For k - 1-50- find a- b- c such that PPT - I where- P - - 23 3k a- -13 -4k b- 23 -5k c -

The correct option is B a = ±13/15√(2), b = ±16/15√(2), c = ±1/3√(2) Given, #160; k = 1/√(50) , PP^T = I where, P = [ 23 amp; 3k ...

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Standard XII

Mathematics

Factorization Method Form to Remove Indeterminate Form

For k = 1/√...

Question

For $k = \frac{1}{\sqrt{50}}$ , find $a$ , $b$ , $c$ such that $P P^{T} = I$ where, $P = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{2}{3} & 3 k & a - \frac{1}{3} & - 4 k & b \frac{2}{3} & - 5 k & c \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$

a = \pm \frac{13}{15 \sqrt{2}}, b = \pm \frac{16}{15 \sqrt{2}}, c = \frac{1}{3 \sqrt{2}}

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a = \pm \frac{13}{15 \sqrt{2}}, b = \pm \frac{16}{15 \sqrt{2}}, c = \pm \frac{1}{15 \sqrt{2}}

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a = \pm \frac{13}{15 \sqrt{2}}, b = \pm \frac{16}{15 \sqrt{2}}, c = \pm \frac{1}{3 \sqrt{2}}

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Solution

The correct option is B $a = \pm \frac{13}{15 \sqrt{2}}, b = \pm \frac{16}{15 \sqrt{2}}, c = \pm \frac{1}{3 \sqrt{2}}$
Given, $k = \frac{1}{\sqrt{50}}$ ,
$P P^{T} = I$ where, $P = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{2}{3} & 3 k & a - \frac{1}{3} & - 4 k & b \frac{2}{3} & - 5 k & c \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$
$P P^{T} = I$
$\Rightarrow ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{2}{3} & 3 k & a - \frac{1}{3} & - 4 k & b \frac{2}{3} & - 5 k & c \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{2}{3} & - \frac{1}{3} & \frac{2}{3} 3 k & - 4 k & - 5 k a & b & c \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow \frac{4}{9} + 9 k^{2} + a^{2} = 1$
$\Rightarrow a^{2} = 1 - \frac{4}{9} - \frac{9}{50}$
$\Rightarrow a = \pm \frac{13}{15 \sqrt{2}}$
Similarly, $b = \pm \frac{16}{15 \sqrt{2}}$ and $c = \pm \frac{5}{15 \sqrt{2}}$
Hence, option C.

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For k=1√50, find a, b, c such that PPT=I where, P=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣233ka−13−4kb23−5kc⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

For $k = \frac{1}{\sqrt{50}}$ , find $a$ , $b$ , $c$ such that $P P^{T} = I$ where, $P = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{2}{3} & 3 k & a - \frac{1}{3} & - 4 k & b \frac{2}{3} & - 5 k & c \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$