a, b, c$ and $x \in R$ , Given that $t r . (A B) = t r . (C) \lor x \in R$ , where $t r . (A)$ denotes trace of $A$ . Find the value of $(a + b + c)$

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Solution

The correct option is C 7
Given $A = ⎡ ⎢ ⎣ \begin{matrix} 3 x^{2} 1 6 x \end{matrix} ⎤ ⎥ ⎦, B = [a, b, c]$
Here, $A B = ⎡ ⎢ ⎣ \begin{matrix} 3 x^{2} 1 6 x \end{matrix} ⎤ ⎥ ⎦ [a, b, c]$
$\Rightarrow A B = ⎡ ⎢ ⎣ \begin{matrix} 3 a x^{2} & 3 b x^{2} & 3 c x^{2} a & b & c 6 a x & 6 b x & 6 c x \end{matrix} ⎤ ⎥ ⎦$
$t r (A B) = 3 a x^{2} + b + 6 c x$
Now, given $C = ⎡ ⎢ ⎢ ⎣ \begin{matrix} (x + 2)^{2} & 5 x^{2} & 2 x 5 x^{2} & 2 x & (x + 2)^{2} 2 x & (x + 2)^{2} & 5 x^{2} \end{matrix} ⎤ ⎥ ⎥ ⎦$
$t r (C) = (x + 2)^{2} + 2 x + 5 x^{2}$
$\Rightarrow t r (C) = 6 x^{2} + 6 x + 4$
Since, we have $t r (A B) = t r (C)$
$\Rightarrow 3 a x^{2} + b + 6 c x = 6 x^{2} + 6 x + 4$
On comparing, we get $a = 2, b = 4, c = 1$
So, $a + b + c = 7$

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Let A=⎡⎢⎣3x216x⎤⎥⎦,B=[a,b,c] and C=⎡⎢ ⎢⎣(x+2)25x22x5x22x(x+2)22x(x+2)25x2⎤⎥ ⎥⎦ be three given matrices, where a,b,c and x∈R, Given that tr.(AB)=tr.(C)∨x∈R, where tr.(A) denotes trace of A. Find the value of (a+b+c)